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Accurate analytic presentation of solution for the spiked harmonic oscillator problem
Annals of Physics 322 (2007) 2211–2232 www.elsevier.com/locate/aop
Accurate analytic presentation of solution for the spiked harmonic oscillator problem E.Z. Liverts *, V.B. Mandelzweig Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Received 19 September 2006; accepted 25 November 2006 Available online 3 January 2007
Abstract High precision approximate analytic expressions of the ground state energies and wave functions for the spiked harmonic oscillator are found by first casting the correspondent Schro¨dinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of parameters. The accuracy ranging between 103 and 107 for the energies and, correspondingly, 102 and 107 for the wave functions in the regions, where they are not extremely small is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems. 2006 Elsevier Inc. All rights reserved. PACS: 03.65.Ca; 03.65.Ge Keywords: Spiked harmonic oscillator; Quasilinearization method; Analytical solutions
*
Corresponding author. E-mail address: [email protected] (E.Z. Liverts).
0003-4916/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2006.11.008
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1. Introduction The spiked harmonic oscillator with the potential U ðxÞ ¼
g 2 x2 k þ a 2x 2
½k > 0; a P 0
ð1Þ
derives its name from the graphical form which has a substantial peak near the origin due to perturbative term 2xka . This peak becomes more pronounced with increase of the coupling constant k and especially with growth of power a which characterizes a degree of the singularity at the origin. The spiked oscillator has a practical importance since it corresponds to different interactions which occur in atomic, molecular, nuclear and particle physics. For a > 2 it is also of relevance to the quantum field theory describing so called supersingular interactions for which matrix elements of the perturbation in the nonperturbed harmonic oscillator quantum states diverge, so that every term in the perturbation series is infinite and the perturbation expansion does not exist [1]. Aside of the physical relevance the spiked harmonic oscillator has very interesting and nontrivial mathematical properties. Indeed, the singular term 2xka provides an infinite repulsive barrier near the origin and therefore could not be neglected even for very small k. Because of this the singular term in potential could not be smoothly turned off with k fi 0, the phenomenon first pointed out by Klauder [2–7]. On the other side, the harmonic oscillator x2 term could not be neglected either, because its absence destroys a possibility of existence of the bound states. Therefore no dominance of one of those potentials could be established, which makes the construction of the perturbation theory rather difficult proposition. This was first stressed by Detwiler and Klauder [1] who pointed out that the usual perturbation theory could not be used for a > 52. Later Aguilera-Navarro and Guardiola [8] confirmed that for a < 52 the ground state energy could be expanded in powers of k while Harrel [9] using especially constructed by him modified singular perturbation theory was able to show that in case of a < 52 for k 1 exists a nonpower series expansion containing together with powers of k also the powers of its logarithm. The works of Klauder and Harrel started the era of intensive study of the spiked harmonic oscillator resulting in many different approaches to the problem [10–39]. Variational computations [13,14] and strong coupling perturbation expansions [14] as well as large order perturbative expansions [17] were employed besides the different specially adjusted numerical procedures [30–39]. In view of this multitude of the computations which concern themselves with different expansions or numerical solutions it is very interesting and helpful to even approximately envision an analytic form of the spiked harmonic oscillator wave function. The goal of this work is to provide a such possibility. Namely, we present in this work the closed analytic presentations of the wave functions for different values of parameters a and k which are accurate to about a few parts of the percent or even better in the region of variable x, where wave functions are not extremely small. Note, that the WKB does not at all provide a good solution for the spiked oscillator, as it was proved by Detwiler and Klauder [1] already many years ago. We use the quasilinearization method (QLM), which have been suggested recently for solving the Schro¨dinger equation after it was converted to the Riccati equation [40–45]. In the QLM the nonlinear differential equation is treated by approximating the nonlinear
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
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terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist [40,46]. Earlier, the accurate analytical presentation of solutions, deduced by using the first QLM iteration were presented for quartic oscillator [47], as well as for the generalized anharmonic oscillator [48] and for the arbitrary physical potential vanishing at large distances [49]. Extension of these results to the spiked harmonic oscillator enables one to make accurate analytical estimates of how variation of different interactions parameters affects relevant physical systems. We derive the QLM equations in Section 2. We analyze the Schro¨dinger equations for different parameters of the spiked harmonic oscillator potentials in Section 3. We summarize our results in Section 4. 2. The quasilinearization method (QLM) Let us start with one-dimensional Schro¨dinger equation in the form w00 ðxÞ ¼ k 2 ðxÞwðxÞ;
ð2Þ
where k 2 ðxÞ ¼ 2½E U ðxÞ:
ð3Þ
The units corresponding to mass m = 1 and h = 1 are used through the paper. Taking into account the simplest mathematical relations for logarithmic derivative y = w 0 /w, 0 2 d w0 ðxÞ w00 w w00 0 y 2 ðxÞ y ðxÞ ¼ ¼ ð4Þ ¼ dx wðxÞ w w w we can transform Schro¨dinger equation into Riccati equation y 0 ðxÞ þ y 2 ðxÞ ¼ k 2 ðxÞ:
ð5Þ
One has y(0) = 0 in case of the Neumann boundary condition w 0 (0) = 0 and y(0) = 1 in case of the Dirichlet boundary condition w(0) = 0, respectively. QLM enables one to solve this equation by iteration procedure using the approximate equation [40,50,51] y 0nþ1 ðxÞ þ 2y nþ1 ðxÞy n ðxÞ ¼ y 2n ðxÞ k 2 ðxÞ
ð6Þ
with the boundary condition yn(0) = y(0). The general solution of this equation (let n = 0, for simplicity) has a form Rx Z x Rx 2 y 0 ðtÞ dt 2 y ðtÞ dt y 1 ðxÞ ¼ C 1 e x0 þ e s 0 ½y 20 ðsÞ k 2 ðsÞds: ð7Þ x2
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Then, for the Neumann boundary condition, we obtain from Eq. (7) Z x2 R x0 2 y ðtÞ dt e s 0 ½y 20 ðsÞ k 2 ðsÞds: C1 ¼
ð8Þ
0
Thus, we have finally in this case Z x Rx 2 y ðtÞ dt y 1 ðxÞ ¼ e s 0 ½y 20 ðsÞ k 2 ðsÞds:
ð9Þ
0
By splitting the integral in the exponent of solution (9), one could present the latter in the form Z x Rs Rx 2 y 0 ðtÞ dt 2 y ðtÞ dt e a e a 0 ½y 20 ðsÞ k 2 ðsÞds; ð10Þ 0
where a is an arbitrary constant. Thus, to avoid an exponential divergence of y1(x) for x fi 1 one has to require Z 1 Rs 2 y ðtÞ dt e a 0 ½y 20 ðsÞ þ 2U ðsÞ 2E0 ds ¼ 0; ð11Þ 0
which yields the following expression for the total energy R 1 2 R s y 0 ðtÞ dt 2 e a ½y 0 ðsÞ þ 2U ðsÞds : E0 ¼ 0 R 1 2 R s y 0 ðtÞ dt ds 2 0 e a
ð12Þ
Multiplying both sides of the Schro¨dinger Eq. (2) by w(x) and then integrating over x in the limits of 0 to 1, one has Z 1 Z 1 00 w ðxÞwðxÞ dx ¼ 2 ½U ðxÞ Ew2 ðxÞ dx: ð13Þ 0
0
Integrating by parts the LHS of Eq. (13) one obtains Z 1 Z 1 w00 ðxÞwðxÞ dx ¼ w0 ðxÞwðxÞj1 ½w0 ðxÞ2 dx: 0 0
ð14Þ
0
Because the wave function or its derivative has to vanish both at the origin and at the infinity, the first term in RHS of Eq. (14) vanishes. Then, using Eqs. (13) and (14), one obtains for the average energy R1 R1 2 U ðxÞw2 ðxÞ dx þ 12 0 ½w0 ðxÞ dx 0 R1 2 E¼ : ð15Þ w ðxÞ dx 0 It is easy to show that the expressions (15) and (12) are identical for any wave function w = w0. Indeed, using the trivial relations y 0 ðtÞ
w00 ðtÞ d ¼ ln w0 ðtÞ; w0 ðtÞ dt
one can transform the exponential from Eq. (12) by the following way: 2 Rs Rs d w ðsÞ w0 ðsÞ 2 lnw 0ðaÞ 2 y 0 ðtÞ dt 2 ln w0 ðtÞ dt 0 a a ¼e ¼e ¼ : e w0 ðaÞ Substitution of Eq. (17) into Eq. (12) leads directly to Eq. (15).
ð16Þ
ð17Þ
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One should emphasize that the expression (15) have been derived from the Schro¨dinger Eq. (2) with using the only condition 1
w0 ðxÞwðxÞj0 ¼ 0;
ð18Þ
whereas expression (12) have been obtained from the (quite different) QLM Eq. (6). Replacing y0 by yn and repeating all of the derivations, we can conclude that the requirement of asymptotic convergence of the logarithmic derivative yn+1 leads to energy R 1 2 R s y n ðtÞ dt 2 e a ½y n ðsÞ þ 2U ðsÞds : ð19Þ En ¼ 0 R 1 2 R s y n ðtÞ dt ds 2 0 e a It is clear from Eq. (15) that this QLM energy presents simply the average energy in the state with the wave function wn(x) and hence with logarithmic derivative of the wave function yn(x) = w 0 n(x)/wn(x). Using relations (16) or (17), let us rewrite the solution (7) of the QLM equation (6) in the form Z x 2 w20 ðx0 Þ w0 ðsÞ 2 þ ½y 0 ðsÞ k 2 ðsÞds: ð20Þ y 1 ðxÞ ¼ C 1 2 2 w0 ðxÞ x2 w0 ðxÞ Multiplying the both sides of this equation by w20 ðxÞ one obtains Z x 2 2 w20 ðsÞ½y 20 ðsÞ k 2 ðsÞds: y 1 ðxÞw0 ðxÞ ¼ C 1 w0 ðx0 Þ þ
ð21Þ
x2
It is clear that also for the wave function with the Dirichlet boundary condition w1(0) = 0 and hence y1(0) = 1, the boundary condition lim y 1 ðxÞw20 ðxÞ lim x!0
x!0
w01 ðxÞw20 ðxÞ ¼0 w1 ðxÞ
ð22Þ
must be valid, if behavior of the functions w1(x) and w0(x) at the origin is the same. Taking the limit at x fi 0 for the both sides of Eq. (21) and using the boundary condition (22), one can write for the unknown quantity C1 Z x2 1 C1 ¼ 2 w20 ðsÞ½y 20 ðsÞ k 2 ðsÞds: ð23Þ w0 ðx0 Þ 0 It is easy to verify that substitution of Eq. (23) for C1 into Eq. (7) with using Eq. (17) leads again to Eq. (9). That is, one can insist that Eq. (9) is valid for the logarithmic derivative (of the wave function) with both the Neumann and Dirichlet boundary conditions y(0) = 0 and y(0) = 1, respectively. Using relations (17), one can represent Eq. (9) in another equivalent form Z x 1 y 1 ðxÞ ¼ 2 w20 ðsÞ½y 20 ðsÞ k 2 ðsÞds: ð24Þ w0 ðxÞ 0 And at last, we can write one more useful representation for y1(x). Toward this, let us perform the following integration by parts: Z x Z x Z x 2 w20 ðsÞy 20 ðsÞ ds ½w00 ðsÞ ds ¼ w0 ðxÞw00 ðxÞ w000 ðsÞw0 ðsÞ ds: ð25Þ 0
0
0
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Boundary condition (18) at the origin was used here. Inserting expression (25) into the RHS of Eq. (24), one obtains Z x 1 y 1 ðxÞ ¼ y 0 ðxÞ 2 w0 ðsÞ½w000 ðsÞ þ k 2 ðsÞw0 ðsÞds: ð26Þ w0 ðxÞ 0 Note, that application of Eq. (15) or Eq. (19) for the energy is equivalent to the following relation: Z 1 w0 ðsÞ½w000 ðsÞ þ k 2 ðsÞw0 ðsÞds ¼ 0: ð27Þ 0
This leads to the alternative representation instead of expression (26) Z 1 1 y 1 ðxÞ ¼ y 0 ðxÞ þ 2 w0 ðsÞ½w000 ðsÞ þ k 2 ðsÞw0 ðsÞds: w0 ðxÞ x
ð28Þ
3. Spiked harmonic oscillator Let us start from emphasizing that the choice of the parameter g = 1 in Eq. (1) does not restrict the framework of our consideration for the exact solutions of Eq. (2). Indeed, performing the scale transformation x fi nx, unitarily implemented on the Hilbert space of states (at first suggested in Refs. [52,53]), one can write for solutions of Schro¨dinger equation (2) with spiked potential (1) g k 2 Eðg; kÞ ¼ n E 2 ; 2a : ð29Þ n n 1 pffiffiffi Putting, in particular, n ¼ k2a or n ¼ g, one obtains the following important relations: 2 g k Eðg; kÞ ¼ k2a E ; 1 ¼ gE 1; ð30Þ 2 2a g2 k2a that enable one to reduce three-parameter problem to the problem of only two parameters k and a. The similar relations could be deduced for the wave functions, as well. Therefore, we shall discuss henceforth only spiked potential (1) with g = 1. Note, that at large values of x we can neglect the perturbative term k/xa as compared with the leading term x2 in Ua(x) corresponding to the simple harmonic oscillator. Hence, asymptotic form of the wave function for the spiked potential Ua(x) must coincide with the well-known asymptotics for the simple harmonic oscillator, i.e. wasymp ðxÞ expðx2 =2Þ:
ð31Þ
The corresponding asymptotic form of the logarithmic derivative has a form y asymp ðxÞ
w0 ðxÞ ¼ x: wðxÞ
ð32Þ
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3.1. Estimation of wave function and its logarithmic derivative near the origin Wave function of the spiked oscillator must satisfy a Dirichlet boundary condition (see [13] and references therein) wð0Þ ¼ 0: ð33Þ Hence, near the origin we could assume the solution of Eq. (2) of the form wðxÞ xl
ð34Þ
with l > 0. Note, that at small x, one can neglect the energy Ea and the harmonic oscillator term as compared with the perturbative term k/xa of the spiked potential Ua(x) in the Schro¨dinger equation (2). Hence, near the origin Eq. (2) can be written in the form w00 ðxÞ ¼
k wðxÞ: xa
ð35Þ
Substitution of the wave function (34) into Eq. (35) can be reduced to the form lðl 1Þ ¼ x2a : k For a = 2 the solution of Eq. (36) is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ 12ð1 þ 1 þ 4kÞ:
ð36Þ
ð37Þ
Eq. (34) with l determined by Eq. (37) presents the correct behavior of the exact analytical solution of the Schro¨dinger Eq. (2) at small x (see, e.g. [13,54]). For the case of a < 2 to satisfy Eq. (36) at x = 0 one should put l ¼ 1:
ð38Þ
For the values of parameter a > 2 the RHS of Eq. (36) approaches infinity at x fi 0. Hence, no finite values of the parameter l can not satisfy Eq. (36) at x = 0. This means that for the case of a > 2 our assumption for the form (34) of the solution of Eq. (2) near the origin, was incorrect. The simplest way to overcome this problem is to use the proper Riccati equation for the logarithmic derivative y(x) = w 0 (x)/w(x) instead of the Schro¨dinger equation (2). For small x one obtains y 0 ðxÞ þ y 2 ðxÞ ¼
k : xa
ð39Þ
Let us assume the following simple form of the logarithmic derivative y(x) near the origin for a > 2: c yðxÞ ¼ d : ð40Þ x Inserting presentation (40) into the LHS of Eq. (39), one obtains
cd c2 k þ ¼ a: dþ1 2d x x x
ð41Þ
Now let us analyze this equation with unknown parameters c and d. If d = 1, then we obtain the well-known case of a = 2 mentioned above. If d < 1, then 2d < d + 1 and hence at the vicinity of x = 0 we can neglect the second term as compared with the first term in the LHS of Eq. (41). The latter equation near
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the origin is then soluble only if one puts d = a 1. However, if a > 2, then d must be greater than unit, that contradicts to our initial assumption. If d > 1, then 2d > d + 1 and hence at the vicinity of x = 0 we can neglect the first term as compared with the second term in the LHS of Eq. (41). The choice of the parameters pffiffiffi c ¼ k and d = a/2 then satisfies Eq. (41) near the origin for the values of a > 2. Additionally, taking into account the fact, that the boundary condition y(0) = + 1 must be valid for all values of parameter a, then one should apply c > 0. As a result, we obtain finally for the logarithmic derivative in the vicinity of x = 0 at a > 2 pffiffiffi k yðxÞ ¼ a=2 : ð42Þ x The corresponding unnormalized wave function can be then written as ! pffiffiffi R k yðxÞ dx x1a=2 : ¼ exp wðxÞ e 1 a=2
ð43Þ
There is another way to deduce the analytic expressions of the wave function and its logarithmic derivative for spiked potential (1) near the origin. The analytical solution of Eq. (35) for a P 0 has the form " ! !# pffiffiffi pffiffiffi pffiffiffi 2 k 2a 2 k 2a x 2 þ C2K 1 x2 wðxÞ ¼ x C 1 I 1 ; ð44Þ j2aj j2aj j2 aj j2 aj where Im(z) and Km(z) are modified Bessel functions of the first and the second kind, respectively, whereas C1 and C2 are arbitrary constants. Note, that expression j2 aj denotes absolute value of the difference (2 a), i.e. j2 aj = 2 a for a < 2 and j2 aj = a 2 for a > 2, respectively. For the special case of a = 2 the solution of Eq. (35) has a form pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 1 wðxÞ ¼ C 1 x2ð1þ 1þ4kÞ þ C 2 x2ð1 1þ4kÞ : ð45Þ To satisfy the boundary condition (33), one should put C2 = 0 in the last equation. This yields, of course, the mentioned above wave function (34) with parameter l given by Eq. (37). The behavior of the general solution (44) near the origin has to be considered for the values of parameter a < 2 and a > 2, separately. pffiffi 2a (i) For the case of a < 2 the argument z 22ak x 2 of the Bessel functions approaches zero as x fi 0. The behavior of Bessel functions at small z is determined by the formulas (see [55], 9.6.7; 9.6.9) z m 1 Im ðzÞ ’ ; ð46Þ Cðm þ 1Þ 2 CðmÞ z m Km ðzÞ ’ : ð47Þ 2 2 According to these representations we should put C2 = 0 in the general solution (44) to satisfy the boundary condition (33). Moreover, from Eq. (46) we have
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
I2a 1
1 ! pffiffiffi pffiffiffi !2a pffiffiffi 2 k 2a 1 k
x2 ’ 1 x: 2a C 2a þ 1 2 a
2219
ð48Þ
Substitution of Eq. (48) into the RHS of the solution (44) (with C2 = 0) yields wðxÞ ’ const1 x;
ð49Þ
that provides yðxÞ ¼ 1x for the corresponding logarithmic pffiffiderivative. 2a (ii) On the contrary, for the case of a > 2 the argument z 2a2k x 2 of the Bessel functions approaches infinity as x fi 0. The behavior of the considered Bessel functions at large z is determined by (see [55], 9.7.1; 9.7.2): ez Im ðzÞ ’ pffiffiffiffiffiffiffi ; 2pz rffiffiffiffiffi p z e : Km ðzÞ ’ 2z
ð50Þ ð51Þ
These expressions mean, that in this case we should put C1 = 0 in the general solution (44) to satisfy the boundary condition (33). Substituting representation (51) into RHS of Eq. (44) (with C1 = 0), we then obtain " pffiffiffi a2 # 2 k 1 2 wðxÞ ’ const2 xa=4 exp : ð52Þ a2 x The corresponding logarithmic derivative has a form pffiffiffi a k yðxÞ ¼ þ a=2 : 4x x
ð53Þ
This result is somewhat more accurate, than Eq. (42). However, it is clear, that at a > 2 the second term in RHS of Eq. (53) is the dominant one, and therefore Eq. (53) reduces to Eq. (42) at x fi 0. 3.2. QLM results for the case of a < 2 Earlier we found, that logarithmic derivative y(x) corresponding to the exact solution of Eq. (2) with spiked potential (1) for a < 2 behaves as 1/x at small x and as x (for g = 1) at large enough x. These results enable us to propose the initial guess function of the form 1 y 0 ðxÞ ¼ þ b x: x
ð54Þ
Parameter b ” bx0 presenting the power function with exponent zero is introduced to improve our initial guess for intermediate values of x. We calculate b by minimizing the energy E0, obtained by using the initial guess (54). Representation (54) yields the following initial guess wave function: 2 R x w0 ðxÞ e y 0 ðxÞ dx x exp þ bx : ð55Þ 2 Using then Eq. (15) or Eq. (19) with w0(x) and y0(x) defined by Eqs. (55) and (54), respectively, one obtains the following expression for the ground state energy:
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E0 ¼
o 1 n 2 pffiffiffi 6b þ 2b3 þ eb pð3 þ 7b2 þ 2b4 Þ½1 þ ErfðbÞ þ 2kAða; bÞ ; T ðbÞ
ð56Þ
where 3a 3a 1 2 a a 3 2 ; ; b þ 2bC 2 M 2 ; ; b ; Aða; bÞ ¼ C M 2 2 2 2 2 2 2 pffiffiffi T ðbÞ ¼ 2b þ eb pð1 þ 2b2 Þ½1 þ ErfðbÞ:
ð57Þ ð58Þ
Here, C(a) is Euler’s Gamma Function, M(a, b, c) is Kummer confluent hypergeometric function, and Erf(a) is Error function. Note, that for the particular case of a = 1 expression (56) reduces to o 1 n 2 pffiffiffi 2ð3b þ b3 þ kÞ þ eb pð3 þ 7b2 þ 2b4 þ 2bkÞ½1 þ Erf ðbÞ : E0 ¼ ð59Þ T ðbÞ Parameter b is found from the equation DEða; k; bÞ oE0 =ob ¼ 0;
ð60Þ
where E0 is defined by the expressions (56)–(58). It is easy to verify, that DEð0; k; bÞ ¼ DEða; 0; bÞ ¼ 2b:
ð61Þ
This means, that for a = 0 or k = 0 the solution of the proper Eq. (60) presents by only root b = 0. Hence, for small enough values of parameter a or k one can use initial guesses (54) and (55) with b = 0, that is with no parameters at all. Putting b = 0 in Eq. (56), one obtains an extremely simple expression for the energy 2k 3a E0 ¼ 3 þ pffiffiffi C : ð62Þ 2 p To derive the analytic expression for the first iteration logarithmic derivative we can use Eq. (28). So, substituting our initial guesses (54) and (55) and formulas Eqs. (56)–(58) for the zero iteration energy into the RHS of Eq. (28), one obtains 2 pffiffiffi 1 k Aða; bÞ ½2ðx þ bÞ þ eðxbÞ pð1 þ 2b2 ÞErfcðx bÞ Bða; b; xÞ ; y 1 ðxÞ ¼ x þ 2 x x 2TðbÞ ð63Þ where 2 2bx
Bða; b; xÞ ¼ ex
Z
1
et
2 þ2bt
t2a dt:
ð64Þ
x
Here, Erfc(a) is complementary error function. For particular case of a = 1 functions A(a, b) and B(a, b, x) have an especially simple form 2 pffiffiffi Að1; bÞ ¼ 1 þ eb b p½1 þ ErfðbÞ; ð65Þ h i 2 pffiffiffi ð66Þ Bð1; b; xÞ ¼ 1 þ eðxbÞ b pErfcðx bÞ =2:
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
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It was mentioned above, that for small enough values of a or k in the spiked potential (1), one can use parameter b = 0. For this special case expression (63) takes the following simple form: pffiffiffi 1 k 3a 3a 3a 2 x2 p ffiffiffi y 1 ðxÞ ¼ x þ 2xC þ e p ErfcðxÞC C ;x ; x 2 2 2 2 p x2 ð67Þ which can be used together with expression (62) for the energy. C(a, b) denotes incomplete Gamma function, as usually. The exact and QLM ground state energies are presented in Table 1. The values of Eexact in Table 1 represent the direct numerical solution of the proper Schro¨dinger equation while E0, E1 represent zero and first iteration values of the ground state energies, calculated by using Eqs. (56)–(58) and Eq. (19) for n = 1, respectively. The parameter b was computed from Eq. (60). Note, that the choice of the maximum value of k in all the Tables are determined by the condition for the energy E < 50. The results, presented in Table 1 show, that the accuracy of energies E0 and E1, calculated by means of Eqs. (56) and (19) for n = 1, respectively, decreases significantly with increasing the parameter a up to its top value, which is equal to 2. To overcome this problem, let us remember, that there is the exact solution of Schro¨dinger equation (2) for the particular case of a = 2 with spiked potential (1). We found earlier, that logarithmic derivpffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ative of the corresponding wave function behaves as l/x with l ¼ ð1 þ 1 þ 4kÞ=2, according to Eq. (37) for small x. Hence, for the values of a close to the top value of 2, it would be natural to assume the logarithmic derivative initial guess of the form y 0 ðxÞ ¼
l x x
ð68Þ
instead of function (54).
Table 1 The exact and QLM zero and first iteration ground state energies E0 and E1 together with values of parameter b for spiked harmonic oscillator potential Ua(x) = x2 + k/xa with a < 2, calculated with the help of initial guesses (54) and (55) a
k
Eexact
E0
b
E1
0.5
0.01 0.1 1 10
3.01022627 3.1021390 4.00920406 12.093130
3.01022629 3.1021399 4.009309 12.1097
0.00117000 0.0116819 0.114959 0.937365
3.01022627 3.1021390 4.00920406 12.093181
1
0.01 0.1 1 10
3.01127600 3.1120669 4.057877 10.577484
3.01127633 3.112099 4.0609 10.6774
0.00272953 0.0270395 0.246789 1.340582
3.01127600 3.1120669 4.057878 10.5815
1.5
0.01 0.1 1 10
3.01379405 3.13505334 4.14189285 9.32417303
3.01379856 3.135482 4.16714 9.54634
0.00535600 0.0519316 0.403194 1.508619
3.01379400 3.13505334 4.14222 9.34747
2222
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For the corresponding wave function we have 2 x w0 ðxÞ xl exp ; 2
ð69Þ
Here, parameter l can be calculated by minimizing the corresponding energy of zero order, which we obtain in the form
kC 12 a2 þ l 2l2
þ E0 ¼ 1 þ ð70Þ 2l 1 C 12 þ l by using Eq. (12) or (15) for w = w0 with initial guess functions (68) or (69), respectively. Note, that mentioned above minimization is equivalent to finding l (for a given a) from the following transcendental equation:
kC 12 a2 þ l 4l l 1 a 1
1 ¼ 0: ð71Þ þ / þl / þl 2l 1 2l 1 2 2 2 C 12 þ l Here, /(z) = C 0 (z)/C(z) is digamma function. Putting l = 1 in the RHS of Eq. (70), we obtain expression (62) as expected. Putting k = 0 in Eq. (71), one obtains minimum of the energy E0 = 3 at l = 1. Note, that expression (70) was obtained by means of the QLM equation (12) for the approximate energy in the zero iteration. However, it is easy to verify, that substitution pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi of the parameters a = 2 and l ¼ ð1 þ 1 þ 4kÞ=2 into the RHS of Eq. (70), yields the pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi exact expression for the ground state energy E ¼ 2 þ 1 þ 4k (see, e.g. [57]). Analytic expression for the logarithmic derivative is obtained by means of general Eq. (28) with using the initial guesses (68) and (69) and the corresponding formula (70) for energy. This leads to "
# 2 2 l 1 k ex C 12 þ l a2 1 1 a 2 2
C þ l; x C þ l ; x y 1 ðxÞ ¼ x þ þ : 2 2 2 2l 1 x 2 x2l C 12 þ l ð72Þ At l = 1 this expression reduces to Eq. (67). It is not difficult to verify that substitution of the top value of a = 2 and the expression (37) for l into the RHS of the QLM formula (72) leads to the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2x2 þ 1 þ 4k y 1 ðxÞ ¼ ; ð73Þ 2x which corresponds to the exact solution pffiffiffiffiffiffiffiffi 1 x2 wðxÞ x2ð1þ 1þ4kÞ e 2
ð74Þ
of the proper Schro¨dinger equation (see, e.g. [57]). So, for the case of a = 2, we obtain the relations y0(x) = y1(x) = yexact(x) with l determined by Eq. (37). Note, that Eq. (74) was found by means of analyzing the behavior of wave function near the origin, and with no knowledge of the exact solution. The QLM ground state energies E0 and E1, calculated by Eqs. (70) and (19) with n = 1, respectively, are presented in Table 2. Comparing the data presented in Tables 1 and 2, one can conclude that initial guesses (68) and (69) yield energies which are more accurate for
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Table 2 The exact and QLM zero and first iteration ground state energies E0 and E1 together with values of parameter l for the spiked harmonic oscillator potential Ua(x) = x2 + k/xa with a < 2, calculated with the help of initial guesses (68) and (69) a
k
Eexact
E0
l
E1
0.5
0.01 0.1 1 10
3.01022627 3.10213909 4.00920406 12.093130
3.01022675 3.10218502 4.013207 12.1569
1.000675 1.006828 1.075876 2.311864
3.01022628 3.10213937 4.00921621 12.1013
1
0.01 0.1 1 10
3.01127600 3.1120669 4.057877 10.577484
3.01127779 3.112236 4.06759 10.6044
1.001735 1.017679 1.206014 3.224482
3.01127604 3.1120669 4.05807 10.57902
1.5
0.01 0.1 1 10
3.01379405 3.13505334 4.14189285 9.32417303
3.01379731 3.13533 4.14791 9.33042
1.003886 1.039387 1.398862 3.577468
3.01379400 3.13505351 4.14207 9.32425
the large values of a and k, than the energies produced by initial guesses of the form (54) and (55). Ground state wave functions for the spiked harmonic oscillator potential for parameters a ¼ 12, 32 and values of k equal to 0.01 and 10 are shown in Figs. 1 and 2. We present the exact wave functions wexact depicted by ‘‘solid’’ line on the upper parts of the graphs. These functions were obtained by means of numerical solution of the proper Schro¨dinger equations. The QLM wave functions depicted in Fig. 1 for the both values of parameter k were calculated with the help of explicit analytical expression (63) for the logarithmic derivative, obtained by using the initial guesses of the form (54) and (55). Our approximate analytical QLM wave functions wQLM are very accurate. The distinctions between them and the exact numerical wave functions cannot be revealed visually. Therefore, we present only relative logarithmic deviations from the exact value
Fig. 1. Ground state wave functions for the spiked harmonic oscillator potential with parameters a = 1/2 and k = 0.01; 10. The exact wave functions are depicted by ‘‘solid’’ line (upper parts of the graphs). Relative logarithmic deviations from the exact value for the QLM approximations are presented in the lower parts of the graphs by ‘‘dash’’ line for the zero order wave function, ‘‘dot’’ line for the first order wave function. Initial guesses of the form (54) and (55) were applied.
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Fig. 2. Ground state wave functions for the spiked harmonic oscillator potential with parameters a = 3/2 and k = 0.01; 10. Notations are the same as in Fig. 1. Initial guesses of the form (68) and (69) were applied.
lnjwQLM/wexact 1j for the QLM approximations in the lower parts of the graphs. The deviations for the zero iteration QLM wave functions are shown by ‘‘dash’’ line, whereas ‘‘dot’’ line is used for deviations of the first iteration QLM wave functions. One can observe in Fig. 1, that in case of a = 1/2 and small k = 0.01 the accuracies of the QLM wave functions are of the order 104 and 107 for the zero and the first iteration, respectively, which present very accurate results. The accuracy of the first iteration wave function for the large k = 10 are of the order 102 103. This is still reasonably good result, in spite of the fact, that it is worse, than for k = 0.01. In Fig. 2 one can see, that in case a ¼ 32 the corresponding logarithmic deviations from the exact values for the QLM first iteration wave functions are of the order 105106 in case of small k = 0.01, and 101103 in case of large k = 10. These results are obtained by means of the explicit analytic expression (72) for the logarithmic derivative, which is based on the initial guesses (68) and (69). 3.3. QLM results for the case of a > 2 As it was shown above, near the origin behavior of logarithmic derivative of the exact wave function for spiked oscillator potential (1) with the values of parameter a > 2 is determined by Eq. (53). Taking additionally into account, that the asymptotic behavior of spiked harmonic oscillator and harmonic oscillator wave functions are identical, we propose the following initial guess pffiffiffi c k y 0 ðxÞ ¼ þ a=2 x: ð75Þ x x The corresponding wave function has a form " # pffiffiffi a2 2 2 2 k 1 x w0 ðxÞ xc exp : a2 x 2
ð76Þ
Free parameter c, which earlier in Eq. (53) was equal to a4, is introduced to considerable improve the accuracy of our results and will be calculated by minimizing the proper energy E0 of a zero order, as it was done earlier in case of parameter b. We cannot derive the explicit analytical expression for E0 using Eq. (12) or (15) for w = w0 with initial guesses (75) or (76), respectively, in the general case of a > 2.
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Nevertheless, it is possible to do this, e.g. for the special cases of a = 5/2, 4, 6 considered in Refs. [13,14]. The corresponding expressions are very cumbersome, except for the case of a = 6, where E0 can be presented in the form pffiffiffi ( ) pffiffiffi 2k1=4 ½3 þ 4 k þ 2ðc 3ÞcK32þc ð2k1=4 Þ 1 2 E0 ¼ pffiffiffi kð12 8cÞ þ 2ð5 2cÞc 3 þ : 4 k K12þc ð2k1=4 Þ ð77Þ Here, Km(z) is modified Bessel function of the second kind. The first order logarithmic derivative function y1(x) can be calculated by means of Eq. (28) with using the initial guesses (75) and (76) for parameter c presenting the solution of equation oE0/ oc = 0 for given value of k. The required values of energy E0 can be obtained by using expression (77) for a = 6 or by using the more complicated expressions for a = 5/2, 4 obtained by Eq. (12). Inserting the obtained function y1(x) into the RHS of Eq. (19) for n = 1, one can find the proper first order energies E1 for the different values of the coupling parameter k. The exact and the QLM zero and first iteration ground state energies E0 and E1 are presented in Table 3. One can see that the accuracy of the QLM energy values is very high, with the first iteration values reproducing all seven digits of the exact energy values. One can notice that in Table 3 the results for large k are given only for a = 6, but not for smaller ones. The reason is easy to understand from analyzing of expression (76) for our initial guess wave function. Indeed, when constructing this expression, we assumed that the first term represents the behavior of wave function at small x, whereas the second term is responsible for the asymptotic behavior at large x. The point xeq, where the both terms are equal, could be calculated from the equation 2 pffiffiffi !ðaþ2 Þ 4 k xeq ¼ : ð78Þ a2
Table 3 The exact and QLM zero and first iteration ground state energies E0 and E1 together with values of parameter c for the spiked harmonic oscillator potential Ua(x) = x2 + k/xa, calculated with the help of initial guesses (75) and (76) for a = 5/2, 4, 6 a
k
Eexact
E0
c
E1
6
0.01 0.1 1 10 100 1000
3.505452 3.915665 4.659940 6.003209 8.413358 12.718617
3.506562 3.915687 4.660697 6.006564 8.420191 12.728722
1.309974 1.469066 1.732133 2.180871 2.962335 4.339169
3.505453 3.915665 4.659940 6.003212 8.413369 12.718639
4
0.01 0.1 1 10
3.2050675 3.575552 4.494178 6.606622
3.205364 3.576828 4.497696 6.613107
1.034424 1.100409 1.270806 1.674225
3.2050675 3.575552 4.494182 6.606633
5/2
0.01 0.1 0.5
3.036729 3.266873 3.848553
3.037492 3.267918 3.849740
0.905375 0.822995 0.790506
3.036733 3.266874 3.848554
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At x < xeq the absolute value of the first term is larger than the absolute value of the second one while at x > xeq the absolute value of the second term is larger than the absolute value of the first one. Initial guess of the form (76) will be valid only if xeq is close enough to the origin. However, from Eq. (78) follows that xeq increases with increase of k and with decrease of a (a > 2). The values of xeq for different values of parameters a and k are presented in Table 4. It is easy to verify, that for 2 6 a 6 6 and 0.01 6 k 6 1000 the exact ground state spiked wave functions are essentially differ from zero only for x < 8 (see the upper graphs of Figs. 1–4). Hence, Table 4 shows, that the mentioned above condition for using the initial guess (76) is not valid for large enough k and small enough a. For such combinations pffiffiffi of parameters k and a we propose simply to discard the problematic term kxa=2 in Table 4 Values xeq for initial guess (76) a/k
0.1
1
10
100
1000
5/2 4 6
1.51 0.86 0.75
2.52 1.26 1
4.20 1.85 1.33
7.01 2.71 1.78
11.7 3.98 2.37
Fig. 3. Ground state wave functions for the spiked harmonic oscillator potential with parameters a = 5/2 and k = 0.01; 1000. Notations are the same as in Fig. 1. Expressions (69) and (86) were used for constructing the approximate wave functions of the zero and the first order, respectively.
Fig. 4. Ground state wave functions for the spiked harmonic oscillator potential with parameters a = 6 and k = 0.01; 1000. Notations are the same as in Fig. 1. Initial guesses of the form (75) and (76) were applied.
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the initial guess (75) that is to return to already considered simple initial guesses (68) and (69). The corresponding analytic expressions (70) and (72) for the ground state energy and logarithmic derivative of wave function, respectively, thus could be used for arbitrary values of parameters. The calculations show that the energy minimum is reached at the large values of parameter l. For example, for a = 5/2 and k = 1000 one obtains l . 24.4. This leads in the RHS of Eq. (72) to incomplete Gamma functions C(a, z) with a large values of order a. The expression in the square brackets of Eq. (72) thus presents a difference of two very large quantities, which are very close to each other. To avoid this difficulty, let us substitute C(a, z) by the another well-known variation of incomplete Gamma function c(a, z) = C(a) C(a, z) in Eq. (72). Then, we obtain 2 l 1 k y 1 ðxÞ ¼ x þ þ Xðl; a; xÞ; ð79Þ 2l 1 x 2 where
"
# 2 C 12 þ l a2 ex 1 a 2 1
c þ l; x2 : Xðl; a; xÞ ¼ 2l c þ l ; x 2 2 2 x C 12 þ l
Substitution of series expansion (see, 6.5.4; 6.5.29 in Ref. [55]) 1 X zi cða; zÞ ¼ za ez CðaÞ Cða þ i þ 1Þ i¼0 for the both incomplete c-functions in the RHS of Eq. (80), thus yields 1 a Xðl; a; xÞ ¼ C þ l ½X1 ðl; a; xÞ X1 ðl; 0; xÞ; 2 2
ð80Þ
ð81Þ
ð82Þ
where X1 ðl; a; xÞ ¼
1 X i¼0
x2iþ1a 3
: C 2 þ l a2 þ i
ð83Þ
It is very important, that one can perform integration of the RHS of Eq. (82) over the variable x. Indeed, carrying out term by term integration of the both sums and then performing summation of the resulting series, we obtain Z Hðl; a; xÞ Xðl; a; xÞ dx
1; 1 a2 ; 32 þ l a2 ; 2 a2 ; x2 C 12 þ l a2 2 22 F 2
¼x þ xa ða 2Þð2l þ 1 aÞ 2C 32 þ l 3 2 F 2 f1; 1g; 2; þ l ; x2 : ð84Þ 2 Here 2 F 2 ðfa1 ; a2 g; fb1 ; b2 g; zÞ
¼
1 X ða1 Þk ða2 Þk zk ðb1 Þk ðb2 Þk k! k¼0
is generalized hypergeometric series [56].
ð85Þ
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Using Eqs. (84) and (79) we can write explicit expression for the ground state wave function of the first QLM iteration 2 R l2 x k y ðxÞ dx ð86Þ w1 ða; k; l; xÞ e 1 ¼ C N x2l1 exp Hðl; a; xÞ ; 2 2 where CN is normalization factor, whereas function H(l, a, x) is defined by Eq. (84). Note, that one can calculate w1(a, k, l; x) by means of Eq. (86) for a = 5/2 without any problem. However, one cannot do this for a = 4 or a = 6, because the first hypergeometric series in the RHS of Eq. (84) is divergent in these cases. The problem is that for the even integer values of a there will be the term with i = a/2 1 in the sum X1(l, a, x) of Eqs. (82) and (83). This term being proportional to 1/x yields ln (x) after integration for obtaining w1. To overcome this difficulty, let us divide the sum X1(l, a, x) into three parts with 0 < i < a/2 1, i = a/2 1 and i > a/2 1, correspondingly. Thus, we obtain a
X1 ðl; a; xÞ ¼
2 2 X i¼0
1 X x2iþ1a 1 x2iþ1a
3
3
: þ þ a 1 C 2þl2þi C 2 þ l a2 þ i xC l þ 2 i¼a
ð87Þ
2
It is easy to verify, that the third sum in the RHS of Eq. (87) reduces to X1(l, 0, x) by means of substitution k = i a/2. Thus, for the even integer values of a one obtains instead of Eq. (82) # " a22 1 a X x2iþ1a 1
:
þ Xðl; a; xÞ ¼ C þ l ð88Þ 2 2 xC l þ 12 C 32 þ l a2 þ i i¼0 Integration of the RHS of Eq. (79) with X(l, a, x) defined by Eq. (88) then yields w1 ða; k; l; xÞ ¼ C N x
l2 þ 2l1
ð Þ ð Þ
kC 1þla 2 2 2C 1þl 2
"
x2 kx2a k 1 a C þl exp 2 2 2 ða 2Þð2l a þ 1Þ 2 # a2 2 X x2iþ2a
; a ¼ 4; 6; 8; . . . ða 2i 2ÞC 32 þ l a2 þ i i¼1
ð89Þ
In particular, for the cases of a = 4, 6 under consideration we have 2 1 2k x k l2 þ2l3 ð Þ 2l1 exp w1 ð4; k; l; xÞ ¼ C N x ; ð90Þ 2 2ð2l 3Þx2 h i 2 1 4k l2 þð2l3Þð2l5Þ x k k 2l1 exp : w1 ð6; k; l; xÞ ¼ C N x 2 ð2l 3Þð2l 5Þx2 4ð2l 5Þx4 ð91Þ One can obtain the same result for even integer a also by applying the recursion relation c(a + 1, z) = ac(a, z) zaez (see, e.g. 6.5.22 in Ref. [55]) to Eq. (80). Since we derive the explicit expressions (86) and (89)–(91) for the first QLM iteration wave function w1(a, k, l; x), we can calculate parameter l by minimizing energy E1 as a
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Table 5 The exact and QLM zero and first iteration ground state energies E0 and E1 for the spiked harmonic potential Ua(x) = x2 + k/xa, calculated with the help of the first iteration wave functions (86) for a = 5/2, and by means of expressions (90) and (91) for a = 4, 6, respectively a
k
Eexact
E0
l1
E1
6
10 100 1000
6.003209 8.413358 12.718617
4.027186 5.770743 8.963489
6.332650 8.676420 12.942326
5.099990 7.458157 12.530918
6.013322 8.417318 12.725446
4
10 100 1000
6.606622 11.265080 21.369462
3.795513 6.888364 13.61853
6.693269 11.332546 21.427793
4.613350 8.768451 14.050578
6.607357 11.265871 21.370201
5/2
0.01 0.1 0.5 1 10 100 1000
3.036729 3.266873 3.848553 4.317311 7.735111 17.541890 44.955485
1.036739 1.210457 1.564245 1.832529 3.741461 9.191431 24.421591
3.037845 3.273543 3.857331 4.325682 7.740873 17.546306 44.959424
l0
3.036813 3.267401 3.848998 4.315508 7.735166 17.541903 44.955491
function of l. Thus, parameter l can be obtained as a root of the equation oE1/ol = 0 with given values of a and k, where energy E1 is defined by Eq. (15) with w(x) presented by Eq. (86) or Eqs. (89)–(91). The results of the exact and QLM ground state energies calculations, performed by means of Eq. (15) with using the wave function expressions ((86), (89)–(91)) are presented in Table 5. Note, that for the case of a = 5/2 we used Eqs. (84) and (86) with parameter l = l0 defined as a root of equation oE0/ol = 0, whereas for a = 4,6 parameter l = l1 was determined as a root of equation oE1/ol = 0 for computing E1 by Eq. (15) and Eqs. (90) and (91). The QLM ground state wave functions for parameter a > 2 are presented in Figs. 3 and 4. The notations of the curve style are the same as were declared in the previous Section for the Fig. 1. As one can observe in Fig. 3 for parameters a = 5/2, the logarithmic deviations from the exact value for the QLM first iteration wave functions are of the order 102 104 for both small k = 0.01 and large k = 1000. Remind, that the choice of the maximum value of k = 1000 is determined by the condition for the energy E < 50. For constructing the zero and the first order (iteration) QLM wave functions, we used expressions (69) and (86), respectively. Accuracy of the QLM wave functions for the spiked harmonic oscillator potential with parameter a = 6 is presented in Fig. 4. The corresponding deviations from the exact values for the QLM first iteration wave functions are of the order 102.5 104 in case of small k = 0.01 and 101.5 103.5 in case of large k = 1000. For constructing the QLM wave functions we employed the general analytic formula (28) with using the initial guesses (75) and (76) and parameters c, presented in Table 3. 4. Conclusions We used the quasilinearization method (QLM) [40–45] to approximate analytically solutions of the Schro¨dinger equation for the spiked harmonic oscillator. We found
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accurate analytic presentation for the ground state energies and wave functions. These high precision approximate analytic expressions are obtained by first casting the Schro¨dinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the method. In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries.
Table 6 Expectation values Æx2æi and Æxaæi of long and short range parts of the spiked harmonic oscillator potential Æx2æexact
Æxaæexact
Æxaæ0
Æxaæ1
3.87415 5.59353 8.80994
0.0771101 0.0138033 0.00246215
0.0734586 0.0135825 0.0024493
0.0712529 0.0140438 0.00255314
4.29551 7.38836 14.1185
3.98755 7.08293 13.7918
0.133341 0.0291972 0.00629284
0.132190 0.0290505 0.00629021
0.132725 0.0294062 0.0063079
1.53674 1.71046 2.06424 2.33253 4.24146 9.69143 24.9216
1.52337 1.66369 1.99964 2.26492 4.17462 9.62764 24.8593
3.40460 2.08577 1.11662 0.806490 0.242319 0.0682645 0.0190275
3.53308 2.11200 1.11636 0.805540 0.242229 0.0682611 0.0190274
3.35067 2.02994 1.09707 0.796386 0.241551 0.0682061 0.0190224
Æx2æ0
a
k
6
10 100 1000
3.77271 5.58701 8.82146
4.52719 6.27074 9.46349
4
10 100 1000
3.97002 7.08790 13.8312
5/2
0.01 0.1 0.5 1 10 100 1000
1.52262 1.65951 1.99407 2.25947 4.17045 9.62425 24.8562
Æx2æ1
Calculations were performed with the exact (i = exact) and QLM wave functions (iteration order i = 0, 1), based on the initial guesses (68) and (69). First order QLM wave functions were computed by Eq. (86) for a = 5/2 or by means of expressions (90) and (91) for a = 4, 6, respectively.
Table 7 Expectation values Æx2æi and Æxaæi of long and short range parts of the spiked harmonic oscillator potential a
k
Æx2æexact
Æx2æ0
Æx2æ1
Æxaæexact
Æxaæ0
Æxaæ1
6
0.01 0.1 1 10 100 1000
1.88289 2.19445 2.75840 3.77271 5.58701 8.82146
1.90599 2.19775 2.73885 3.73174 5.52896 8.75131
1.88338 2.19446 2.75864 3.77360 5.58855 8.82344
13.0168 2.36615 0.428429 0.0771101 0.0138033 0.00246215
13.8256 2.37943 0.419511 0.0750321 0.0134844 0.00242144
12.9760 2.36606 0.428067 0.0769335 0.0137646 0.00245613
4
0.01 0.1 1 10
1.65003 1.91202 2.54393 3.97002
1.63630 1.88394 2.49799 3.90827
1.65010 1.91228 2.54451 3.97096
9.49951 2.48497 0.593678 0.133341
9.25290 2.41504 0.579689 0.131238
9.49359 2.48106 0.592219 0.133015
5/2
0.01 0.1 0.5
1.52262 1.65951 1.99407
1.50552 1.63461 1.96497
1.52300 1.65984 1.99428
3.40460 2.08577 1.11662
3.30069 2.05885 1.11005
3.38889 2.08177 1.11559
Calculations were performed with the exact (i = exact) and QLM wave functions of the ith order (i = 0, 1), based on the initial guesses (75) and (76).
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The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist [46–49]. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems. Using Eq.(9) or Eqs.(24), (26), and (28) alternatively, one can write the analytical expressions for the wave function logarithmic derivatives of the arbitrary physical potential. The zero and first iteration ground state energies E0 and E1 can be calculated with the help of the general formulas (12) and (15) with w = w0, and (19) with n = 1, respectively. In order to determine the accuracy of the QLM solutions, we also found the direct numerical solutions. The quality of the wave functions is illustrated by Figs. 1–4. In order to check the overall accuracy of the wave functions we calculated also the expectation values i and i of long and short range parts of the spiked harmonic oscillator potential for the different initial guesses. These results are presented in Tables 6 and 7. In case of the spiked harmonic oscillator (1) with parameter a < 2, the explicit analytic presentations for the logarithmic derivatives of the wave functions are given by Eq.(63), whereas analytic expressions for the zero iteration energy E0 is presented by Eqs. (56)–(58). These presentations for both energy and wave functions are especially accurate for the small values of parameter a. For 1 < a < 2 analytic presentation of the form (86) can be effectively applied. And at last for the case of small values of k at a > 2, the mentioned above general formulas with using the initial guesses (75) and (76) enable us to obtain very accurate results. Expressions (86) and (89), as it was explained in the previous section are applicable for larger values of k. The first QLM iterate given by mentioned above analytic expression for the spiked harmonic oscillator allows to estimate analytically the role of different parameters and the influence of their variation on different characteristics of the relevant quantum systems. The next iterates display very fast quadratic convergence, so that accuracy of both energies and wave functions obtained after a few iterations could be extremely high, as it was shown in Refs. [58,59] on the examples of different widely used physical potentials. Acknowledgments The discussion and correspondence with Dr. R. Guardiola are greatfully acknowledged. The research was supported by Grant No. 2004106 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L.C. Detwiler, J.R. Klauder, Phys. Rev. D 11 (1975) 1436. J.R. Klauder, Acta Phys. Austriaca (Suppl. 11) (1973) 341. J.R. Klauder, Phys. Lett. B 47 (1973) 523. J.R. Klauder, Science 199 (1978) 735. H. Ezawa, J.R. Klauder, L.A. Shepp, J. Math. Phys. 16 (1975) 783. B. Simon, J. Funct. Anal. 14 (1973) 295. B. DeFacio, C.L. Hammer, J. Math. Phys. 15 (1974) 1071. V.C. Aguilera-Navarro, R. Guardiola, J. Math. Phys. 32 (1991) 2135. E.M. Harrel, Ann. Phys. 105 (1977) 379.
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R. Hall, N. Saad, A. von Keviczky, J. Math. Phys. 39 (1998) 6345. R. Hall, N. Saad, J. Phys. A 33 (2000) 569. R. Hall, N. Saad, J. Phys. A 33 (2000) 5531. R. Hall, N. Saad, A. von Keviczky, J. Phys. A 34 (2001) 1169; J. Math. Phys. 43 (2002) 94. V.C. Aguilera-Navarro, G.A. Estez, R. Guardiola, J. Math. Phys. 31 (1990) 99. R.M. Spector, J. Math. Phys. 8 (1967) 2357. M. de Llano, Rev. Mex. Fis. 27 (1981) 243. V.C. Aguilera-Navarro, F.M. Ferndez, R. Guardiola, J. Ros, J. Phys. A 253 (1992) 6379. V.C. Aguilera-Navarro, A.L. Coelho, Nazakat Ullah, Phys. Rev. A 49 (1994) 1477. M. Znojil, J. Phys. A 15 (1982) 2111. M. Znojil, Phys. Lett. A 101 (1984) 66. M. Znojil, Phys. Lett. A 158 (1991) 436. M. Znojil, P.G.L. Leach, J. Math. Phys. 33 (1992) 2785. W. Solano-Torres, G.A. Estes, F.M. Ferndez, G.C. Groenenboom, J. Phys. A 25 (1992) 3427. F.M. Fermendez, Phys. Lett. A 160 (1991) 51. M.F. Flynn, R. Guardiola, M. Znojil, Czech. J. Phys. 41 (1993) 1019. N. Nag, R. Roychoudhury, Czech. J. Phys. 46 (1996) 343. M. Znojil, R. Roychoudhury, Czech. J. Phys. 48 (1998) 1. E.S. Estez-Bret, G.A. Estez-Bret, J. Math. Phys. 34 (1993) 437. R. Hall, N. Saad, Can. J. Phys. 73 (1995) 493. J. Killingbeck, J. Phys. A 13 (1980) 49. J. Killingbeck, J. Phys. A 13 (1980) L231. J. Killingbeck, J. Phys. B 15 (1982) 829. J. Killingbeck, J. Phys. A 14 (1981) 1005. J. Killingbeck, G. Jolicard, A. Grosjean, J. Phys. A 34 (2001) L367. H.J. Korsch, H. Laurent, J. Phys. B 14 (1981) 4213. M. Znojil, J. Math. Phys. 30 (1989) 23. M. Znojil, J. Math. Phys. 31 (1990) 108. M. Znojil, J. Math. Phys. 34 (1993) 4914. H.G. Miller, J. Math. Phys. 35 (1994) 2229. V.B. Mandelzweig, J. Math. Phys. Com. 40 (1999) 6266. R. Krivec, V.B. Mandelzweig, Comput. Phys. Commun. 138 (2001) 69. V.B. Mandelzweig, F. Tabakin, Comput. Phys. Commun. 141 (2001) 268. R. Krivec, V.B. Mandelzweig, Comput. Phys. Commun. 152 (2003) 165. V.B. Mandelzweig, Phys. Atom. Nuclei 68 (2005) 1227. V.B. Mandelzweig, Ann. Phys., in press. R. Krivec, V.B. Mandelzweig, Comput. Phys. Commun. 138 (2001) 69. E.Z. Liverts, V.B. Mandelzweig, F. Tabakin, J. Math. Phys. 47 (2006) 062109. E.Z. Liverts, V.B. Mandelzweig, Accurate analytic presentation of solutions of the Schro¨dinger equation for the generalized anharmonic oscillator, submitted for publication. E.Z. Liverts, V.B. Mandelzweig, Accurate analytic presentation of solution of the Schro¨dinger equation with arbitrary physical potential and its applications, submitted for publication. V.B. Mandelzweig, F. Tabakin, Comput. Phys. Commun. 141 (2001) 268. K. Raghunathan, R. Vasudevan, J. Phys. A 20 (1987) 839. K. Symanzik, Commun. Math. Phys. 181 (1970) 227. C.G. Callan, Phys. Rev. D 2 (1970) 1541. L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, Pergamon Press, New York, 1991. M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York and London, 1965. R.L. Hall, N. Saad, A.B. von Keviczky, J. Phys. A 34 (2001) 1169. R. Krivec, V.B. Mandelzweig, Phys. Lett. A 337 (2005) 354. R. Krivec, V.B. Mandelzweig, Comput. Phys. Commun. 174 (2006) 119.
Accurate analytic presentation of solution for the spiked harmonic oscillator problem E.Z. Liverts *, V.B. Mandelzweig Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Received 19 September 2006; accepted 25 November 2006 Available online 3 January 2007
Abstract High precision approximate analytic expressions of the ground state energies and wave functions for the spiked harmonic oscillator are found by first casting the correspondent Schro¨dinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of parameters. The accuracy ranging between 103 and 107 for the energies and, correspondingly, 102 and 107 for the wave functions in the regions, where they are not extremely small is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems. 2006 Elsevier Inc. All rights reserved. PACS: 03.65.Ca; 03.65.Ge Keywords: Spiked harmonic oscillator; Quasilinearization method; Analytical solutions
*
Corresponding author. E-mail address: [email protected] (E.Z. Liverts).
0003-4916/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2006.11.008
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1. Introduction The spiked harmonic oscillator with the potential U ðxÞ ¼
g 2 x2 k þ a 2x 2
½k > 0; a P 0
ð1Þ
derives its name from the graphical form which has a substantial peak near the origin due to perturbative term 2xka . This peak becomes more pronounced with increase of the coupling constant k and especially with growth of power a which characterizes a degree of the singularity at the origin. The spiked oscillator has a practical importance since it corresponds to different interactions which occur in atomic, molecular, nuclear and particle physics. For a > 2 it is also of relevance to the quantum field theory describing so called supersingular interactions for which matrix elements of the perturbation in the nonperturbed harmonic oscillator quantum states diverge, so that every term in the perturbation series is infinite and the perturbation expansion does not exist [1]. Aside of the physical relevance the spiked harmonic oscillator has very interesting and nontrivial mathematical properties. Indeed, the singular term 2xka provides an infinite repulsive barrier near the origin and therefore could not be neglected even for very small k. Because of this the singular term in potential could not be smoothly turned off with k fi 0, the phenomenon first pointed out by Klauder [2–7]. On the other side, the harmonic oscillator x2 term could not be neglected either, because its absence destroys a possibility of existence of the bound states. Therefore no dominance of one of those potentials could be established, which makes the construction of the perturbation theory rather difficult proposition. This was first stressed by Detwiler and Klauder [1] who pointed out that the usual perturbation theory could not be used for a > 52. Later Aguilera-Navarro and Guardiola [8] confirmed that for a < 52 the ground state energy could be expanded in powers of k while Harrel [9] using especially constructed by him modified singular perturbation theory was able to show that in case of a < 52 for k 1 exists a nonpower series expansion containing together with powers of k also the powers of its logarithm. The works of Klauder and Harrel started the era of intensive study of the spiked harmonic oscillator resulting in many different approaches to the problem [10–39]. Variational computations [13,14] and strong coupling perturbation expansions [14] as well as large order perturbative expansions [17] were employed besides the different specially adjusted numerical procedures [30–39]. In view of this multitude of the computations which concern themselves with different expansions or numerical solutions it is very interesting and helpful to even approximately envision an analytic form of the spiked harmonic oscillator wave function. The goal of this work is to provide a such possibility. Namely, we present in this work the closed analytic presentations of the wave functions for different values of parameters a and k which are accurate to about a few parts of the percent or even better in the region of variable x, where wave functions are not extremely small. Note, that the WKB does not at all provide a good solution for the spiked oscillator, as it was proved by Detwiler and Klauder [1] already many years ago. We use the quasilinearization method (QLM), which have been suggested recently for solving the Schro¨dinger equation after it was converted to the Riccati equation [40–45]. In the QLM the nonlinear differential equation is treated by approximating the nonlinear
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terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist [40,46]. Earlier, the accurate analytical presentation of solutions, deduced by using the first QLM iteration were presented for quartic oscillator [47], as well as for the generalized anharmonic oscillator [48] and for the arbitrary physical potential vanishing at large distances [49]. Extension of these results to the spiked harmonic oscillator enables one to make accurate analytical estimates of how variation of different interactions parameters affects relevant physical systems. We derive the QLM equations in Section 2. We analyze the Schro¨dinger equations for different parameters of the spiked harmonic oscillator potentials in Section 3. We summarize our results in Section 4. 2. The quasilinearization method (QLM) Let us start with one-dimensional Schro¨dinger equation in the form w00 ðxÞ ¼ k 2 ðxÞwðxÞ;
ð2Þ
where k 2 ðxÞ ¼ 2½E U ðxÞ:
ð3Þ
The units corresponding to mass m = 1 and h = 1 are used through the paper. Taking into account the simplest mathematical relations for logarithmic derivative y = w 0 /w, 0 2 d w0 ðxÞ w00 w w00 0 y 2 ðxÞ y ðxÞ ¼ ¼ ð4Þ ¼ dx wðxÞ w w w we can transform Schro¨dinger equation into Riccati equation y 0 ðxÞ þ y 2 ðxÞ ¼ k 2 ðxÞ:
ð5Þ
One has y(0) = 0 in case of the Neumann boundary condition w 0 (0) = 0 and y(0) = 1 in case of the Dirichlet boundary condition w(0) = 0, respectively. QLM enables one to solve this equation by iteration procedure using the approximate equation [40,50,51] y 0nþ1 ðxÞ þ 2y nþ1 ðxÞy n ðxÞ ¼ y 2n ðxÞ k 2 ðxÞ
ð6Þ
with the boundary condition yn(0) = y(0). The general solution of this equation (let n = 0, for simplicity) has a form Rx Z x Rx 2 y 0 ðtÞ dt 2 y ðtÞ dt y 1 ðxÞ ¼ C 1 e x0 þ e s 0 ½y 20 ðsÞ k 2 ðsÞds: ð7Þ x2
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Then, for the Neumann boundary condition, we obtain from Eq. (7) Z x2 R x0 2 y ðtÞ dt e s 0 ½y 20 ðsÞ k 2 ðsÞds: C1 ¼
ð8Þ
0
Thus, we have finally in this case Z x Rx 2 y ðtÞ dt y 1 ðxÞ ¼ e s 0 ½y 20 ðsÞ k 2 ðsÞds:
ð9Þ
0
By splitting the integral in the exponent of solution (9), one could present the latter in the form Z x Rs Rx 2 y 0 ðtÞ dt 2 y ðtÞ dt e a e a 0 ½y 20 ðsÞ k 2 ðsÞds; ð10Þ 0
where a is an arbitrary constant. Thus, to avoid an exponential divergence of y1(x) for x fi 1 one has to require Z 1 Rs 2 y ðtÞ dt e a 0 ½y 20 ðsÞ þ 2U ðsÞ 2E0 ds ¼ 0; ð11Þ 0
which yields the following expression for the total energy R 1 2 R s y 0 ðtÞ dt 2 e a ½y 0 ðsÞ þ 2U ðsÞds : E0 ¼ 0 R 1 2 R s y 0 ðtÞ dt ds 2 0 e a
ð12Þ
Multiplying both sides of the Schro¨dinger Eq. (2) by w(x) and then integrating over x in the limits of 0 to 1, one has Z 1 Z 1 00 w ðxÞwðxÞ dx ¼ 2 ½U ðxÞ Ew2 ðxÞ dx: ð13Þ 0
0
Integrating by parts the LHS of Eq. (13) one obtains Z 1 Z 1 w00 ðxÞwðxÞ dx ¼ w0 ðxÞwðxÞj1 ½w0 ðxÞ2 dx: 0 0
ð14Þ
0
Because the wave function or its derivative has to vanish both at the origin and at the infinity, the first term in RHS of Eq. (14) vanishes. Then, using Eqs. (13) and (14), one obtains for the average energy R1 R1 2 U ðxÞw2 ðxÞ dx þ 12 0 ½w0 ðxÞ dx 0 R1 2 E¼ : ð15Þ w ðxÞ dx 0 It is easy to show that the expressions (15) and (12) are identical for any wave function w = w0. Indeed, using the trivial relations y 0 ðtÞ
w00 ðtÞ d ¼ ln w0 ðtÞ; w0 ðtÞ dt
one can transform the exponential from Eq. (12) by the following way: 2 Rs Rs d w ðsÞ w0 ðsÞ 2 lnw 0ðaÞ 2 y 0 ðtÞ dt 2 ln w0 ðtÞ dt 0 a a ¼e ¼e ¼ : e w0 ðaÞ Substitution of Eq. (17) into Eq. (12) leads directly to Eq. (15).
ð16Þ
ð17Þ
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One should emphasize that the expression (15) have been derived from the Schro¨dinger Eq. (2) with using the only condition 1
w0 ðxÞwðxÞj0 ¼ 0;
ð18Þ
whereas expression (12) have been obtained from the (quite different) QLM Eq. (6). Replacing y0 by yn and repeating all of the derivations, we can conclude that the requirement of asymptotic convergence of the logarithmic derivative yn+1 leads to energy R 1 2 R s y n ðtÞ dt 2 e a ½y n ðsÞ þ 2U ðsÞds : ð19Þ En ¼ 0 R 1 2 R s y n ðtÞ dt ds 2 0 e a It is clear from Eq. (15) that this QLM energy presents simply the average energy in the state with the wave function wn(x) and hence with logarithmic derivative of the wave function yn(x) = w 0 n(x)/wn(x). Using relations (16) or (17), let us rewrite the solution (7) of the QLM equation (6) in the form Z x 2 w20 ðx0 Þ w0 ðsÞ 2 þ ½y 0 ðsÞ k 2 ðsÞds: ð20Þ y 1 ðxÞ ¼ C 1 2 2 w0 ðxÞ x2 w0 ðxÞ Multiplying the both sides of this equation by w20 ðxÞ one obtains Z x 2 2 w20 ðsÞ½y 20 ðsÞ k 2 ðsÞds: y 1 ðxÞw0 ðxÞ ¼ C 1 w0 ðx0 Þ þ
ð21Þ
x2
It is clear that also for the wave function with the Dirichlet boundary condition w1(0) = 0 and hence y1(0) = 1, the boundary condition lim y 1 ðxÞw20 ðxÞ lim x!0
x!0
w01 ðxÞw20 ðxÞ ¼0 w1 ðxÞ
ð22Þ
must be valid, if behavior of the functions w1(x) and w0(x) at the origin is the same. Taking the limit at x fi 0 for the both sides of Eq. (21) and using the boundary condition (22), one can write for the unknown quantity C1 Z x2 1 C1 ¼ 2 w20 ðsÞ½y 20 ðsÞ k 2 ðsÞds: ð23Þ w0 ðx0 Þ 0 It is easy to verify that substitution of Eq. (23) for C1 into Eq. (7) with using Eq. (17) leads again to Eq. (9). That is, one can insist that Eq. (9) is valid for the logarithmic derivative (of the wave function) with both the Neumann and Dirichlet boundary conditions y(0) = 0 and y(0) = 1, respectively. Using relations (17), one can represent Eq. (9) in another equivalent form Z x 1 y 1 ðxÞ ¼ 2 w20 ðsÞ½y 20 ðsÞ k 2 ðsÞds: ð24Þ w0 ðxÞ 0 And at last, we can write one more useful representation for y1(x). Toward this, let us perform the following integration by parts: Z x Z x Z x 2 w20 ðsÞy 20 ðsÞ ds ½w00 ðsÞ ds ¼ w0 ðxÞw00 ðxÞ w000 ðsÞw0 ðsÞ ds: ð25Þ 0
0
0
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Boundary condition (18) at the origin was used here. Inserting expression (25) into the RHS of Eq. (24), one obtains Z x 1 y 1 ðxÞ ¼ y 0 ðxÞ 2 w0 ðsÞ½w000 ðsÞ þ k 2 ðsÞw0 ðsÞds: ð26Þ w0 ðxÞ 0 Note, that application of Eq. (15) or Eq. (19) for the energy is equivalent to the following relation: Z 1 w0 ðsÞ½w000 ðsÞ þ k 2 ðsÞw0 ðsÞds ¼ 0: ð27Þ 0
This leads to the alternative representation instead of expression (26) Z 1 1 y 1 ðxÞ ¼ y 0 ðxÞ þ 2 w0 ðsÞ½w000 ðsÞ þ k 2 ðsÞw0 ðsÞds: w0 ðxÞ x
ð28Þ
3. Spiked harmonic oscillator Let us start from emphasizing that the choice of the parameter g = 1 in Eq. (1) does not restrict the framework of our consideration for the exact solutions of Eq. (2). Indeed, performing the scale transformation x fi nx, unitarily implemented on the Hilbert space of states (at first suggested in Refs. [52,53]), one can write for solutions of Schro¨dinger equation (2) with spiked potential (1) g k 2 Eðg; kÞ ¼ n E 2 ; 2a : ð29Þ n n 1 pffiffiffi Putting, in particular, n ¼ k2a or n ¼ g, one obtains the following important relations: 2 g k Eðg; kÞ ¼ k2a E ; 1 ¼ gE 1; ð30Þ 2 2a g2 k2a that enable one to reduce three-parameter problem to the problem of only two parameters k and a. The similar relations could be deduced for the wave functions, as well. Therefore, we shall discuss henceforth only spiked potential (1) with g = 1. Note, that at large values of x we can neglect the perturbative term k/xa as compared with the leading term x2 in Ua(x) corresponding to the simple harmonic oscillator. Hence, asymptotic form of the wave function for the spiked potential Ua(x) must coincide with the well-known asymptotics for the simple harmonic oscillator, i.e. wasymp ðxÞ expðx2 =2Þ:
ð31Þ
The corresponding asymptotic form of the logarithmic derivative has a form y asymp ðxÞ
w0 ðxÞ ¼ x: wðxÞ
ð32Þ
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3.1. Estimation of wave function and its logarithmic derivative near the origin Wave function of the spiked oscillator must satisfy a Dirichlet boundary condition (see [13] and references therein) wð0Þ ¼ 0: ð33Þ Hence, near the origin we could assume the solution of Eq. (2) of the form wðxÞ xl
ð34Þ
with l > 0. Note, that at small x, one can neglect the energy Ea and the harmonic oscillator term as compared with the perturbative term k/xa of the spiked potential Ua(x) in the Schro¨dinger equation (2). Hence, near the origin Eq. (2) can be written in the form w00 ðxÞ ¼
k wðxÞ: xa
ð35Þ
Substitution of the wave function (34) into Eq. (35) can be reduced to the form lðl 1Þ ¼ x2a : k For a = 2 the solution of Eq. (36) is pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ 12ð1 þ 1 þ 4kÞ:
ð36Þ
ð37Þ
Eq. (34) with l determined by Eq. (37) presents the correct behavior of the exact analytical solution of the Schro¨dinger Eq. (2) at small x (see, e.g. [13,54]). For the case of a < 2 to satisfy Eq. (36) at x = 0 one should put l ¼ 1:
ð38Þ
For the values of parameter a > 2 the RHS of Eq. (36) approaches infinity at x fi 0. Hence, no finite values of the parameter l can not satisfy Eq. (36) at x = 0. This means that for the case of a > 2 our assumption for the form (34) of the solution of Eq. (2) near the origin, was incorrect. The simplest way to overcome this problem is to use the proper Riccati equation for the logarithmic derivative y(x) = w 0 (x)/w(x) instead of the Schro¨dinger equation (2). For small x one obtains y 0 ðxÞ þ y 2 ðxÞ ¼
k : xa
ð39Þ
Let us assume the following simple form of the logarithmic derivative y(x) near the origin for a > 2: c yðxÞ ¼ d : ð40Þ x Inserting presentation (40) into the LHS of Eq. (39), one obtains
cd c2 k þ ¼ a: dþ1 2d x x x
ð41Þ
Now let us analyze this equation with unknown parameters c and d. If d = 1, then we obtain the well-known case of a = 2 mentioned above. If d < 1, then 2d < d + 1 and hence at the vicinity of x = 0 we can neglect the second term as compared with the first term in the LHS of Eq. (41). The latter equation near
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the origin is then soluble only if one puts d = a 1. However, if a > 2, then d must be greater than unit, that contradicts to our initial assumption. If d > 1, then 2d > d + 1 and hence at the vicinity of x = 0 we can neglect the first term as compared with the second term in the LHS of Eq. (41). The choice of the parameters pffiffiffi c ¼ k and d = a/2 then satisfies Eq. (41) near the origin for the values of a > 2. Additionally, taking into account the fact, that the boundary condition y(0) = + 1 must be valid for all values of parameter a, then one should apply c > 0. As a result, we obtain finally for the logarithmic derivative in the vicinity of x = 0 at a > 2 pffiffiffi k yðxÞ ¼ a=2 : ð42Þ x The corresponding unnormalized wave function can be then written as ! pffiffiffi R k yðxÞ dx x1a=2 : ¼ exp wðxÞ e 1 a=2
ð43Þ
There is another way to deduce the analytic expressions of the wave function and its logarithmic derivative for spiked potential (1) near the origin. The analytical solution of Eq. (35) for a P 0 has the form " ! !# pffiffiffi pffiffiffi pffiffiffi 2 k 2a 2 k 2a x 2 þ C2K 1 x2 wðxÞ ¼ x C 1 I 1 ; ð44Þ j2aj j2aj j2 aj j2 aj where Im(z) and Km(z) are modified Bessel functions of the first and the second kind, respectively, whereas C1 and C2 are arbitrary constants. Note, that expression j2 aj denotes absolute value of the difference (2 a), i.e. j2 aj = 2 a for a < 2 and j2 aj = a 2 for a > 2, respectively. For the special case of a = 2 the solution of Eq. (35) has a form pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 1 wðxÞ ¼ C 1 x2ð1þ 1þ4kÞ þ C 2 x2ð1 1þ4kÞ : ð45Þ To satisfy the boundary condition (33), one should put C2 = 0 in the last equation. This yields, of course, the mentioned above wave function (34) with parameter l given by Eq. (37). The behavior of the general solution (44) near the origin has to be considered for the values of parameter a < 2 and a > 2, separately. pffiffi 2a (i) For the case of a < 2 the argument z 22ak x 2 of the Bessel functions approaches zero as x fi 0. The behavior of Bessel functions at small z is determined by the formulas (see [55], 9.6.7; 9.6.9) z m 1 Im ðzÞ ’ ; ð46Þ Cðm þ 1Þ 2 CðmÞ z m Km ðzÞ ’ : ð47Þ 2 2 According to these representations we should put C2 = 0 in the general solution (44) to satisfy the boundary condition (33). Moreover, from Eq. (46) we have
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
I2a 1
1 ! pffiffiffi pffiffiffi !2a pffiffiffi 2 k 2a 1 k
x2 ’ 1 x: 2a C 2a þ 1 2 a
2219
ð48Þ
Substitution of Eq. (48) into the RHS of the solution (44) (with C2 = 0) yields wðxÞ ’ const1 x;
ð49Þ
that provides yðxÞ ¼ 1x for the corresponding logarithmic pffiffiderivative. 2a (ii) On the contrary, for the case of a > 2 the argument z 2a2k x 2 of the Bessel functions approaches infinity as x fi 0. The behavior of the considered Bessel functions at large z is determined by (see [55], 9.7.1; 9.7.2): ez Im ðzÞ ’ pffiffiffiffiffiffiffi ; 2pz rffiffiffiffiffi p z e : Km ðzÞ ’ 2z
ð50Þ ð51Þ
These expressions mean, that in this case we should put C1 = 0 in the general solution (44) to satisfy the boundary condition (33). Substituting representation (51) into RHS of Eq. (44) (with C1 = 0), we then obtain " pffiffiffi a2 # 2 k 1 2 wðxÞ ’ const2 xa=4 exp : ð52Þ a2 x The corresponding logarithmic derivative has a form pffiffiffi a k yðxÞ ¼ þ a=2 : 4x x
ð53Þ
This result is somewhat more accurate, than Eq. (42). However, it is clear, that at a > 2 the second term in RHS of Eq. (53) is the dominant one, and therefore Eq. (53) reduces to Eq. (42) at x fi 0. 3.2. QLM results for the case of a < 2 Earlier we found, that logarithmic derivative y(x) corresponding to the exact solution of Eq. (2) with spiked potential (1) for a < 2 behaves as 1/x at small x and as x (for g = 1) at large enough x. These results enable us to propose the initial guess function of the form 1 y 0 ðxÞ ¼ þ b x: x
ð54Þ
Parameter b ” bx0 presenting the power function with exponent zero is introduced to improve our initial guess for intermediate values of x. We calculate b by minimizing the energy E0, obtained by using the initial guess (54). Representation (54) yields the following initial guess wave function: 2 R x w0 ðxÞ e y 0 ðxÞ dx x exp þ bx : ð55Þ 2 Using then Eq. (15) or Eq. (19) with w0(x) and y0(x) defined by Eqs. (55) and (54), respectively, one obtains the following expression for the ground state energy:
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E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
E0 ¼
o 1 n 2 pffiffiffi 6b þ 2b3 þ eb pð3 þ 7b2 þ 2b4 Þ½1 þ ErfðbÞ þ 2kAða; bÞ ; T ðbÞ
ð56Þ
where 3a 3a 1 2 a a 3 2 ; ; b þ 2bC 2 M 2 ; ; b ; Aða; bÞ ¼ C M 2 2 2 2 2 2 2 pffiffiffi T ðbÞ ¼ 2b þ eb pð1 þ 2b2 Þ½1 þ ErfðbÞ:
ð57Þ ð58Þ
Here, C(a) is Euler’s Gamma Function, M(a, b, c) is Kummer confluent hypergeometric function, and Erf(a) is Error function. Note, that for the particular case of a = 1 expression (56) reduces to o 1 n 2 pffiffiffi 2ð3b þ b3 þ kÞ þ eb pð3 þ 7b2 þ 2b4 þ 2bkÞ½1 þ Erf ðbÞ : E0 ¼ ð59Þ T ðbÞ Parameter b is found from the equation DEða; k; bÞ oE0 =ob ¼ 0;
ð60Þ
where E0 is defined by the expressions (56)–(58). It is easy to verify, that DEð0; k; bÞ ¼ DEða; 0; bÞ ¼ 2b:
ð61Þ
This means, that for a = 0 or k = 0 the solution of the proper Eq. (60) presents by only root b = 0. Hence, for small enough values of parameter a or k one can use initial guesses (54) and (55) with b = 0, that is with no parameters at all. Putting b = 0 in Eq. (56), one obtains an extremely simple expression for the energy 2k 3a E0 ¼ 3 þ pffiffiffi C : ð62Þ 2 p To derive the analytic expression for the first iteration logarithmic derivative we can use Eq. (28). So, substituting our initial guesses (54) and (55) and formulas Eqs. (56)–(58) for the zero iteration energy into the RHS of Eq. (28), one obtains 2 pffiffiffi 1 k Aða; bÞ ½2ðx þ bÞ þ eðxbÞ pð1 þ 2b2 ÞErfcðx bÞ Bða; b; xÞ ; y 1 ðxÞ ¼ x þ 2 x x 2TðbÞ ð63Þ where 2 2bx
Bða; b; xÞ ¼ ex
Z
1
et
2 þ2bt
t2a dt:
ð64Þ
x
Here, Erfc(a) is complementary error function. For particular case of a = 1 functions A(a, b) and B(a, b, x) have an especially simple form 2 pffiffiffi Að1; bÞ ¼ 1 þ eb b p½1 þ ErfðbÞ; ð65Þ h i 2 pffiffiffi ð66Þ Bð1; b; xÞ ¼ 1 þ eðxbÞ b pErfcðx bÞ =2:
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
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It was mentioned above, that for small enough values of a or k in the spiked potential (1), one can use parameter b = 0. For this special case expression (63) takes the following simple form: pffiffiffi 1 k 3a 3a 3a 2 x2 p ffiffiffi y 1 ðxÞ ¼ x þ 2xC þ e p ErfcðxÞC C ;x ; x 2 2 2 2 p x2 ð67Þ which can be used together with expression (62) for the energy. C(a, b) denotes incomplete Gamma function, as usually. The exact and QLM ground state energies are presented in Table 1. The values of Eexact in Table 1 represent the direct numerical solution of the proper Schro¨dinger equation while E0, E1 represent zero and first iteration values of the ground state energies, calculated by using Eqs. (56)–(58) and Eq. (19) for n = 1, respectively. The parameter b was computed from Eq. (60). Note, that the choice of the maximum value of k in all the Tables are determined by the condition for the energy E < 50. The results, presented in Table 1 show, that the accuracy of energies E0 and E1, calculated by means of Eqs. (56) and (19) for n = 1, respectively, decreases significantly with increasing the parameter a up to its top value, which is equal to 2. To overcome this problem, let us remember, that there is the exact solution of Schro¨dinger equation (2) for the particular case of a = 2 with spiked potential (1). We found earlier, that logarithmic derivpffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ative of the corresponding wave function behaves as l/x with l ¼ ð1 þ 1 þ 4kÞ=2, according to Eq. (37) for small x. Hence, for the values of a close to the top value of 2, it would be natural to assume the logarithmic derivative initial guess of the form y 0 ðxÞ ¼
l x x
ð68Þ
instead of function (54).
Table 1 The exact and QLM zero and first iteration ground state energies E0 and E1 together with values of parameter b for spiked harmonic oscillator potential Ua(x) = x2 + k/xa with a < 2, calculated with the help of initial guesses (54) and (55) a
k
Eexact
E0
b
E1
0.5
0.01 0.1 1 10
3.01022627 3.1021390 4.00920406 12.093130
3.01022629 3.1021399 4.009309 12.1097
0.00117000 0.0116819 0.114959 0.937365
3.01022627 3.1021390 4.00920406 12.093181
1
0.01 0.1 1 10
3.01127600 3.1120669 4.057877 10.577484
3.01127633 3.112099 4.0609 10.6774
0.00272953 0.0270395 0.246789 1.340582
3.01127600 3.1120669 4.057878 10.5815
1.5
0.01 0.1 1 10
3.01379405 3.13505334 4.14189285 9.32417303
3.01379856 3.135482 4.16714 9.54634
0.00535600 0.0519316 0.403194 1.508619
3.01379400 3.13505334 4.14222 9.34747
2222
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
For the corresponding wave function we have 2 x w0 ðxÞ xl exp ; 2
ð69Þ
Here, parameter l can be calculated by minimizing the corresponding energy of zero order, which we obtain in the form
kC 12 a2 þ l 2l2
þ E0 ¼ 1 þ ð70Þ 2l 1 C 12 þ l by using Eq. (12) or (15) for w = w0 with initial guess functions (68) or (69), respectively. Note, that mentioned above minimization is equivalent to finding l (for a given a) from the following transcendental equation:
kC 12 a2 þ l 4l l 1 a 1
1 ¼ 0: ð71Þ þ / þl / þl 2l 1 2l 1 2 2 2 C 12 þ l Here, /(z) = C 0 (z)/C(z) is digamma function. Putting l = 1 in the RHS of Eq. (70), we obtain expression (62) as expected. Putting k = 0 in Eq. (71), one obtains minimum of the energy E0 = 3 at l = 1. Note, that expression (70) was obtained by means of the QLM equation (12) for the approximate energy in the zero iteration. However, it is easy to verify, that substitution pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi of the parameters a = 2 and l ¼ ð1 þ 1 þ 4kÞ=2 into the RHS of Eq. (70), yields the pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi exact expression for the ground state energy E ¼ 2 þ 1 þ 4k (see, e.g. [57]). Analytic expression for the logarithmic derivative is obtained by means of general Eq. (28) with using the initial guesses (68) and (69) and the corresponding formula (70) for energy. This leads to "
# 2 2 l 1 k ex C 12 þ l a2 1 1 a 2 2
C þ l; x C þ l ; x y 1 ðxÞ ¼ x þ þ : 2 2 2 2l 1 x 2 x2l C 12 þ l ð72Þ At l = 1 this expression reduces to Eq. (67). It is not difficult to verify that substitution of the top value of a = 2 and the expression (37) for l into the RHS of the QLM formula (72) leads to the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2x2 þ 1 þ 4k y 1 ðxÞ ¼ ; ð73Þ 2x which corresponds to the exact solution pffiffiffiffiffiffiffiffi 1 x2 wðxÞ x2ð1þ 1þ4kÞ e 2
ð74Þ
of the proper Schro¨dinger equation (see, e.g. [57]). So, for the case of a = 2, we obtain the relations y0(x) = y1(x) = yexact(x) with l determined by Eq. (37). Note, that Eq. (74) was found by means of analyzing the behavior of wave function near the origin, and with no knowledge of the exact solution. The QLM ground state energies E0 and E1, calculated by Eqs. (70) and (19) with n = 1, respectively, are presented in Table 2. Comparing the data presented in Tables 1 and 2, one can conclude that initial guesses (68) and (69) yield energies which are more accurate for
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
2223
Table 2 The exact and QLM zero and first iteration ground state energies E0 and E1 together with values of parameter l for the spiked harmonic oscillator potential Ua(x) = x2 + k/xa with a < 2, calculated with the help of initial guesses (68) and (69) a
k
Eexact
E0
l
E1
0.5
0.01 0.1 1 10
3.01022627 3.10213909 4.00920406 12.093130
3.01022675 3.10218502 4.013207 12.1569
1.000675 1.006828 1.075876 2.311864
3.01022628 3.10213937 4.00921621 12.1013
1
0.01 0.1 1 10
3.01127600 3.1120669 4.057877 10.577484
3.01127779 3.112236 4.06759 10.6044
1.001735 1.017679 1.206014 3.224482
3.01127604 3.1120669 4.05807 10.57902
1.5
0.01 0.1 1 10
3.01379405 3.13505334 4.14189285 9.32417303
3.01379731 3.13533 4.14791 9.33042
1.003886 1.039387 1.398862 3.577468
3.01379400 3.13505351 4.14207 9.32425
the large values of a and k, than the energies produced by initial guesses of the form (54) and (55). Ground state wave functions for the spiked harmonic oscillator potential for parameters a ¼ 12, 32 and values of k equal to 0.01 and 10 are shown in Figs. 1 and 2. We present the exact wave functions wexact depicted by ‘‘solid’’ line on the upper parts of the graphs. These functions were obtained by means of numerical solution of the proper Schro¨dinger equations. The QLM wave functions depicted in Fig. 1 for the both values of parameter k were calculated with the help of explicit analytical expression (63) for the logarithmic derivative, obtained by using the initial guesses of the form (54) and (55). Our approximate analytical QLM wave functions wQLM are very accurate. The distinctions between them and the exact numerical wave functions cannot be revealed visually. Therefore, we present only relative logarithmic deviations from the exact value
Fig. 1. Ground state wave functions for the spiked harmonic oscillator potential with parameters a = 1/2 and k = 0.01; 10. The exact wave functions are depicted by ‘‘solid’’ line (upper parts of the graphs). Relative logarithmic deviations from the exact value for the QLM approximations are presented in the lower parts of the graphs by ‘‘dash’’ line for the zero order wave function, ‘‘dot’’ line for the first order wave function. Initial guesses of the form (54) and (55) were applied.
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Fig. 2. Ground state wave functions for the spiked harmonic oscillator potential with parameters a = 3/2 and k = 0.01; 10. Notations are the same as in Fig. 1. Initial guesses of the form (68) and (69) were applied.
lnjwQLM/wexact 1j for the QLM approximations in the lower parts of the graphs. The deviations for the zero iteration QLM wave functions are shown by ‘‘dash’’ line, whereas ‘‘dot’’ line is used for deviations of the first iteration QLM wave functions. One can observe in Fig. 1, that in case of a = 1/2 and small k = 0.01 the accuracies of the QLM wave functions are of the order 104 and 107 for the zero and the first iteration, respectively, which present very accurate results. The accuracy of the first iteration wave function for the large k = 10 are of the order 102 103. This is still reasonably good result, in spite of the fact, that it is worse, than for k = 0.01. In Fig. 2 one can see, that in case a ¼ 32 the corresponding logarithmic deviations from the exact values for the QLM first iteration wave functions are of the order 105106 in case of small k = 0.01, and 101103 in case of large k = 10. These results are obtained by means of the explicit analytic expression (72) for the logarithmic derivative, which is based on the initial guesses (68) and (69). 3.3. QLM results for the case of a > 2 As it was shown above, near the origin behavior of logarithmic derivative of the exact wave function for spiked oscillator potential (1) with the values of parameter a > 2 is determined by Eq. (53). Taking additionally into account, that the asymptotic behavior of spiked harmonic oscillator and harmonic oscillator wave functions are identical, we propose the following initial guess pffiffiffi c k y 0 ðxÞ ¼ þ a=2 x: ð75Þ x x The corresponding wave function has a form " # pffiffiffi a2 2 2 2 k 1 x w0 ðxÞ xc exp : a2 x 2
ð76Þ
Free parameter c, which earlier in Eq. (53) was equal to a4, is introduced to considerable improve the accuracy of our results and will be calculated by minimizing the proper energy E0 of a zero order, as it was done earlier in case of parameter b. We cannot derive the explicit analytical expression for E0 using Eq. (12) or (15) for w = w0 with initial guesses (75) or (76), respectively, in the general case of a > 2.
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
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Nevertheless, it is possible to do this, e.g. for the special cases of a = 5/2, 4, 6 considered in Refs. [13,14]. The corresponding expressions are very cumbersome, except for the case of a = 6, where E0 can be presented in the form pffiffiffi ( ) pffiffiffi 2k1=4 ½3 þ 4 k þ 2ðc 3ÞcK32þc ð2k1=4 Þ 1 2 E0 ¼ pffiffiffi kð12 8cÞ þ 2ð5 2cÞc 3 þ : 4 k K12þc ð2k1=4 Þ ð77Þ Here, Km(z) is modified Bessel function of the second kind. The first order logarithmic derivative function y1(x) can be calculated by means of Eq. (28) with using the initial guesses (75) and (76) for parameter c presenting the solution of equation oE0/ oc = 0 for given value of k. The required values of energy E0 can be obtained by using expression (77) for a = 6 or by using the more complicated expressions for a = 5/2, 4 obtained by Eq. (12). Inserting the obtained function y1(x) into the RHS of Eq. (19) for n = 1, one can find the proper first order energies E1 for the different values of the coupling parameter k. The exact and the QLM zero and first iteration ground state energies E0 and E1 are presented in Table 3. One can see that the accuracy of the QLM energy values is very high, with the first iteration values reproducing all seven digits of the exact energy values. One can notice that in Table 3 the results for large k are given only for a = 6, but not for smaller ones. The reason is easy to understand from analyzing of expression (76) for our initial guess wave function. Indeed, when constructing this expression, we assumed that the first term represents the behavior of wave function at small x, whereas the second term is responsible for the asymptotic behavior at large x. The point xeq, where the both terms are equal, could be calculated from the equation 2 pffiffiffi !ðaþ2 Þ 4 k xeq ¼ : ð78Þ a2
Table 3 The exact and QLM zero and first iteration ground state energies E0 and E1 together with values of parameter c for the spiked harmonic oscillator potential Ua(x) = x2 + k/xa, calculated with the help of initial guesses (75) and (76) for a = 5/2, 4, 6 a
k
Eexact
E0
c
E1
6
0.01 0.1 1 10 100 1000
3.505452 3.915665 4.659940 6.003209 8.413358 12.718617
3.506562 3.915687 4.660697 6.006564 8.420191 12.728722
1.309974 1.469066 1.732133 2.180871 2.962335 4.339169
3.505453 3.915665 4.659940 6.003212 8.413369 12.718639
4
0.01 0.1 1 10
3.2050675 3.575552 4.494178 6.606622
3.205364 3.576828 4.497696 6.613107
1.034424 1.100409 1.270806 1.674225
3.2050675 3.575552 4.494182 6.606633
5/2
0.01 0.1 0.5
3.036729 3.266873 3.848553
3.037492 3.267918 3.849740
0.905375 0.822995 0.790506
3.036733 3.266874 3.848554
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At x < xeq the absolute value of the first term is larger than the absolute value of the second one while at x > xeq the absolute value of the second term is larger than the absolute value of the first one. Initial guess of the form (76) will be valid only if xeq is close enough to the origin. However, from Eq. (78) follows that xeq increases with increase of k and with decrease of a (a > 2). The values of xeq for different values of parameters a and k are presented in Table 4. It is easy to verify, that for 2 6 a 6 6 and 0.01 6 k 6 1000 the exact ground state spiked wave functions are essentially differ from zero only for x < 8 (see the upper graphs of Figs. 1–4). Hence, Table 4 shows, that the mentioned above condition for using the initial guess (76) is not valid for large enough k and small enough a. For such combinations pffiffiffi of parameters k and a we propose simply to discard the problematic term kxa=2 in Table 4 Values xeq for initial guess (76) a/k
0.1
1
10
100
1000
5/2 4 6
1.51 0.86 0.75
2.52 1.26 1
4.20 1.85 1.33
7.01 2.71 1.78
11.7 3.98 2.37
Fig. 3. Ground state wave functions for the spiked harmonic oscillator potential with parameters a = 5/2 and k = 0.01; 1000. Notations are the same as in Fig. 1. Expressions (69) and (86) were used for constructing the approximate wave functions of the zero and the first order, respectively.
Fig. 4. Ground state wave functions for the spiked harmonic oscillator potential with parameters a = 6 and k = 0.01; 1000. Notations are the same as in Fig. 1. Initial guesses of the form (75) and (76) were applied.
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
2227
the initial guess (75) that is to return to already considered simple initial guesses (68) and (69). The corresponding analytic expressions (70) and (72) for the ground state energy and logarithmic derivative of wave function, respectively, thus could be used for arbitrary values of parameters. The calculations show that the energy minimum is reached at the large values of parameter l. For example, for a = 5/2 and k = 1000 one obtains l . 24.4. This leads in the RHS of Eq. (72) to incomplete Gamma functions C(a, z) with a large values of order a. The expression in the square brackets of Eq. (72) thus presents a difference of two very large quantities, which are very close to each other. To avoid this difficulty, let us substitute C(a, z) by the another well-known variation of incomplete Gamma function c(a, z) = C(a) C(a, z) in Eq. (72). Then, we obtain 2 l 1 k y 1 ðxÞ ¼ x þ þ Xðl; a; xÞ; ð79Þ 2l 1 x 2 where
"
# 2 C 12 þ l a2 ex 1 a 2 1
c þ l; x2 : Xðl; a; xÞ ¼ 2l c þ l ; x 2 2 2 x C 12 þ l
Substitution of series expansion (see, 6.5.4; 6.5.29 in Ref. [55]) 1 X zi cða; zÞ ¼ za ez CðaÞ Cða þ i þ 1Þ i¼0 for the both incomplete c-functions in the RHS of Eq. (80), thus yields 1 a Xðl; a; xÞ ¼ C þ l ½X1 ðl; a; xÞ X1 ðl; 0; xÞ; 2 2
ð80Þ
ð81Þ
ð82Þ
where X1 ðl; a; xÞ ¼
1 X i¼0
x2iþ1a 3
: C 2 þ l a2 þ i
ð83Þ
It is very important, that one can perform integration of the RHS of Eq. (82) over the variable x. Indeed, carrying out term by term integration of the both sums and then performing summation of the resulting series, we obtain Z Hðl; a; xÞ Xðl; a; xÞ dx
1; 1 a2 ; 32 þ l a2 ; 2 a2 ; x2 C 12 þ l a2 2 22 F 2
¼x þ xa ða 2Þð2l þ 1 aÞ 2C 32 þ l 3 2 F 2 f1; 1g; 2; þ l ; x2 : ð84Þ 2 Here 2 F 2 ðfa1 ; a2 g; fb1 ; b2 g; zÞ
¼
1 X ða1 Þk ða2 Þk zk ðb1 Þk ðb2 Þk k! k¼0
is generalized hypergeometric series [56].
ð85Þ
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Using Eqs. (84) and (79) we can write explicit expression for the ground state wave function of the first QLM iteration 2 R l2 x k y ðxÞ dx ð86Þ w1 ða; k; l; xÞ e 1 ¼ C N x2l1 exp Hðl; a; xÞ ; 2 2 where CN is normalization factor, whereas function H(l, a, x) is defined by Eq. (84). Note, that one can calculate w1(a, k, l; x) by means of Eq. (86) for a = 5/2 without any problem. However, one cannot do this for a = 4 or a = 6, because the first hypergeometric series in the RHS of Eq. (84) is divergent in these cases. The problem is that for the even integer values of a there will be the term with i = a/2 1 in the sum X1(l, a, x) of Eqs. (82) and (83). This term being proportional to 1/x yields ln (x) after integration for obtaining w1. To overcome this difficulty, let us divide the sum X1(l, a, x) into three parts with 0 < i < a/2 1, i = a/2 1 and i > a/2 1, correspondingly. Thus, we obtain a
X1 ðl; a; xÞ ¼
2 2 X i¼0
1 X x2iþ1a 1 x2iþ1a
3
3
: þ þ a 1 C 2þl2þi C 2 þ l a2 þ i xC l þ 2 i¼a
ð87Þ
2
It is easy to verify, that the third sum in the RHS of Eq. (87) reduces to X1(l, 0, x) by means of substitution k = i a/2. Thus, for the even integer values of a one obtains instead of Eq. (82) # " a22 1 a X x2iþ1a 1
:
þ Xðl; a; xÞ ¼ C þ l ð88Þ 2 2 xC l þ 12 C 32 þ l a2 þ i i¼0 Integration of the RHS of Eq. (79) with X(l, a, x) defined by Eq. (88) then yields w1 ða; k; l; xÞ ¼ C N x
l2 þ 2l1
ð Þ ð Þ
kC 1þla 2 2 2C 1þl 2
"
x2 kx2a k 1 a C þl exp 2 2 2 ða 2Þð2l a þ 1Þ 2 # a2 2 X x2iþ2a
; a ¼ 4; 6; 8; . . . ða 2i 2ÞC 32 þ l a2 þ i i¼1
ð89Þ
In particular, for the cases of a = 4, 6 under consideration we have 2 1 2k x k l2 þ2l3 ð Þ 2l1 exp w1 ð4; k; l; xÞ ¼ C N x ; ð90Þ 2 2ð2l 3Þx2 h i 2 1 4k l2 þð2l3Þð2l5Þ x k k 2l1 exp : w1 ð6; k; l; xÞ ¼ C N x 2 ð2l 3Þð2l 5Þx2 4ð2l 5Þx4 ð91Þ One can obtain the same result for even integer a also by applying the recursion relation c(a + 1, z) = ac(a, z) zaez (see, e.g. 6.5.22 in Ref. [55]) to Eq. (80). Since we derive the explicit expressions (86) and (89)–(91) for the first QLM iteration wave function w1(a, k, l; x), we can calculate parameter l by minimizing energy E1 as a
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
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Table 5 The exact and QLM zero and first iteration ground state energies E0 and E1 for the spiked harmonic potential Ua(x) = x2 + k/xa, calculated with the help of the first iteration wave functions (86) for a = 5/2, and by means of expressions (90) and (91) for a = 4, 6, respectively a
k
Eexact
E0
l1
E1
6
10 100 1000
6.003209 8.413358 12.718617
4.027186 5.770743 8.963489
6.332650 8.676420 12.942326
5.099990 7.458157 12.530918
6.013322 8.417318 12.725446
4
10 100 1000
6.606622 11.265080 21.369462
3.795513 6.888364 13.61853
6.693269 11.332546 21.427793
4.613350 8.768451 14.050578
6.607357 11.265871 21.370201
5/2
0.01 0.1 0.5 1 10 100 1000
3.036729 3.266873 3.848553 4.317311 7.735111 17.541890 44.955485
1.036739 1.210457 1.564245 1.832529 3.741461 9.191431 24.421591
3.037845 3.273543 3.857331 4.325682 7.740873 17.546306 44.959424
l0
3.036813 3.267401 3.848998 4.315508 7.735166 17.541903 44.955491
function of l. Thus, parameter l can be obtained as a root of the equation oE1/ol = 0 with given values of a and k, where energy E1 is defined by Eq. (15) with w(x) presented by Eq. (86) or Eqs. (89)–(91). The results of the exact and QLM ground state energies calculations, performed by means of Eq. (15) with using the wave function expressions ((86), (89)–(91)) are presented in Table 5. Note, that for the case of a = 5/2 we used Eqs. (84) and (86) with parameter l = l0 defined as a root of equation oE0/ol = 0, whereas for a = 4,6 parameter l = l1 was determined as a root of equation oE1/ol = 0 for computing E1 by Eq. (15) and Eqs. (90) and (91). The QLM ground state wave functions for parameter a > 2 are presented in Figs. 3 and 4. The notations of the curve style are the same as were declared in the previous Section for the Fig. 1. As one can observe in Fig. 3 for parameters a = 5/2, the logarithmic deviations from the exact value for the QLM first iteration wave functions are of the order 102 104 for both small k = 0.01 and large k = 1000. Remind, that the choice of the maximum value of k = 1000 is determined by the condition for the energy E < 50. For constructing the zero and the first order (iteration) QLM wave functions, we used expressions (69) and (86), respectively. Accuracy of the QLM wave functions for the spiked harmonic oscillator potential with parameter a = 6 is presented in Fig. 4. The corresponding deviations from the exact values for the QLM first iteration wave functions are of the order 102.5 104 in case of small k = 0.01 and 101.5 103.5 in case of large k = 1000. For constructing the QLM wave functions we employed the general analytic formula (28) with using the initial guesses (75) and (76) and parameters c, presented in Table 3. 4. Conclusions We used the quasilinearization method (QLM) [40–45] to approximate analytically solutions of the Schro¨dinger equation for the spiked harmonic oscillator. We found
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accurate analytic presentation for the ground state energies and wave functions. These high precision approximate analytic expressions are obtained by first casting the Schro¨dinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the method. In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries.
Table 6 Expectation values Æx2æi and Æxaæi of long and short range parts of the spiked harmonic oscillator potential Æx2æexact
Æxaæexact
Æxaæ0
Æxaæ1
3.87415 5.59353 8.80994
0.0771101 0.0138033 0.00246215
0.0734586 0.0135825 0.0024493
0.0712529 0.0140438 0.00255314
4.29551 7.38836 14.1185
3.98755 7.08293 13.7918
0.133341 0.0291972 0.00629284
0.132190 0.0290505 0.00629021
0.132725 0.0294062 0.0063079
1.53674 1.71046 2.06424 2.33253 4.24146 9.69143 24.9216
1.52337 1.66369 1.99964 2.26492 4.17462 9.62764 24.8593
3.40460 2.08577 1.11662 0.806490 0.242319 0.0682645 0.0190275
3.53308 2.11200 1.11636 0.805540 0.242229 0.0682611 0.0190274
3.35067 2.02994 1.09707 0.796386 0.241551 0.0682061 0.0190224
Æx2æ0
a
k
6
10 100 1000
3.77271 5.58701 8.82146
4.52719 6.27074 9.46349
4
10 100 1000
3.97002 7.08790 13.8312
5/2
0.01 0.1 0.5 1 10 100 1000
1.52262 1.65951 1.99407 2.25947 4.17045 9.62425 24.8562
Æx2æ1
Calculations were performed with the exact (i = exact) and QLM wave functions (iteration order i = 0, 1), based on the initial guesses (68) and (69). First order QLM wave functions were computed by Eq. (86) for a = 5/2 or by means of expressions (90) and (91) for a = 4, 6, respectively.
Table 7 Expectation values Æx2æi and Æxaæi of long and short range parts of the spiked harmonic oscillator potential a
k
Æx2æexact
Æx2æ0
Æx2æ1
Æxaæexact
Æxaæ0
Æxaæ1
6
0.01 0.1 1 10 100 1000
1.88289 2.19445 2.75840 3.77271 5.58701 8.82146
1.90599 2.19775 2.73885 3.73174 5.52896 8.75131
1.88338 2.19446 2.75864 3.77360 5.58855 8.82344
13.0168 2.36615 0.428429 0.0771101 0.0138033 0.00246215
13.8256 2.37943 0.419511 0.0750321 0.0134844 0.00242144
12.9760 2.36606 0.428067 0.0769335 0.0137646 0.00245613
4
0.01 0.1 1 10
1.65003 1.91202 2.54393 3.97002
1.63630 1.88394 2.49799 3.90827
1.65010 1.91228 2.54451 3.97096
9.49951 2.48497 0.593678 0.133341
9.25290 2.41504 0.579689 0.131238
9.49359 2.48106 0.592219 0.133015
5/2
0.01 0.1 0.5
1.52262 1.65951 1.99407
1.50552 1.63461 1.96497
1.52300 1.65984 1.99428
3.40460 2.08577 1.11662
3.30069 2.05885 1.11005
3.38889 2.08177 1.11559
Calculations were performed with the exact (i = exact) and QLM wave functions of the ith order (i = 0, 1), based on the initial guesses (75) and (76).
E.Z. Liverts, V.B. Mandelzweig / Annals of Physics 322 (2007) 2211–2232
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The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist [46–49]. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems. Using Eq.(9) or Eqs.(24), (26), and (28) alternatively, one can write the analytical expressions for the wave function logarithmic derivatives of the arbitrary physical potential. The zero and first iteration ground state energies E0 and E1 can be calculated with the help of the general formulas (12) and (15) with w = w0, and (19) with n = 1, respectively. In order to determine the accuracy of the QLM solutions, we also found the direct numerical solutions. The quality of the wave functions is illustrated by Figs. 1–4. In order to check the overall accuracy of the wave functions we calculated also the expectation values
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