On the Canonical Transformation of Time-Dependent Harmonic Oscillator

Abstract

We performed a two-variable canonical transformation on the time momentum operator, and without loss of generality we carried out a three-variable transformation on the coordinate and momentum space operators to trivialize the Hamiltonian operator of the system. Fortunately, this operation separates the time-coordinate and space coordinate naturally, and the wave function of the time-dependent Harmonic Oscillator is evaluated via the generator.

1. Introduction

The method of Canonical transformations (CT) has proved to be a fruitful approach in treating quantum systems [1]. It is often used in describing systems with Hamiltonian that are quadratic either in co-ordinate and momentum or equivalently in Boson creation and annihilation operators. One major advantage of the method of (CT) consists in reducing the Hamiltonian 𝐻 of the system 𝑆 to a Hamiltonian 𝐻^′ of some simple system 𝑆^′ with known solution, that is, 𝐻=𝐶𝐻^′𝐶⁻¹. These two systems are canonically equivalent since their Hamiltonian is related by means of a CT. However, analytical approaches to isolate conserved quantities for a given physical system are a major objective in the realm of Hamiltonian theory [2]. For autonomous, where the Hamiltonian is time independent, one conserved quantity is immediately found: the Hamiltonian which represents the total energy of the system is a constant of motion.

Rather unfortunately the Hamiltonian of most real physical systems is explicitly time-dependent and does not provide direct conserved quantity. In classical mechanics, the equation of motion governed the dynamical behavior of the system and this second-order differential equation is solved directly by the trivialization of the conjugate variables through CT [3]. Park [3] asserted that because of the non-commutability of operators, CT has not been fully realized in quantum mechanics. However, CT is usually defined as a change of the noncommutating variables that preserved the commutation relations in quantum mechanics [4].

In literature [4, 5] three types of transformations have been identified such as interchange, similarity, and point transformations. Different models of oscillators have been evaluated via the CT [6, 7]. The applications of quantum canonical transformation to a harmonic oscillator in which the angular frequency and equilibrium position are time dependent have been reviewed [3]. In this paper we extend the work of [3] by performing a three-variable transformation on both the space co-ordinate and momentum co-ordinate operators.

2. Time-Dependent Harmonic Oscillator

The Time-Dependent Schrodinger Wave equations have been solved explicitly by the method of Quantum Canonical Transformation (QCT). The QCT is also used in solving time-dependent frequency [3]. However, the solutions of time-dependent harmonic oscillator have been obtained through various methods including invariant operator [8], Path Integral [9, 10], and the space-time transformation [11, 12]. We write the Schrödinger equation to be solved as 𝐻₁Ψ₁(𝑞,𝑡)=0, (1) where

𝐻₁=𝑃_𝑡+𝑃²+Ω²(𝑡)(𝑞−𝑥₀(𝑡)), (2) with 𝑃_𝑡=−𝒾𝜕/𝜕𝑡, and 𝑃=−𝒾𝜕/𝜕𝑞, as the momentum operators in time and in the co-ordinate. It is worthy to note that the angular frequency Ω(𝑡) and the equilibrium position 𝑥₀(𝑡) are all functions of time.

We make time-variable similarity transformation on the momentum time-coordinate and three-variable transformations on space momentum operators as 𝑃_𝑡→𝑃_𝑡−𝒾2(𝜕𝑓𝜕𝑡)𝑞²𝒾,𝑃→𝑃−𝒾𝑓𝑞−2(𝜕𝑓𝜕𝑞). (3) And the Hamiltonian Operator becomes 𝐻₂=𝑃_𝑡−𝒾2(𝜕𝑓𝜕𝑡)𝑞²𝒾+(𝑃−𝒾𝑓𝑞−2(𝜕𝑓𝜕𝑞))²+Ω²(𝑡)(𝑞−𝑥₀(𝑡))²=𝑃_𝑡+𝑃²+(Ω²𝒾(𝑡)−2(𝜕𝑓𝜕𝑡)−𝑓²)𝑞²−(𝒾𝑃𝜕𝑓𝜕𝑞+𝑓𝑞𝜕𝑓+1𝜕𝑞4(𝜕𝑓)𝜕𝑞²)−2Ω²(𝑡)𝑥₀(𝑡)𝑞+Ω²(𝑡)𝑥₀(𝑡)−𝑓. (4) The condition for the differential in (4) to be cancelled is 4𝜕²𝑓𝜕𝑞²+4𝑓𝑞(𝜕𝑓𝜕𝑞)+(𝜕𝑓)𝜕𝑞²=0. (5) And we solve (5) by series method as

𝑓(𝑞)=_𝑘𝐶_𝑘𝑞^𝑘+𝛽, (6) where 𝐶_𝑘 is the expansion coefficient and 𝐾 and 𝛽 are arbitrary constant.

Substituting (6) into (5) yields 𝑓(𝑞)=𝑞²+2𝒾√2𝑞+8. (7) Imposing boundary condition on (7), we obtain 𝑓(0)=8 and (7) becomes

𝑓(𝑞)=𝑞²+2𝒾√2𝑞+𝑓(0). (8) And we recaste (8) in the form 𝑓(𝑞)=𝑞²+2𝒾√2𝑞, (9) where 𝑓(0) has been absorbed into 𝑓(𝑞). Now multiplying the first term of the LHS of (9) by (−1/2)𝑚𝜔² and replacing the magnitude 2𝒾√2 by 𝑓(𝑞) and replacing 𝑓→𝐿−(1/2)𝑚𝑞̇², (9) becomes

1𝐿=2𝑚𝑞̇²−12𝑚𝜔²𝑞²+𝐹(𝑞)𝑞, (10) and (10) is the Lagrangian of the system whose classical equation is 𝑚𝑞̈+𝑚𝜔²𝑞−𝜕𝑉𝜕𝑞=0. (11) The result of (10) shows that the cancellation of the differential in (4) generates the Lagrangian describing the harmonic oscillator of the system. Another condition of (4) is the cancellation of quadratic term in co-ordinate and these yields 𝒾𝜕𝑓𝜕𝑡+2𝑓²=2Ω²(𝑡), (12) however (12) is a first-order nonlinear equation and can be linearized by setting 𝒾𝑓=2𝜕𝜕𝑡ln𝜙(𝑡), (13) and (12) becomes

𝜕²𝜕𝑡²𝜙(𝑡)+4Ω²(𝑡)𝜙(𝑡)=0. (14) Solving (14) gives 𝜙(𝑡)=𝜙(0)sin(𝜔𝑡+𝛿), (15) where 𝛿 is the phase factor. Putting (15) into (13) leads to

𝑓(𝑡)=𝒾𝜔2tan(𝜔𝑡+𝛿). (16) With (16) and (9), we can determine the generator𝑒^{−𝑓(𝑞)}𝑒^{−𝑓(𝑡)(𝑞2/2)}.

The second Canonical transformation is carried out this time to remove the quadratic term in momentum and the associated differential in co-ordinate:

𝑃_𝑡→𝑃_𝑡−𝒾2(𝜕𝑔𝜕𝑡)𝑃²𝒾,𝑞→𝑞−𝒾𝑔𝑃+2(𝜕𝑔𝜕𝑞). (17) Substituting (17) into (4) yields 𝐻₃=𝑃_𝑡+(2+4𝑔𝑓−𝒾𝜕𝑔)𝑃𝜕𝑡2²𝜕−(𝒾𝑓²𝑔𝜕𝑞²+𝒾Ω²(𝑡)𝜕𝑔)𝜕𝑞−2𝒾𝑓𝑔𝑝−2𝒾Ω²𝑥₀(𝑡)𝑔𝑃−2Ω²𝑥₀(𝑡)𝑃+Ω²𝑥₀²−𝑓. (18) We eliminate the quadratic term in momentum by setting

2+4𝑔𝑓−𝒾𝜕𝑔𝜕𝑡=0, (19) and solving (19), we obtain 𝑔(𝑡)=−2𝒾𝑒^{−4𝒾∫𝑓(𝑡)𝑑𝑡}𝑒^{4𝒾∫𝑓(𝑡)𝑑𝑡}+𝐾𝑒^{−4𝒾∫𝑓(𝑡)𝑑𝑡}𝑑𝑡. (20) Similarly, the differential in co-ordinate is eliminated as

𝜕²𝑔𝜕𝑞²+Ω²(𝑡)𝑓𝜕𝑔𝜕𝑞=0, (21) with 𝑓 being known, and we determined (21) as

𝑔(𝑡)=𝐷+𝐹𝑒^{−𝛼2(𝑞)𝜔2𝑞}, (22) where 𝐷 and 𝐹 are arbitrary constants and 𝛼(𝑞)=1/𝑓(𝑞).

The next transformation is carried out to eliminate the linear term in position 𝑞 and the associated differential in co-ordinate:

𝑃_𝑡→𝑃_𝑡−𝒾(𝜕ℎ𝒾𝜕𝑡)𝑞,𝑞→𝑞−𝒾ℎ+2(𝜕ℎ𝜕𝑞), (23) and the Hamiltonian becomes

𝐻₄=𝑃_𝑡−(𝒾𝜕ℎ𝜕𝑡+2𝑓ℎ+2Ω²𝑥₀(𝑡))𝑞−(𝑓𝑞𝜕ℎ𝜕𝑞+Ω²𝑥₀𝑔𝜕ℎ)𝜕𝑞−2𝒾𝑓𝑞𝑝−2𝒾Ω²𝑥₀𝑔𝑝−2Ω²𝑥₀𝑔ℎ+Ω²x₀²−𝑓. (24) The condition for the linear term in 𝑞 and the differential to be cancelled leads to the solution of ℎ(𝑡) as

ℎ(𝑡)=2𝒾𝑒^{2𝒾∫𝑓(𝑡)𝑑𝑡}Ω²𝑥₀𝑒^{−2𝒾∫𝑓(𝑡)𝑑𝑡}𝑑𝑡, (25) and the ℎ(𝑞)as

ℎ(𝑞)=𝐶𝑒^{(∫𝑓(𝑔)[ln(𝑞)]−1/𝑔Ω2)𝑑𝑞}, (26) where 𝐶 is an arbitrary constant.

Another two- andthree-variable transformation is carried out on the Hamiltonian to eliminate the linear term in momentum and the associated differential in co-ordinate:

𝑃_𝑡→𝑃_𝑡−𝒾2(𝜕𝑟𝒾𝜕𝑡)𝑃,𝑞→𝑞−𝒾𝑟+2(𝜕𝑟𝜕𝑞), (27) and the Hamiltonian becomes

𝐻₅=𝑃_𝑡𝒾−(2𝜕𝑟𝜕𝑡+2𝒾Ω²𝑥₀𝜕𝑔−2𝑓𝑟)𝑃−2𝒾𝑓𝑞𝑝+𝒾𝑓²𝑟𝜕𝑞²−2Ω²𝑥₀𝑔ℎ+Ω²x₀²−𝑓. (28) The linear term in 𝑃 in (28) is eliminated if 𝒾2(𝜕𝑟𝜕𝑡)+2𝒾Ω²𝑥₀𝑔−2𝑓𝑟=0, (29) and solving (29) yields

𝑟(𝑡)=2𝑒^{2𝒾∫𝑓(𝑡)𝑑𝑡}Ω²𝑥₀𝑔𝑒^{−2𝒾∫𝑓(𝑡)𝑑𝑡}𝑑𝑡+𝐶𝑒^{−2𝒾∫𝑓(𝑡)𝑑𝑡}, (30) and the condition for the differential in co-ordinate to cancelled yields

𝜕²𝑟(𝑞)𝜕𝑞²=0. (31) However (31) is a second-order differential Equation and we rewrite (31) in the form

𝒾𝑓𝑞²𝑑𝑑𝑞{𝑞²𝜕𝑟𝜕𝑞}=0 (32) or

𝜕²𝑟𝜕𝑞²+1𝑞𝜕𝑟𝜕𝑞=0. (33) Multiplying (33) by 𝑞² gives

𝑞²𝜕²𝑟𝜕𝑞²𝜕+𝑞²𝑟𝜕𝑞=0, (34) and comparing (34) with (35) we have

𝑞²𝑦^′′+𝑥𝑦^′−𝜆²𝑦=0. (35) This shows that (34) is a Bessel equation with 𝜆=0. The general solution of (33) becomes

𝑟(𝑞)=𝐴₁𝐽_𝑜(𝑚𝑞)+𝐴₂𝑌_𝑜(𝑚𝑞), (36) where 𝑚 is an integer, 𝐴₁, 𝐴₂ are arbitrary constant, and 𝐽_𝑜(𝑞) and 𝑌_𝑜(𝑞), respectively, represent Bessel functions of first and second kind.

Again a point transformation is done to eliminate the co-ordinate in the Hamiltonian of (28) [3]:

𝑃→𝑒^−𝑞𝑃,𝑞→𝑒^𝑞. (37) Substituting (37) into (28) will be

𝐻₆=𝑃_𝑡−2𝒾𝑓𝑝−2Ω²𝑥₀𝑔ℎ+Ω²x₀²−𝑓. (38) In order to trivialize the Hamiltonian of (38), we carried out a two-variable similarity transformation: 𝑃_𝑡→𝑃_𝑡+2𝒾𝑓𝑝+2Ω²𝑥₀𝑔ℎ−Ω²x₀²+𝑓, (39) and (38) becomes

𝐻₇=𝑃_𝑡. (40) The Schrodinger equation associated with the trivialized Hamiltonian of (40) is

𝑃_𝑡Ψ₇^(𝑞)=𝐻₇Ψ₇^(𝑞), (41) where Ψ₇^(𝑞) being the time-dependent Wave function originates from the differential in the canonical transformed Hamiltonian Operators.

The complete wave function Ψ₁(𝑞,𝑡) is obtained as

Ψ₁(𝑞,𝑡)=Ψ(𝑞)Ψ(𝑡)=e^𝑓(𝑞)e^𝑔(𝑝)e^ℎ(𝑝)e^𝛾(𝑝)e^{𝑓(𝑡)𝑞2/2}e^{𝑔(𝑡)𝑝2/2}e^{ℎ(𝑡)𝑃}e^𝛾(𝑡)P×P_ln𝑔e^{−2𝑃∫𝑓(𝑡)𝑑𝑡+𝒾∫(2Ω2𝑥0𝑔ℎ−Ω2x02+𝑓)𝑑𝑡}. (42) The first four terms of (42) are the times Independent Wave Function arising naturally from the cancellation of the differential in co-ordinate in the Hamiltonian Operator while the other terms are the time-dependent wave function gotten as the consequences of the cancellation in the co-ordinates and momentum.

Solving (42) completely yields the result as [3] without the ambiguity of Ψ₁(𝑞) which they encountered because Ψ₁(𝑞) is obtained directly from the cancellation of the differentials in the Hamiltonian.

We obtained after a simple algebra the unnormalized wave function from (11) as

Ψ₁(𝑞,𝑡)=𝑁𝑔^3𝑛/2e^{−(1/2)𝑔ℎ2}exp(𝒾[2𝜔²𝑥₀𝑔ℎ−𝜔²𝑥₀²+(2𝑛+1)𝑓]𝑑𝑡)×𝑒^{(1/2)𝑓𝑞2+ℎ𝑞}𝐻_𝑛(𝑔2(𝑞−𝒾𝛾𝑔−ℎ)), (43) where 𝑁 is the normalization constant and 𝐻_𝑛(𝑔(𝑞)/2(𝑞−𝒾𝛾(𝑞)/Ω−ℎ(𝑞))) is the Hermite polynomial of order 𝑛.

3. Conclusion

The introduction of a three-variable CT on the co-ordinate and the momentum space removed the ambiguity encountered in [3] in evaluating the time-independent Schrödinger wave function. In (41), we obtain the generalized wave function of the time dependent harmonic oscillator with the first four terms arising from the introduction of the three canonical variables. These terms were simply given as Ψ₁ in [3] without its explicit values as we have obtained above. Rather fortunately the three-variable transformations separate naturally the time-dependent and time-independent function explicitly. This separation aids us to solve the Schrödinger equation completely by trivializing the time-dependent Hamiltonian Operator. In addition our result can be viewed as a generalized and explicit case of that obtained in [3].

The extension of this work to different time dependent systems with different potential is possible. Thus canonical transformation is a good tool in solving time-dependent systems.

Acknowledgment

The authors acknowledge the Nandy-Leabio Foundation for partial financial support in this work.