6.B
Nuclear Physics 24 (1961) 313--317 ; © North-Holland Publishing Co ., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
A NEW VARIATIONAL METHOD FOR CALCULATING EXCITED ENERGY LEVELS AND ITS APPLICATION TO THE ANHARMONIC OSCILLATOR KAZUO YAMAZAKI Max-Planck-Institut für Physik und Astrophysik, München, Gertnany Received 21 November 1960 Abstract : A simple variational method for the calculation of the excited levels of a Iquantal system is developed and as an illustration it is applied to the anharmonic oscillator. For this case it agrees in the weak coupling limit with the perturbation calculation and for the strong coupling limit it also agrees with the exact values within a 1 % error.
1 . General Method We here propose a new variational method for the determination of the excited levels and apply it to the anharmonic oscillator as an example. The method is quite simple and, for the anharmonic oscillator, it agrees with the perturbation calculation in the weak coupling limit ; in the strong coupling limit it also agrees quite well, say up to about 1 %, with the exact value obtained by numerically solving the Schr6dinger equation with a computer . For the n-th excited level it is in principle possible to use the usual variational method, if we know the exact 0-th, . . ., (n-1 }-th state vectors ; among the state vectors which are orthogonal to these exact 0-th, . . . (n-1)-th state vectors, we choose a suitable trial function and find the best one by the variational method; then we can find an upper bound for the n-th excited level. But in the real problems it is almost always impossible to know the exact state vectors. If we use approximate 0-th to (n-1)-th state vectors, we are in trouble, especially since we know that rather good eigenvalues can be obtained with rather poor associated state vectors. This is the difficulty usually encountered in applying the variational method to excited levels . Our idea is the following . We choose a suitable simple trial function and the variational parameters are so chosen that the expression -- 2 becomes as small as possible t ; it is, of course, zero for the exact n-th state . This requirement has another nice feature, namely it gives us a possibility to estimate the error in Em by computing tt 2
2 2
A similar variational method was used in another context by H. PreuSS 4 ) . I am indebted to Dr. Pr. euss for this information . tt The factor j is due to the squaring . t
313
KAZUO YAMAZAKI
31 4
However, this is generally an overestimate of the error in En , because the above expression is the ratio of the sum of the non-diagonal elements and the diagonal element, thus it refers rather to the error of the state vector In>. (Compare with table 1 at the end of this paper) . It is regrettable that even though we use here the variational method we cannot say that the values so obtained give an upper bound of the true value (see table 1) . This is so, because we give up the requirement of the orthogonality of the state vectors, as remarked above. Because of its simplicity, the possibility of the estimate of errors and the rather surprising agreement with the exact values, we can hope that our method may have some application in the problems of field theory, as for example in that of mass determination in the Heisenberg theory of elementary particles'), or in the domain of solid state physics . 2. Anharmonic Oscillator The Hamiltonian of the anharmonic oscillator is given by (we use the notation of Heisenberg 2) ) Let us now put
H = J(p 2+x2) +-4~
with
[P, x]
- -i (a-a*), x .- 1/Ja(a+a *), 1/2x
(1)
(2)
where a* and a are creation and annihilation operators satisfying the usual commutation relations [a, a*] == 1,
[a, a] = [a*, a*] - 0.
The point is here that we introduce a variational parameter a which is adjusted differently for each level. If we wont to calcuïate the n-th excited level E , the corresponding a. is determines from the condition where
6(-1nlHIn>2) = 0,
ln> =
a*n - I0>,
1/n !
aj0> --- 0.
(4) (5 )
The state Ink seems to be just the usual free harmonic oscillators's n-th excited state but through (2) this In> is implicitly a function of the parameter an. The various In> are not orthogonal to each other, as we can see from the fact that the x-represensative is the Hermite polynomial with the argument xâ1~ A method in spirit similar to the above was proposed by Kaiser 3) in the framework of the new Tamm-Dancoff approximation. In some cases it may
n
METHOD FOR CALCULATING ENERGY LEVELS
31 5
have advantages because of the possible ease with which it is extended to field theory. The superiority of our method is its simplicity, the possibility of the estimate of errors and, for the anharmonic oscillator, a better agreement with the exact values, as already emphasized above. To carry out the actual calculation we proceed as follows. First we put (2) into (1) and express H and H2 in terms of a, a* and the parameter a, and then perform the ordering of all the operators in H and H2 in the normal product form of Wick (the a stand to the right and the a* stand to the left) . The following formula is useful for the ordering of the operators x8, a,4 . . . . (a±a*)2n
-
j
n
([a
*, n -n a ) Bn-23 (±1) +'(2n)! a* )2n-2i-i . ai (vr (2n--2i-Z) ! i!
(6
By taking the expectation values of the well-ordered H and H2 in the state In> given by (5), the non-diagonal parts with respect to the number operator a* a drop out. It is quite easy to re-express these well-ordered operators in terms of the number operator a* a . Thereupon we use (5) with = n. After all this we find, for arbitrary n,
=4
OC+ 1
oc -lnlHln ;2
(2n+ 1) +-
3Âoc- (2n2+ 2n+ 1), 16
a, 1 )2 (n2+n+1)+ ~,a2 a- 1
=s a a A2 a4 256
8 (
)
(2n3+3n2+7n+3)
(34n4+68n3+278f22+244n+96) .
From (4) and (7) an is seen to satisfy the equation
3) (2n3 +3n2 +7n+3) 2 (n +n+ 1) + (Aan (3oa n2- 1) -}-4(Aa3)2(17n 4 -}-34n3 +139n2 +122n+48) = 0.
2(an4 -1)
This equation fixes the value of an as function of A (the redundant solution can be simply rejected ty the requirements that an be real and positive and an -> 1, for A -> 0) . Inserting this value of a,, into the expression in (7) we get out energy level En in the form E
1
1
3l~.an 2 (2n2+2n+1) .
_ 4 an+ (2n+1)+ ?a 16 an
(9)
For the case A < 1 the solution of (8) is clearly an = 1+0(A)+ . . . ; thus our energy levels are given by En
=
31 (2n2+2n+1)+O(A2)+ n+2)+ ` 16
. . .,
A < 1,
(10)
KAZUO YAMAZAKI
31 6
which agrees with the usual perturbation calculation. In the strong coupling limit, A > 1, the first four a become a3 1 = 1.219Af, a31 = 1 .41311, a271 = 1.537AI, a31 = 1 .648A1. (11) The corresponding E., their error by comparison with the exact values, and the rough estimate of error < njH2j n> __ 2/22 are tabulated in table 1 . We see the agreement is excellent . For further comparison we have also quoted the values of Heisenberg 2) and of Kaiser 3), which are obtained with the new Tamm-Dancoff method and its modification, respectively. In the last column of table 1 we have also quoted the values obtained by the simple variation method which will be discussed below. TABLE t Comparison of the calculated energy values with the exact ones e
Exact value 2)
En eq . (9)
error
I
- 2 f2
En SN.M.
Heisenberg Kaiser 1)
2)
2
0.4292
I
0.421 a)
(
E1aH
1.5079 1 1 .5293 I +1.4%1
0.027
I
1.574
11 .566
1 .5269
E21-lr
2.9587 1 2.9531 ß -0.2%1
0.020
I
2.889
( 3.008
E3;.-1
4.6210
5.311
4.659
2.9513 1 14.5933
a)
0.129
4.6100
-0.2%
0.015
0.421
a) i
E(,;.-i ( 0.4208 1 0.4309 I +2.2%1
The Eo value cannot be calculated, in principle, by the new Tamm-Dancoff method . Here we
take the exact value as reference level for the excited states .
3. Comparison with Simple Variational Method
Lastly we want to compare our method with the simple variational method (quoted in the following as S.V.M.), namely, that in which the same trial function (5) is used but the variational parameters a are determined simply from 6 = 0, instead of (4). Let us denote the n-th energy level obtained by S.V.M., that given by our eq. (9), and the exact one, by Env, E and Ene, respectively. Then it is clear that En < En (for arbitraly n), because Env is the minimum among all possible En (a) . Now, for the ground state, clearly Eô s Eov. The 1st excited state is an odd function owing to the special symmetry of our Hamiltonian (1), H(x) = H(-x), the exact ground state function is an even function. As our trial function is an odd function, it is orthogonal to the exact ground state function. Thus it is clear that E1e < E1v. From this argument we know without any explicit calculation that Eae
METHOD FOR CALCULATING ENERGY LEVELS
317
This means that for the ground and 1st excited state, S.V.M. gives better results than our eq. (9). But for the higher excited state there is no inequality, E,av. In these cases our method should give better results than S .V.M . E,e The numerical results summarized in table I show that this is actually the case, although numerically the difference is small. The above comparison with the S.V.M. shows clearly the validity of our method . For the ground state it is always poor, while for the excited state it is in general excellent. In general our method and S.V.M . are complementary in the sense that if we know some general symmetry properties of the systems we can use them for the orthogonality of the trial function and exact state vector ; then S.V.M. is better ; otherwise, for the general excited state, our method is better.
s
The author wishes to thank Professor W. Heisenberg for the interest and encouragement in this work, and also Dr. R. M. Ahrens for the kind reading of the manuscript . References 1) H. P. Dürr, W. Heisenberg, H. Mitter, S. Schlieder and K. Yamazaki, Zelts . f. Naturf . 14 (1959) 441 2) W. Heisenberg, Nachr. d. G6tt . Akad. d. 'Wiss. (1953) p. 111 3) H. J. Kaiser, Thesis, Dresden 1959; Ann . d. Phys . 6 (1960) 131 4) H. Preuss, Zeits. f. Naturf. 13 (1958) 439