- Email: [email protected]
Dynamics of zero-energy nonspreading non-Gaussian wave packets for a class of central potentials
Annals of Physics 337 (2013) 25–33
Contents lists available at SciVerse ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
Dynamics of zero-energy nonspreading non-Gaussian wave packets for a class of central potentials Adam J. Makowski ∗ , Piotr Pepłowski Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland
highlights • • • •
Central potentials are considered. Nonspreading, non-Gaussian wave packets are constructed. Time evolution of the zero-energy packets is studied. Quantum–classical correspondence is discussed.
article
info
abstract
Article history: Received 4 February 2013 Accepted 20 June 2013 Available online 27 June 2013
Zero-energy wave packets, coherent states, are constructed in such a way that they retain their shape during the time evolution for a large class of central potentials. The packets are not of the Gaussian type with −r 2 dependence but, instead, their shape is determined
Keywords: Zero-energy wave packet Central potential Quantum–classical correspondence
by −r µ+1/2 with −1/2 < µ < 1/2. A very close quantum–classical correspondence is also shown, i.e., the well localized states travel along suitable classical trajectories. © 2013 Elsevier Inc. All rights reserved.
1
1. Introduction The most classical among pure states of quantum mechanics, called after Glauber [1] the coherent states, are used in many areas of physics from solid state physics through quantum optics to modern cosmology. The problem of construction of such well-localized, nonspreading states was first addressed in 1926 by Schrödinger who successfully solved it for the harmonic oscillator [2], however,
∗
Corresponding author. Tel.: +48 56 6113300. E-mail addresses: [email protected] (A.J. Makowski), [email protected] (P. Pepłowski).
0003-4916/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.aop.2013.06.011
26
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
without a similar result for the hydrogen atom. Since then, numerous methods were developed to derive localized states, wave packets, which travel along classical orbits for variety of potentials [3–7]. Three groups of methods are among the most popular ones. In Barut’s approach [3] coherent states are constructed as eigenstates of the lowering operator. They can also be found by acting on a reference state, often chosen as the ground state, with a suitably defined displacement operator. These are Perelomov type coherent states [4]. In the third approach, let us call it Nieto’s method [5], the states are constructed as minimum-uncertainty wave packets. Only for the harmonic oscillator the three methods lead to the same wave packet. Otherwise, the methods are not equivalent. A comprehensive review of other approaches can be found in [6,7]. Experimentally, the packets were first created some 25 years ago [8,9] by photoexcitation using ultrashort pulses. The technique was further developed by Jones et al. [10] by using half-cycle pulses. A procedure to produce well-localized Bohr-like wave packets to mimic an electron moving in a Kepler orbit was suggested in [11] and successfully applied in [12]. Some other interesting experimental papers on Rydberg wave packets in atoms can be found in [13] and those for molecules in [14]. In this paper, we construct spatially well-localized and nonspreading packets using the method described in [15]. In the method, one uses superpositions of states with well-defined angular momentum corresponding to the total energy E = 0. Calculations are performed in the twodimensional (2D) configural space for the potentials in the form Vµ (r ) =
λ 2µ + 1
2
λ2 /2 2µ−1
(r /r0 ) µ+1/2
Q
−
2µ
(r /r0 ) µ+1/2
,
(1)
where the parameters λ, r0 , Q are positively determined and µ ̸= −1/2. It was shown in [16] that these potentials have normalizable E = 0 bound states, and that the corresponding wave functions can be deduced from E ̸= 0 solutions of the 3D harmonic oscillator. This was done with the help of some point transformations. Our aim is to derive wave packets that move precisely along classical trajectories for the potentials in Eq. (1) for some distinguished values of µ. To this end, in Section 2, we will first find and discuss classical trajectories of the problem. Since for the central potentials the angular momentum is a conserved quantity, the orbits are flat, and therefore we can consider only the 2D case. Then, in Section 3, we shall derive square-integrable wave functions for the radial 2D Schrödinger equation with the potential Vµ (r ). In Section 4, the functions are used for the construction of a stationary nonGaussian wave packet. Its motion along classical solutions is discussed in Section 5, and finally, our conclusions are given in Section 6. 2. Classical solutions The E = 0 orbits in the potential in Eq. (1) can be easily found using the polar coordinates r and ϕ . For the angular momentum L and the threshold energy, we now have L = mr 2 ϕ, ˙
(2)
0 = (m/2)(˙r + r ϕ˙ ) + Vµ (r ), 2
2
2
(3)
where the dots over symbols mean time derivatives. From these equations, we get dr dϕ
=
−r 2 −
2mr 4 L2
Vµ (r ).
(4)
Eq. (4) can be integrated if the transformation ρ = r /r0 = Z µ+1/2 is made. In this way, we obtain
µ+1/2
ρµ (ϕ) =
2 B−(
B2
1/2
− 4A)
sin
ϕ−ϕ0 µ+1/2
,
(5)
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
27
where B2 ≥ 4A and the constants A and B read as mλ2 r02
A=
B=
L2 2mQr02
λ 2µ + 1
2
λ 2µ + 1
2
L2
,
(6)
.
(7)
√
They are obviously dimensionless since λ2 and Q in the potential Vµ (r ) are measured in units of J, where J = kg m2 /s2 stands for the joule. The general E = 0 solution for ρµ (ϕ), with B2 ̸= 4A and µ ̸= −1/2, admits a variety of orbits of either periodic ones, when µ is a rational number, or of nonrepeating orbits, if µ is an irrational number. For the special µ = 0 case, the solution represents trajectories which are either circles or ellipses for the 2D radial harmonic oscillator for which the parameter Q λ2 in Vµ (r ) plays now 2 the role of the total √ energy E ̸= 0 for the oscillator. When µ ̸= −1/2 and B = 4A, i.e. when Q = L(2µ+ 1)/(r0 m), we get a family of circles ρµ = (2/B)µ+1/2 , all corresponding to the minimum of effective potential Veff (ρ) = L2 /(2mr02 ρ 2 ) + Vµ (ρ). In Fig. 1 we show some of the variety of trajectories for the considered class of potentials. The classically inaccessible regions, shaded in the figure, are determined by the relations: ρ ≤ ρ1 and ρ ≥ ρ2 , where ρi (i = 1, 2) are classical turning points in Veff (ρ) = L2 /(2mr02 ρ 2 ) + Vµ (ρ) and are found to be ρ1,2 = [(B ∓ harmonic oscillator.
√
B2 − 4A)/2A]µ+1/2 . In the middle of the figure there is an ellipse for the
3. Zero-energy wave functions We consider now the radial part of the two-dimensional (2D) Schrödinger equation for the potentials in Eq. (1):
d2
1 d
l2
2mr02
λ 2µ + 1
2
λ2 /2
Q
− R(ρ) = 0, (8) 2µ−1 2µ ρ µ+1/2 ρ µ+1/2 √ where, as previously, ρ = r /r0 = (x/r0 )2 + (y/r0 )2 = X 2 + Y 2 , Q > 0, λ is a constant parameter, dρ 2
+
ρ dρ
−
ρ2
−
h¯ 2
and m and l stand respectively for the mass of the particle and angular momentum quantum numbers. Eq. (8) possesses square-integrable solutions provided that the coupling constant Q is chosen in a proper way. To see that, we change the independent variable ρ in the following way 2µ+1
λ
ρ=
h¯ 2
µ+21/2
mr02
uµ+1/2 .
(9)
Then, Eq. (8) simplifies to the following one
d2 du2
+
1 d u du
−
l2 (µ + 1/2)2 u2
1
+ Q
mr02 1 h¯ 2 u
2
1
− R(u) = 0.
(10)
4
Its solution is proposed to be R(u) = e−u/2 ul(µ+1/2) F (u),
µ > −1/2,
(11)
and then, we can obtain
u
d2 F du2
+ [2l(µ + 1/2) + 1 − u]
dF du
1
+ Q 2
mr02 h¯ 2
−
1 2
− l(µ + 1/2) F = 0.
(12)
28
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
Fig. 1. (Color online) A gallery of closed trajectories obtained from Eq. (5). In the figures, Y = y/r0 is plotted versus X = x/r0 , the origin of coordinates is in the middle of each square, and classically inaccessible regions are shaded.
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
29
γ
This is an equation for the generalized Laguerre polynomials Ln (u) = F (u) if Q is ‘quantized’ according to the rule
Q =
h¯ 2 mr02
[2n + 1 + 2l(µ + 1/2)].
(13)
In this way, the normalizable solutions of Eq. (8), for µ > −1/2, have the final form
(µ)
1
(µ)
Rnl (ρ) = Nnl exp − λ 2
2
mr02 h¯ 2
ρ
1 µ+1/2
ρ
l 2l(µ+1/2) Ln
λ
2
mr02 h¯ 2
ρ
1 µ+1/2
,
(14)
(µ)
where the symbol Nnl stands for the normalization constant. Its explicit form is given in the Appendix and the quantum numbers n and l take the values of zero and positive integers, i.e., n ≥ 0 and l ≥ 0. 4. Stationary wave packets We now define a spatially well-localized coherent state, being a combination of the functions (µ)
(µ)
Φnl (ρ, ϕ) = Nϕ exp(ilϕ)Rnl (ρ),
(15)
in the form (µ)
Ψαβ (ρ, ϕ) = N (α, β)
∞ ∞
(µ)
Nϕ Nnl
l =0 n =0
−1 α n β l (µ) Φnl (ρ, ϕ). l!
(16)
The parameters α and β are taken to be real without loss of generality. In the above formulas (µ) N (α, β), Nϕ and Nnl denote the normalization constants of corresponding states. With the help of Eqs. (14) and (15), and of the generating function formula for Laguerre polynomials [17]
∞
1 (µ+1/2) 2 α n L2l (λ mr02 / h¯ 2 ρ µ+1/2 ) = n
n=0
exp
αλ2 α−1
1
mr02 / h¯ 2 ρ µ+1/2
(1 − α)2l(µ+1/2)+1
,
|α| < 1,
(17)
we obtain (µ)
Ψαβ (ρ, ϕ) =
N (α, β) 1−α
exp
1 −λ2 (1 + α) β eiϕ ρ . mr02 / h¯ 2 ρ µ+1/2 + 2(1 − α) (1 − α)2µ+1
(18)
Thus, for the probability density we can write finally (µ) |Ψαβ (ρ, ϕ)|2
N 2 (α, β)
1 −λ2 (1 + α) 2β cos(ϕ) 2 µ+1/2 2 mr0 / h¯ ρ + = exp ρ . (1 − α)2 1−α (1 − α)2µ+1
(19)
This packet is of the Gaussian type only for µ = 0, and then, it reduces to the known functional form as, e.g., that considered in [18]. Nevertheless, it is very well localized when −1/2 < µ < 1/2. Its image is given in Fig. 2 and is only slightly sensitive to the chosen value of µ. In the next section we shall show how to set the packet in motion along some of the orbits presented in Fig. 1. 5. Setting packets in motion The question how to set in motion the stationary packet in Eq. (19) has no simple answer. One should take into consideration that the packet is built of the Hamiltonian eigenstates all corresponding to the total energy E = 0, i.e. H Φ (ρ, ϕ) = 0. If H is a Hermitian operator then, according to the rules of quantum mechanics, the time-evolution of a state for time-independent problems is governed by the unitary operator exp(−iHt /h¯ ). In our case, no immediate application of the rule is possible. However, we can utilize the relationship of Eq. (8), for E = 0 with the potential in Eq. (1), to the 2D harmonic
30
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
Fig. 2. (Color online) The normalized to unity wave packet from Eq. (19) plotted for α = 0.625, β = 1.525,
mr02 / h¯ 2 =
1, λ = 1, µ = −1/6, and N (α, β) = 0.14. The coordinates X and Y are dimensionless as explained in the text.
oscillator with E ̸= 0. To this end, let us make in Eq. (8) the point transformation ρ = v 2µ+1 . Then Eq. (8) can be written in the form
d2 dv 2
+
1 d
v dv
−
−2mr02 2 Λ2 mr02 4 2 λ v R (v) = λ QR(v). − v2 h¯ 2 h¯ 2
(20)
We have thus obtained the radial Schrödinger equation for the harmonic oscillator with total energy Enl = λ2 Qnl = h¯ ω[2n + 1 + Λ] and scaled angular momentum Λ, where ω
(21)
=
λ4 /(mr02 ) and Λ = (2µ + 1)l. In this
connection, we expect that the oscillator’s Hamiltonian can be considered as a proper generator for (µ) the time–displacement operator also for our packet |Ψαβ (ρ, ϕ)|2 . Formally, its time-dependence, i.e. (µ)
|Ψαβ (ρ, ϕ, τ )|2 , can be easily found if in the summands of Eq. (16), the parameter α is replaced by α exp(−2iτ ) and β by β exp[−i(2µ + 1)τ ], where λ2 h¯ 2 τ= t (22) h¯ mr02 is a dimensionless scaled time. Then, performing calculations as in Section 4, we will get N (α, β, τ ) 2
(µ)
|Ψαβ (ρ, ϕ, τ )|2 =
1 + α 2 − 2α cos(2τ )
+
exp
−λ2 (1 − α 2 ) mr02 / h¯ 2 1 + α 2 − 2α cos(2τ )
2β cos[C (ϕ, µ, τ )]
ρ , [1 + α 2 − 2α cos(2τ )]µ+1/2
1
ρ µ+1/2
(23)
where C (ϕ, µ, τ ) = ϕ − (2µ + 1)τ − (2µ + 1) tan−1
α sin(2τ ) . 1 − α cos(2τ )
(24)
As it should be, the packet reduces to the stationary one when τ = 0. That the time-dependence of our packet is correctly defined can be observed in Fig. 3. We have plotted the cut of packet for some fixed instants of time. It moves precisely along classical orbits as expected. Note also that the packet still ‘looks’ at the same point at every instant of time with no rotation at all. The same excellent quantum–classical correspondence can be shown for other values of µ > −1/2 including, of course, the µ = 0 case for the harmonic oscillator. We do not illustrate this case since it was considered in a number of papers [18–20] though in a different context. One can also choose µ to be an irrational number and again the correspondence is perfect.
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
a
b
c
d
31
Fig. 3. (Color online) Motion of the wave packet from Eq. (23) for some chosen values of µ. The contour of constant probability density is plotted for some fixed instants of time. Its initial position, for τ = 0, is marked by the shaded area. Dots represent the positions of the packet’s maxima and the full line is the classical orbit obtained from Eq. (5). In each case the cut of packet (µ)
corresponds to max(0.99|Ψαβ (ρ, ϕ)|2 ), λ2 = 1 and
mr02 / h¯ 2 = 1. The four figures are prepared using the following sets of
parameters: (a) α = 0.71, β = 0.99; (b) α = 0.66, β = 0.93; (c) α = 0.525, β = 0.665; (d) α = 0.5, β = 0.595. The coordinates X and Y are dimensionless.
A few comments on normalization of the packet are in order now. When µ = 0, the functions in Eq. (14) are all orthogonal each other and the packet’s norm in Eq. (23) is preserved in time. When µ ̸= 0 the problem is much more subtle. Namely, one should remember that the functions (µ) Rnl (ρ) are all eigenstates of the Hamiltonian to the total energy of E = 0. They are not orthogonal (µ)
each other and, during the packet’s |Ψαβ (ρ, ϕ, τ )|2 evolution its norm is not preserved. The states (µ) Rnl (ρ) would become orthogonal if were integrated with the additional weight function w(ρ) = −2µ . Preparing images in Fig. 3, the factor N (α, β, τ ) in Eq. (23) is chosen such that the integral ρ (µ) |Ψαβ (ρ, ϕ, τ )|2 ρ dρ dϕ = 1 be obeyed. We can also consider another way of reasoning. Treat our 2D, E = 0, Schrödinger equation for the potential Vµ (r ), as the following eigenvalue problem (µ)
(µ)
ˆ Φnl (ρ, ϕ) = Qnl Φnl (ρ, ϕ), O
(25)
32
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
where
− h¯ 2
ˆ ≡ O
2µ + 1
2
λ
2mr02
2µ
ρ µ+1/2
∂2 1 ∂ 1 ∂2 + + ∂ρ 2 ρ ∂ρ ρ 2 ∂ϕ 2
+
λ2 2
1
ρ µ+1/2 ,
(26)
(µ)
the functions Φnl (ρ, ϕ) are defined in Eq. (15), and the eigenvalues Qnl in Eqs. (13) or (21). Then, the (µ)
functions χnl (ρ, ϕ, τ ) = exp(−iQnl τ )Φnl (ρ, ϕ) obey the generalized equation
ˆ −i O
∂ ∂τ
χnl (ρ, ϕ, τ ) = 0.
(27)
ˆ becomes Hermitian if the orthonormality In the space of functions χnl (ρ, ϕ, τ ), the operator O condition
ρ −2µ χn∗′ l′ χnl ρ dρ dϕ = δn′ n δl′ l
(28)
is imposed. This is a Sturm–Liouville approach to the problem that supports the above made choice for the packet’s time-dependence. 6. Conclusion In this paper, we have been concerned with the construction and dynamics of two-dimensional non-Gaussian wave packets for a large class of central potentials. The method we have used guarantees that the maximum of the packets follows the corresponding classical orbits with very high accuracy. The point that needs some further study is the problem of how to set in motion packets constructed from Hamiltonian zero-energy eigenstates. The close relation of the used here potentials to the harmonic oscillator, and a possibility of using a Sturm–Liouville approach, were a fruitful advice in that direction and the results presented in Fig. 3 confirm the ideas. Appendix The wave function in Eq. (14) can be normalized in the standard way, i.e. we require
(µ)
1 = Nnl
∞
2
(µ)
[Rnl (ρ)]2 ρ dρ.
(A.1)
0
Changing the variable of integration
Z = λ2
mr02 h¯ 2
1
ρ µ+1/2 ,
(A.2)
we will get
(µ)
1 = Nnl
2
µ + 1/2
λ(2µ+1)(2l+2)
h¯ 2 mr02
(1/2+µ)(l+1)
∞
(µ+1/2) e−Z Z 2µ(l+1)+l L2l (Z ) n
2
dZ .
(A.3)
0
This integral issimple for the 2D harmonic oscillator, µ = 0, and it is equal to (n + l)!/n!. Then, assuming λ = mw 2 /r02 , the normalization constant has the well-known form
(0)
Nnl =
2(mw/h¯ )l+1 n!
(n + l)!
.
(A.4)
In the general case, for µ > −1/2, we can utilize the formula 2.19.14.(15) in Ref. [21]. Now, the integral in Eq. (A.3) is equal to I = S (µ, l, n)3 F2 (−n, 2µ(l + 1) + l + 1, 2µ + 1; 2l(µ + 1/2) + 1, 2µ + 1 − n; 1),
(A.5)
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
33
where 3 F2 means the generalized hypergeometric function and S (µ, l, n) =
(1 + 2l(µ + 1/2))n (−2µ)n Γ (2µ(l + 1) + l + 1) , [n!]2
(A.6)
whereas (a)k = Γ (a + k)/Γ (a). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
R.J. Glauber, Phys. Rev. 131 (1963) 2766. E. Schrödinger, Naturwisseschaften 14 (1926) 664. A.O. Barut, L. Girardello, Comm. Math. Phys. 21 (1971) 41. A.M. Perelomov, Comm. Math. Phys. 26 (1972) 222. M.M. Nieto, L.M. Simmons Jr., Phys. Rev. Lett. 41 (1978) 207. V.V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1. A. Buchleiter, D. Delande, J. Zakrzewski, Phys. Rep. 368 (2002) 409. A. ten Wolde, L.D. Noordam, A. Lagendijk, H.B. van den Heuvell, Phys. Rev. Lett. 61 (1988) 2099. J.A. Yeazell, M. Mallalieu, J. Parker, C.R. Stroud Jr., Phys. Rev. A 40 (1989) 5040. R.R. Jones, D. You, P.H. Bucksbaum, Phys. Rev. Lett. 70 (1993) 1236. Z.D. Gaeta, M.W. Noel, C.R. Stroud Jr., Phys. Rev. Lett. 73 (1994) 636. J.J. Mestayer, B. Wyker, F.B. Dunning, C.O. Reinhold, S. Yoshida, J. Burgdörfer, Phys. Rev. A 78 (2008) 045401. Z.D. Gaeta, C.R. Stroud Jr., Phys. Rev. A 42 (1990) 6308. H.H. Fielding, Annu. Rev. Phys. Chem. 56 (2005) 91. A.J. Makowski, P. Pepłowski, Phys. Rev. A 86 (2012) 042117. A.D. Alhaidari, Int. J. Mod. Phys. A 17 (2002) 4551. H. Bateman, A. Erdélyi, Tables of Integral Transforms, Vol. II, Mc Graw-Hill, Inc., New York, 1954 (Chapter 10.12). E. Colavita, S. Hacyan, Phys. Lett. A 337 (2005) 183. F. Arickx, J. Broeckhove, P. van Leuven, Il Nuovo Cimento B 104 (1989) 629. F. Arickx, J. Broeckhove, P. van Leuven, Phys. Rev. A 43 (1991) 1211. A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series: Special Functions, Vol. II, Fizmatlit, Moscow, 2003 (in Russian).
Contents lists available at SciVerse ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
Dynamics of zero-energy nonspreading non-Gaussian wave packets for a class of central potentials Adam J. Makowski ∗ , Piotr Pepłowski Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland
highlights • • • •
Central potentials are considered. Nonspreading, non-Gaussian wave packets are constructed. Time evolution of the zero-energy packets is studied. Quantum–classical correspondence is discussed.
article
info
abstract
Article history: Received 4 February 2013 Accepted 20 June 2013 Available online 27 June 2013
Zero-energy wave packets, coherent states, are constructed in such a way that they retain their shape during the time evolution for a large class of central potentials. The packets are not of the Gaussian type with −r 2 dependence but, instead, their shape is determined
Keywords: Zero-energy wave packet Central potential Quantum–classical correspondence
by −r µ+1/2 with −1/2 < µ < 1/2. A very close quantum–classical correspondence is also shown, i.e., the well localized states travel along suitable classical trajectories. © 2013 Elsevier Inc. All rights reserved.
1
1. Introduction The most classical among pure states of quantum mechanics, called after Glauber [1] the coherent states, are used in many areas of physics from solid state physics through quantum optics to modern cosmology. The problem of construction of such well-localized, nonspreading states was first addressed in 1926 by Schrödinger who successfully solved it for the harmonic oscillator [2], however,
∗
Corresponding author. Tel.: +48 56 6113300. E-mail addresses: [email protected] (A.J. Makowski), [email protected] (P. Pepłowski).
0003-4916/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.aop.2013.06.011
26
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
without a similar result for the hydrogen atom. Since then, numerous methods were developed to derive localized states, wave packets, which travel along classical orbits for variety of potentials [3–7]. Three groups of methods are among the most popular ones. In Barut’s approach [3] coherent states are constructed as eigenstates of the lowering operator. They can also be found by acting on a reference state, often chosen as the ground state, with a suitably defined displacement operator. These are Perelomov type coherent states [4]. In the third approach, let us call it Nieto’s method [5], the states are constructed as minimum-uncertainty wave packets. Only for the harmonic oscillator the three methods lead to the same wave packet. Otherwise, the methods are not equivalent. A comprehensive review of other approaches can be found in [6,7]. Experimentally, the packets were first created some 25 years ago [8,9] by photoexcitation using ultrashort pulses. The technique was further developed by Jones et al. [10] by using half-cycle pulses. A procedure to produce well-localized Bohr-like wave packets to mimic an electron moving in a Kepler orbit was suggested in [11] and successfully applied in [12]. Some other interesting experimental papers on Rydberg wave packets in atoms can be found in [13] and those for molecules in [14]. In this paper, we construct spatially well-localized and nonspreading packets using the method described in [15]. In the method, one uses superpositions of states with well-defined angular momentum corresponding to the total energy E = 0. Calculations are performed in the twodimensional (2D) configural space for the potentials in the form Vµ (r ) =
λ 2µ + 1
2
λ2 /2 2µ−1
(r /r0 ) µ+1/2
Q
−
2µ
(r /r0 ) µ+1/2
,
(1)
where the parameters λ, r0 , Q are positively determined and µ ̸= −1/2. It was shown in [16] that these potentials have normalizable E = 0 bound states, and that the corresponding wave functions can be deduced from E ̸= 0 solutions of the 3D harmonic oscillator. This was done with the help of some point transformations. Our aim is to derive wave packets that move precisely along classical trajectories for the potentials in Eq. (1) for some distinguished values of µ. To this end, in Section 2, we will first find and discuss classical trajectories of the problem. Since for the central potentials the angular momentum is a conserved quantity, the orbits are flat, and therefore we can consider only the 2D case. Then, in Section 3, we shall derive square-integrable wave functions for the radial 2D Schrödinger equation with the potential Vµ (r ). In Section 4, the functions are used for the construction of a stationary nonGaussian wave packet. Its motion along classical solutions is discussed in Section 5, and finally, our conclusions are given in Section 6. 2. Classical solutions The E = 0 orbits in the potential in Eq. (1) can be easily found using the polar coordinates r and ϕ . For the angular momentum L and the threshold energy, we now have L = mr 2 ϕ, ˙
(2)
0 = (m/2)(˙r + r ϕ˙ ) + Vµ (r ), 2
2
2
(3)
where the dots over symbols mean time derivatives. From these equations, we get dr dϕ
=
−r 2 −
2mr 4 L2
Vµ (r ).
(4)
Eq. (4) can be integrated if the transformation ρ = r /r0 = Z µ+1/2 is made. In this way, we obtain
µ+1/2
ρµ (ϕ) =
2 B−(
B2
1/2
− 4A)
sin
ϕ−ϕ0 µ+1/2
,
(5)
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
27
where B2 ≥ 4A and the constants A and B read as mλ2 r02
A=
B=
L2 2mQr02
λ 2µ + 1
2
λ 2µ + 1
2
L2
,
(6)
.
(7)
√
They are obviously dimensionless since λ2 and Q in the potential Vµ (r ) are measured in units of J, where J = kg m2 /s2 stands for the joule. The general E = 0 solution for ρµ (ϕ), with B2 ̸= 4A and µ ̸= −1/2, admits a variety of orbits of either periodic ones, when µ is a rational number, or of nonrepeating orbits, if µ is an irrational number. For the special µ = 0 case, the solution represents trajectories which are either circles or ellipses for the 2D radial harmonic oscillator for which the parameter Q λ2 in Vµ (r ) plays now 2 the role of the total √ energy E ̸= 0 for the oscillator. When µ ̸= −1/2 and B = 4A, i.e. when Q = L(2µ+ 1)/(r0 m), we get a family of circles ρµ = (2/B)µ+1/2 , all corresponding to the minimum of effective potential Veff (ρ) = L2 /(2mr02 ρ 2 ) + Vµ (ρ). In Fig. 1 we show some of the variety of trajectories for the considered class of potentials. The classically inaccessible regions, shaded in the figure, are determined by the relations: ρ ≤ ρ1 and ρ ≥ ρ2 , where ρi (i = 1, 2) are classical turning points in Veff (ρ) = L2 /(2mr02 ρ 2 ) + Vµ (ρ) and are found to be ρ1,2 = [(B ∓ harmonic oscillator.
√
B2 − 4A)/2A]µ+1/2 . In the middle of the figure there is an ellipse for the
3. Zero-energy wave functions We consider now the radial part of the two-dimensional (2D) Schrödinger equation for the potentials in Eq. (1):
d2
1 d
l2
2mr02
λ 2µ + 1
2
λ2 /2
Q
− R(ρ) = 0, (8) 2µ−1 2µ ρ µ+1/2 ρ µ+1/2 √ where, as previously, ρ = r /r0 = (x/r0 )2 + (y/r0 )2 = X 2 + Y 2 , Q > 0, λ is a constant parameter, dρ 2
+
ρ dρ
−
ρ2
−
h¯ 2
and m and l stand respectively for the mass of the particle and angular momentum quantum numbers. Eq. (8) possesses square-integrable solutions provided that the coupling constant Q is chosen in a proper way. To see that, we change the independent variable ρ in the following way 2µ+1
λ
ρ=
h¯ 2
µ+21/2
mr02
uµ+1/2 .
(9)
Then, Eq. (8) simplifies to the following one
d2 du2
+
1 d u du
−
l2 (µ + 1/2)2 u2
1
+ Q
mr02 1 h¯ 2 u
2
1
− R(u) = 0.
(10)
4
Its solution is proposed to be R(u) = e−u/2 ul(µ+1/2) F (u),
µ > −1/2,
(11)
and then, we can obtain
u
d2 F du2
+ [2l(µ + 1/2) + 1 − u]
dF du
1
+ Q 2
mr02 h¯ 2
−
1 2
− l(µ + 1/2) F = 0.
(12)
28
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
Fig. 1. (Color online) A gallery of closed trajectories obtained from Eq. (5). In the figures, Y = y/r0 is plotted versus X = x/r0 , the origin of coordinates is in the middle of each square, and classically inaccessible regions are shaded.
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
29
γ
This is an equation for the generalized Laguerre polynomials Ln (u) = F (u) if Q is ‘quantized’ according to the rule
Q =
h¯ 2 mr02
[2n + 1 + 2l(µ + 1/2)].
(13)
In this way, the normalizable solutions of Eq. (8), for µ > −1/2, have the final form
(µ)
1
(µ)
Rnl (ρ) = Nnl exp − λ 2
2
mr02 h¯ 2
ρ
1 µ+1/2
ρ
l 2l(µ+1/2) Ln
λ
2
mr02 h¯ 2
ρ
1 µ+1/2
,
(14)
(µ)
where the symbol Nnl stands for the normalization constant. Its explicit form is given in the Appendix and the quantum numbers n and l take the values of zero and positive integers, i.e., n ≥ 0 and l ≥ 0. 4. Stationary wave packets We now define a spatially well-localized coherent state, being a combination of the functions (µ)
(µ)
Φnl (ρ, ϕ) = Nϕ exp(ilϕ)Rnl (ρ),
(15)
in the form (µ)
Ψαβ (ρ, ϕ) = N (α, β)
∞ ∞
(µ)
Nϕ Nnl
l =0 n =0
−1 α n β l (µ) Φnl (ρ, ϕ). l!
(16)
The parameters α and β are taken to be real without loss of generality. In the above formulas (µ) N (α, β), Nϕ and Nnl denote the normalization constants of corresponding states. With the help of Eqs. (14) and (15), and of the generating function formula for Laguerre polynomials [17]
∞
1 (µ+1/2) 2 α n L2l (λ mr02 / h¯ 2 ρ µ+1/2 ) = n
n=0
exp
αλ2 α−1
1
mr02 / h¯ 2 ρ µ+1/2
(1 − α)2l(µ+1/2)+1
,
|α| < 1,
(17)
we obtain (µ)
Ψαβ (ρ, ϕ) =
N (α, β) 1−α
exp
1 −λ2 (1 + α) β eiϕ ρ . mr02 / h¯ 2 ρ µ+1/2 + 2(1 − α) (1 − α)2µ+1
(18)
Thus, for the probability density we can write finally (µ) |Ψαβ (ρ, ϕ)|2
N 2 (α, β)
1 −λ2 (1 + α) 2β cos(ϕ) 2 µ+1/2 2 mr0 / h¯ ρ + = exp ρ . (1 − α)2 1−α (1 − α)2µ+1
(19)
This packet is of the Gaussian type only for µ = 0, and then, it reduces to the known functional form as, e.g., that considered in [18]. Nevertheless, it is very well localized when −1/2 < µ < 1/2. Its image is given in Fig. 2 and is only slightly sensitive to the chosen value of µ. In the next section we shall show how to set the packet in motion along some of the orbits presented in Fig. 1. 5. Setting packets in motion The question how to set in motion the stationary packet in Eq. (19) has no simple answer. One should take into consideration that the packet is built of the Hamiltonian eigenstates all corresponding to the total energy E = 0, i.e. H Φ (ρ, ϕ) = 0. If H is a Hermitian operator then, according to the rules of quantum mechanics, the time-evolution of a state for time-independent problems is governed by the unitary operator exp(−iHt /h¯ ). In our case, no immediate application of the rule is possible. However, we can utilize the relationship of Eq. (8), for E = 0 with the potential in Eq. (1), to the 2D harmonic
30
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
Fig. 2. (Color online) The normalized to unity wave packet from Eq. (19) plotted for α = 0.625, β = 1.525,
mr02 / h¯ 2 =
1, λ = 1, µ = −1/6, and N (α, β) = 0.14. The coordinates X and Y are dimensionless as explained in the text.
oscillator with E ̸= 0. To this end, let us make in Eq. (8) the point transformation ρ = v 2µ+1 . Then Eq. (8) can be written in the form
d2 dv 2
+
1 d
v dv
−
−2mr02 2 Λ2 mr02 4 2 λ v R (v) = λ QR(v). − v2 h¯ 2 h¯ 2
(20)
We have thus obtained the radial Schrödinger equation for the harmonic oscillator with total energy Enl = λ2 Qnl = h¯ ω[2n + 1 + Λ] and scaled angular momentum Λ, where ω
(21)
=
λ4 /(mr02 ) and Λ = (2µ + 1)l. In this
connection, we expect that the oscillator’s Hamiltonian can be considered as a proper generator for (µ) the time–displacement operator also for our packet |Ψαβ (ρ, ϕ)|2 . Formally, its time-dependence, i.e. (µ)
|Ψαβ (ρ, ϕ, τ )|2 , can be easily found if in the summands of Eq. (16), the parameter α is replaced by α exp(−2iτ ) and β by β exp[−i(2µ + 1)τ ], where λ2 h¯ 2 τ= t (22) h¯ mr02 is a dimensionless scaled time. Then, performing calculations as in Section 4, we will get N (α, β, τ ) 2
(µ)
|Ψαβ (ρ, ϕ, τ )|2 =
1 + α 2 − 2α cos(2τ )
+
exp
−λ2 (1 − α 2 ) mr02 / h¯ 2 1 + α 2 − 2α cos(2τ )
2β cos[C (ϕ, µ, τ )]
ρ , [1 + α 2 − 2α cos(2τ )]µ+1/2
1
ρ µ+1/2
(23)
where C (ϕ, µ, τ ) = ϕ − (2µ + 1)τ − (2µ + 1) tan−1
α sin(2τ ) . 1 − α cos(2τ )
(24)
As it should be, the packet reduces to the stationary one when τ = 0. That the time-dependence of our packet is correctly defined can be observed in Fig. 3. We have plotted the cut of packet for some fixed instants of time. It moves precisely along classical orbits as expected. Note also that the packet still ‘looks’ at the same point at every instant of time with no rotation at all. The same excellent quantum–classical correspondence can be shown for other values of µ > −1/2 including, of course, the µ = 0 case for the harmonic oscillator. We do not illustrate this case since it was considered in a number of papers [18–20] though in a different context. One can also choose µ to be an irrational number and again the correspondence is perfect.
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
a
b
c
d
31
Fig. 3. (Color online) Motion of the wave packet from Eq. (23) for some chosen values of µ. The contour of constant probability density is plotted for some fixed instants of time. Its initial position, for τ = 0, is marked by the shaded area. Dots represent the positions of the packet’s maxima and the full line is the classical orbit obtained from Eq. (5). In each case the cut of packet (µ)
corresponds to max(0.99|Ψαβ (ρ, ϕ)|2 ), λ2 = 1 and
mr02 / h¯ 2 = 1. The four figures are prepared using the following sets of
parameters: (a) α = 0.71, β = 0.99; (b) α = 0.66, β = 0.93; (c) α = 0.525, β = 0.665; (d) α = 0.5, β = 0.595. The coordinates X and Y are dimensionless.
A few comments on normalization of the packet are in order now. When µ = 0, the functions in Eq. (14) are all orthogonal each other and the packet’s norm in Eq. (23) is preserved in time. When µ ̸= 0 the problem is much more subtle. Namely, one should remember that the functions (µ) Rnl (ρ) are all eigenstates of the Hamiltonian to the total energy of E = 0. They are not orthogonal (µ)
each other and, during the packet’s |Ψαβ (ρ, ϕ, τ )|2 evolution its norm is not preserved. The states (µ) Rnl (ρ) would become orthogonal if were integrated with the additional weight function w(ρ) = −2µ . Preparing images in Fig. 3, the factor N (α, β, τ ) in Eq. (23) is chosen such that the integral ρ (µ) |Ψαβ (ρ, ϕ, τ )|2 ρ dρ dϕ = 1 be obeyed. We can also consider another way of reasoning. Treat our 2D, E = 0, Schrödinger equation for the potential Vµ (r ), as the following eigenvalue problem (µ)
(µ)
ˆ Φnl (ρ, ϕ) = Qnl Φnl (ρ, ϕ), O
(25)
32
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
where
− h¯ 2
ˆ ≡ O
2µ + 1
2
λ
2mr02
2µ
ρ µ+1/2
∂2 1 ∂ 1 ∂2 + + ∂ρ 2 ρ ∂ρ ρ 2 ∂ϕ 2
+
λ2 2
1
ρ µ+1/2 ,
(26)
(µ)
the functions Φnl (ρ, ϕ) are defined in Eq. (15), and the eigenvalues Qnl in Eqs. (13) or (21). Then, the (µ)
functions χnl (ρ, ϕ, τ ) = exp(−iQnl τ )Φnl (ρ, ϕ) obey the generalized equation
ˆ −i O
∂ ∂τ
χnl (ρ, ϕ, τ ) = 0.
(27)
ˆ becomes Hermitian if the orthonormality In the space of functions χnl (ρ, ϕ, τ ), the operator O condition
ρ −2µ χn∗′ l′ χnl ρ dρ dϕ = δn′ n δl′ l
(28)
is imposed. This is a Sturm–Liouville approach to the problem that supports the above made choice for the packet’s time-dependence. 6. Conclusion In this paper, we have been concerned with the construction and dynamics of two-dimensional non-Gaussian wave packets for a large class of central potentials. The method we have used guarantees that the maximum of the packets follows the corresponding classical orbits with very high accuracy. The point that needs some further study is the problem of how to set in motion packets constructed from Hamiltonian zero-energy eigenstates. The close relation of the used here potentials to the harmonic oscillator, and a possibility of using a Sturm–Liouville approach, were a fruitful advice in that direction and the results presented in Fig. 3 confirm the ideas. Appendix The wave function in Eq. (14) can be normalized in the standard way, i.e. we require
(µ)
1 = Nnl
∞
2
(µ)
[Rnl (ρ)]2 ρ dρ.
(A.1)
0
Changing the variable of integration
Z = λ2
mr02 h¯ 2
1
ρ µ+1/2 ,
(A.2)
we will get
(µ)
1 = Nnl
2
µ + 1/2
λ(2µ+1)(2l+2)
h¯ 2 mr02
(1/2+µ)(l+1)
∞
(µ+1/2) e−Z Z 2µ(l+1)+l L2l (Z ) n
2
dZ .
(A.3)
0
This integral issimple for the 2D harmonic oscillator, µ = 0, and it is equal to (n + l)!/n!. Then, assuming λ = mw 2 /r02 , the normalization constant has the well-known form
(0)
Nnl =
2(mw/h¯ )l+1 n!
(n + l)!
.
(A.4)
In the general case, for µ > −1/2, we can utilize the formula 2.19.14.(15) in Ref. [21]. Now, the integral in Eq. (A.3) is equal to I = S (µ, l, n)3 F2 (−n, 2µ(l + 1) + l + 1, 2µ + 1; 2l(µ + 1/2) + 1, 2µ + 1 − n; 1),
(A.5)
A.J. Makowski, P. Pepłowski / Annals of Physics 337 (2013) 25–33
33
where 3 F2 means the generalized hypergeometric function and S (µ, l, n) =
(1 + 2l(µ + 1/2))n (−2µ)n Γ (2µ(l + 1) + l + 1) , [n!]2
(A.6)
whereas (a)k = Γ (a + k)/Γ (a). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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