PRAMANA — journal of
c Indian Academy of Sciences °
physics
Vol. 73, No. 5 November 2009 pp. 821–828
Do the generalized Fock-state wave functions have some relations with classical initial condition? JEONG RYEOL CHOI1 , KYU HWANG YEON2,∗ , IN HYUN NAHM3 and SEOK SEONG KIM2 1
School of Electrical Engineering and Computer Science, Kyungpook National University, 1370 Sankyuk-dong, Buk-gu, Daegu 702-701, Republic of Korea 2 BK21 Physics Program and Department of Physics, College of Natural Science, Chungbuk National University, Cheongju, Chungbuk 361-763, Republic of Korea 3 Department of Information Display, Sunmoon University, Asan 336-708, Republic of Korea ∗ Corresponding author. E-mail:
[email protected] MS received 10 February 2009; revised 3 June 2009; accepted 22 June 2009 Abstract. We examine whether the general type Fock-state wave functions for the harmonic oscillator have some relations with the classical initial condition. Keywords. Wave functions; harmonic oscillator; classical initial condition. PACS Nos 03.65.-w; 03.65.Ge; 03.65.Ca
1. Introduction It is known that the generalized quantum solutions of mechanical systems can be represented in terms of classical solutions (see, for instance, ref. [1]). Up to now, research in connection with this problem is mainly performed for systems that are described by time-dependent Hamiltonian. For the (time-dependent) harmonic oscillator, there usually need two independent homogeneous classical solutions in order to express general wave functions. Some researchers [2–4] took c-number solutions as such classical solutions and the others [5–13] took real solutions. For the case that real solutions are chosen, several researchers [5–9] have represented the wave functions in terms of classical solutions constructed by considering classical initial condition (CIC) while other relevant researchers [10–13] have not considered it. In this paper, we are interested in physically acceptable solutions considered CIC rather than the mathematically allowed one. In particular, some researchers [5–7] investigated de Broglie–Bohm quantum theory with the consideration of CIC by taking advantage of the Feynman path integral method. Indeed, the classical limit in the quantum mechanics has been a matter of rigorous debate ever since
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Jeong Ryeol Choi et al the theory of quantum mechanics is introduced due to the comparative differences between the classical and quantum descriptions of physical systems [14]. In this paper, we shall investigate whether the generalized Fock-state wave functions have some relations with CIC for mechanical systems. For this purpose, their corresponding quantum behaviour in the coherent state and quantum fluctuation of canonical variables in the Fock state will be examined. To attain this objective, it may not be necessary to consider the complicated dynamical system described by time-dependent Hamiltonian even if many previous researchers have dealt with time-dependent Hamiltonian system. Rather, it seems satisfactory to consider a simple harmonic oscillator. From now on, let us call it just harmonic oscillator for convenience. Harmonic oscillator is one of the most familiar model in physics. The motion of charge in LC circuit, the analysis of electromagnetic field, and many other elementary physical situations can be described by harmonic oscillator. Whenever we investigate the behaviour of a physical system in the vicinity of a stable equilibrium position, such as swinging pendulum, the potential energy can usually be quite well approximated to that of simple harmonic motion in the limit of small oscillations. Hence, even if most of the real restoring forces in nature are more complicated than the simple Hooke’s law force, the study of the essential properties associated with harmonic oscillator is still valuable. The importance of the harmonic oscillator as a stepping-stone for the development of theoretical physics probably became obvious with the appearance of quantum mechanics [15]. This paper is organized as follows. In §2, we construct generalized Fock-state wave functions with the consideration of CIC for the harmonic oscillator. To investigate possible relationship between these wave functions and CIC, the quantum behaviour associated with these wave functions will be checked in §§3 and 4 by examining the motion of the oscillator in the coherent state and by analysing the fluctuations of the canonical variables in the Fock state, respectively. The summary and concluding remarks are given in §5. 2. Wave functions considered CIC Since it is enough to consider simple mechanical system in order to achieve our purpose, as mentioned in the previous section, the system we employ is harmonic oscillator. Let us first start from the exact analytical classical solution of the harmonic oscillator with frequency ω, that is given by X(t) = x0 cos ωt +
v0 sin ωt, ω
(1)
where x0 and v0 are the position and velocity at t = 0, namely, the two components of CIC. Thus, this familiar expression is considered CIC. The two terms of the above equation can be regarded as two independent homogeneous classical solutions. Now we introduce an annihilation operator considered CIC in the form à ! r r f˙(t) mω f (t) 1−i x ˆ+i pˆ, (2) a ˆ= 2~f (t) 2ω 2mω~ 822
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Generalized Fock-state wave functions where m is the mass of the oscillator and f (t) is a time function given by µ ¶ ω v2 x20 cos2 ωt + 02 sin2 ωt . f (t) = x0 v0 ω
(3)
Equation (2) is unique and its Hermitian adjoint is of course the creation operator, a ˆ† . Particularly, for v0 = ωx0 , a ˆ and a ˆ† reduce to the well-known conventional formulae. Since the time derivative of eq. (2) results in dˆ a/dt = −(iω/f )ˆ a, the time evolution of a ˆ becomes a ˆ(t) = a ˆ(0)e−iω
Rt 0
f −1 (t0 )dt0
.
(4)
This means that the absolute value of a ˆ does not vary with time. From straightforward calculation, we readily see that a ˆ and a ˆ† satisfy the boson commutation relation: [ˆ a, a ˆ† ] = 1. The wave functions therefore can be evaluated by applying the usual method with the aid of a ˆ and a ˆ† . From the relation a ˆψ0 = 0, the ground state wave function can be generated. Next, by operating a ˆ† to the ground state wave function n times, we get the nth order wave function. Hence, we eventually have µ ¶1/4 µr ¶ mω mω 1 √ ψn (x, t) = Hn x ~πf (t) ~f (t) 2n n! " à ! # R t −1 0 0 m f˙(t) ω−i x2 e−iω(n+1/2) 0 f (t )dt . (5) ×exp − 2~f (t) 2 These are the full wave functions which are considered CIC for the harmonic oscillator. We can easily check that eq. (5) satisfies the Schr¨odinger equation, from direct calculation after inserting it into the Schr¨odinger equation (see Appendix A). The wave function, eq. (5), is normalized and forms a complete set of orthogonal basis for the harmonic oscillator. If we take v0 = ωx0 as an initial condition, the results of these quantization scheme entirely recovers to those of the usual quantization and, consequently, eq. (5) becomes the familiar formula r µr ¶ h mω i mω 1 mω 4 √ ψn (x, t) = Hn x exp − x2 ~π 2n n! ~ 2~ ×e−iωt(n+1/2) . (6) As you can see, eq. (5) depends on time while eq. (6) does not. In the next two sections, we will investigate whether eq. (5) is physically acceptable and whether the generalized type of wave functions really have some relations with CIC as represented in eq. (5). 3. Coherent state analysis Now we examine the quantum behaviour of the harmonic oscillator in the relevant coherent state. The coherent state |αi is the eigenstate of annihilation operator: Pramana – J. Phys., Vol. 73, No. 5, November 2009
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Jeong Ryeol Choi et al a ˆ|αi = α|αi.
(7)
By representing eq. (2) and its conjugate a ˆ† together inversely, we have the expression of canonical variables x ˆ and pˆ such that r ~f (t) (ˆ a+a ˆ† ), (8) x ˆ= 2mω s "à ! à ! # ~mω f˙(t) f˙(t) pˆ = −i 1+i a ˆ− 1−i a ˆ† . (9) 2f (t) 2ω 2ω Then, if we regard eq. (7), the expectation values of these canonical variables can be easily identified as r ~f (t) hα|ˆ x|αi = (α + α∗ ), (10) 2mω s "à ! à ! # f˙(t) f˙(t) ~mω hα|ˆ p|αi = −i 1+i α− 1−i α∗ . (11) 2f (t) 2ω 2ω From the substitution of eq. (2) in eq. (7), we get the eigenvalue α in the form à ! r r mω f (t) f˙(t) ˙ α= 1−i X(t) + i mX(t). (12) 2~f (t) 2ω 2mω~ The execution of some algebra after the further substitution of eq. (1) in the above equation leads to à " !# r X(t) mx0 v0 −1 p α= exp i cos . (13) ~ 2x0 v0 f (t)/ω This is a complex form due to the non-Hermitian property of a ˆ. With the help of the above equation, we can readily see that eqs (10) and (11) become v0 hα|ˆ x|αi = x0 cos ωt + sin ωt, (14) ω µ ¶ dhα|ˆ x|αi hα|ˆ p|αi = −m(ωx0 sin ωt − v0 cos ωt) = m . (15) dt These are exactly the same as the classical coordinate and momentum. Thus we showed that the behaviour of the corresponding coherent state is in accordance with the classical state. If we recall the fact that the coherent states are very close to classical systems as far as quantum mechanics permits, as has been suggested by Glauber [16], the Fock-state wave functions, eq. (5), are acceptable as physical solutions of the harmonic oscillator. However, at this stage, it is still unclear whether the Fock-state wave functions really have some physical relation with CIC or not, since the use of other type wave functions like eq. (6) also yields the same coherent state results (eqs (14) and (15)) [16]. We thus may need further investigation for the quantum behaviour associated with eq. (5) from different points of view. Hence, we will investigate the fluctuations of the canonical variables in Fock state in connection with this problems in the next section. 824
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Generalized Fock-state wave functions 4. Analysis of fluctuations in the Fock state Let us examine the wave functions by investigating the fluctuations of the canonical variables in the Fock state, whose definitions are given by ∆x = [hψn |ˆ x2 |ψn i − (hψn |ˆ x|ψn i)2 ]1/2 ,
(16)
∆p = [hψn |ˆ p2 |ψn i − (hψn |ˆ p|ψn i)2 ]1/2 .
(17)
By making use of eq. (5), these becomes ·
µ ¶ ¸1/2 v2 x20 cos2 ωt + 02 sin2 ωt (2n + 1) , ω " µ ¶−1 m~x0 v0 v2 x20 cos2 ωt + 02 sin2 ωt ∆p = 2 ω #1/2 ¶ µ (v02 − ω 2 x20 )2 2 sin (2ωt) (2n + 1) . × 1+ 4ω 2 x20 v02
∆x =
~ 2mx0 v0
(18)
(19)
Using the conservation law of mechanical energy, it is elementary to show that the classical initial velocity v0 can be determined from x0 : q v0 = ±ω x2m − x20 , (20) where xm is the amplitude of oscillation. By considering this relation, we have plotted the results, eqs (18) and (19), and associated uncertainty products in figure 1. From figures 1a and 1b, we see that the fluctuations depend sensitively on the value of x0 (and, consequently, v0 ). This outcome seems somewhat unreasonable and may not make any physical sense, since it is expected that the fluctuations should not be affected by CIC as long as mechanical energy is conserved. Thus, the supposition imposed tacitly at the beginning of §2, which is that the general type Fock state wave functions should be represented in terms of CIC, may not be acceptable physically. We can therefore consider that the generalized Fock-state wave functions have nothing to do with CIC. 5. Summary and discussion Generalized Fock-state wave functions have been formulated in eq. (5) for the harmonic oscillator under the assumption that they have some relations with CIC. Since these wave functions satisfy the Schr¨odinger equation, they may belong to a class of agreeable quantum solutions. Nevertheless, there would be a possibility that this fact does not fully guarantee that the generalized quantum description of harmonic oscillator has something to do with CIC. It is therefore unclear whether the generalized Fock-state wave functions which can be accepted physically are really related to the CIC even if eq. (5) is expressed in terms of CIC. To investigate Pramana – J. Phys., Vol. 73, No. 5, November 2009
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Figure 1. Fluctuations (a) ∆x and (b) ∆p and their product (c) ∆x∆p with x0 = 1, 2, 3. We used n = 0, xm = 4, ω = 1, ~ = 1 and m = 1.
the relationship between the generalized Fock state wave functions and CIC, we have analyzed the quantum behaviour of the system in the corresponding coherent state and have examined fluctuations of the canonical variables in the Fock state. From the coherent state analysis, we showed that the trajectory of position and momentum in the corresponding coherent state exactly follows the classical trajectory. This fact enables us to confirm that the Fock-state wave functions, eq. (5), are acceptable as physical solutions of the harmonic oscillator. Even so, this fact does not guarantee that the generalized Fock-state wave functions have some relations with CIC, since the coherent state associated to another type of Fock-state wave functions such as eq. (6) equally well follow the classical state. Thus, further check about the quantum behaviour associated to eq. (5) is yet to be done. As a second check, we investigated fluctuations of the canonical variables in the Fock state in order to fix the raised question. From figure 1, we see that the fluctuations of x ˆ and pˆ change significantly depending on the initial conditions x0 and v0 even if all other conditions such as mechanical energy, mass and frequency are the same. This result seems physically unreasonable, since CIC is merely related to the choice of initial time only. Thus, the generalized Fock-state wave functions may have nothing to do with CIC. We therefore should represent eq. (5) in terms of a reformed expression of f (t) such that it cannot contain x0 and v0 . By taking this point into consideration, it may be reasonable to alter f (t) of eq. (3) such that f (t) →
¢ 1 ¡ 2 c1 cos2 ωt + c22 sin2 ωt , c1 c2
(21)
where c1 and c2 are arbitrary real constants which do not have any relevance with 826
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Generalized Fock-state wave functions CIC. Then, under the free choice of c1 and c2 , the Fock-state wave functions can be represented much more flexibly than eq. (6). In conclusion, we can think that the classical solutions used in the description of quantum wave functions are merely auxiliary functions which do not need any consideration of CIC. It is however still obscure why there are numerous quantum results for a system, which corresponds to different choices of the values c1 and/or c2 . Indeed, in future, it may be worthwhile to investigate the origin of these various quantum results and see whether they could be actually realized in laboratory. As shown in Appendix A, all these results completely satisfy the Schr¨odinger equation. We may not be able to exclude the possibility that these quantum mechanical solutions except for a standard one given in eq. (6) are physically meaningless even if they are mathematically possible solutions. Although the quantum mechanics has made startling progress ever since its inception by Planck, quantum mechanics may yet need radical self-examination and verification in connection with these theoretical consequences. By the way, x0 and v0 appeared in eqs (14) and (15) which are expectation values in the coherent state, cannot be replaced by CIC-independent ones. The coherent state thus has something to do with CIC, whereas the Fock-state does not. Appendix A In this Appendix, we will show that the wave function, eq. (5), satisfies the Schr¨odinger equation. The necessary differentiations of eq. (5) can be evaluated as (" µ ¶1/4 ∂ψn mω 1 √ =− f (t)f˙(t) + 2iωf (t)(2n + 1) ∂t π~f (t) 4f 2 (t) 2n n! # µr ¶ mω 2 −[(2ω − if˙(t))f˙(t) + if (t)f¨(t)]mx /~ Hn x ~f (t) ) ¶ µ r r mω mω +4nf (t)f˙(t) xHn−1 x ~f (t) ~f (t) " Ã ! # R t −1 0 0 m f˙(t) × exp − ω−i x2 e−iω(n+1/2) 0 f (t )dt , (A1) 2~f (t) 2 (" µ ¶1/4 ∂ 2 ψn mω m √ = 2if (t)f˙(t) − mx2 [f˙(t) ∂x2 π~f (t) 4f 2 (t)~ 2n n! # µr ¶ mω 2 +2iω] /~ − 4f (t)ω(2n + 1) Hn x ~f (t) µr ¶) r mω mω +8inf (t)f˙(t) xHn−1 x ~f (t) ~f (t)
Pramana – J. Phys., Vol. 73, No. 5, November 2009
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Jeong Ryeol Choi et al "
m × exp − 2~f (t)
Ã
f˙(t) ω−i 2
!
# x
2
e−iω(n+1/2)
Rt 0
f −1 (t0 )dt0
In the above calculations we considered dHn (x) = 2nHn−1 (x), dx Hn+1 (x) − 2xHn (x) + 2nHn−1 (x) = 0. By using eqs (A1) and (A2) with the help of the relation µ ¶ f˙2 (t) 1 2 ¨ f (t) = − 2ω f (t) − , 2f (t) f (t)
.
(A2)
(A3) (A4)
(A5)
which is identifiable from eq. (3), we can directly show that ψn satisfies the following Schr¨odinger equation: µ ¶ ∂ψn (x, t) ~2 ∂ 2 1 2 2 i~ = − + mω x ψn (x, t). (A6) ∂t 2m ∂x2 2
Acknowledgements This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (No. 2009-0077951). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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