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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.64 no.2 México mar./abr. 2018

https://doi.org/10.31349/revmexfis.64.150 

Research

Integrals of the motion and Green function for dual damped oscillators and coupled harmonic oscillators

Surarit Peporea 

aDepartment of Physics, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Rangsit-Nakornayok Road, Pathumthani 12110, Thailand, e-mail: surapepore@gmail.com


Abstract

The application for the integrals of the motion of a quantum system in deriving Green function or propagator is presented. The Green function is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phase space. The exact expressions for the Green functions of the dual damped oscillators and the coupled harmonic oscillators are evaluated in co-ordinate representations. The relation between the integrals of the motion method and other methods such as Feynman path integral and Schwinger method are also presented.

Keywords: Integrals of the motion; Green function; dual damped oscillators

PACS: 03.65.-w

1. Introduction

In non-relativistic quantum mechanics the Green function or propagator gives the probability amplitude for a particle to travel from one spatial point at one time to another spatial point at a later time. The propagator can be calculated by the well-known methods of Feynman path integral 1 and Schwinger method 2-6. In 1975, V.V. Dodonov, I.A. Malkin, and V.I. Man’ko 7 presented the connection between the integrals of the motion of a quantum system and its Green function that is the eigenfunction of the integrals of the motion describing initial points of the system trajectory in the phase space. In 1977, V.V. Dodonov et al. 8 constructed a new method of calculating non-equilibrium density matrices with the aid of the quantum integrals of motion. D.B. Lemeshevskiy and V.I. Man’ko applied the integrals of the motion method to the problem of the driven harmonic oscillator in 2012 9.

The aim of this paper is to calculate the Green functions for the dual damped oscillators and the coupled harmonic oscillators by the integrals of the motion method. The organization of this paper are as follows. In Sec. 2, the Green function for the dual damped oscillators is derived. In Sec. 3, the Green function for the coupled harmonic oscillators is obtained with the aid of the integrals of the motion. Finally, the conclusion is presented in Sec. 4.

2. The Green function for a dual damped oscillators

The Bateman damped harmonic oscillator is described as an open system in which energy is dissipated by interaction with a heat bath 10. Bateman has shown that dissipative systems can be presented as a pair of damped oscillators, the so called dual damped oscillators. This system includes a primary one expressed by q1 variables and its time reversed image by q2 variables. The Hamiltonian operator for a dual damped oscillators can be expressed as

H^t=p^1p^2m+γmq^2q^2-q^1q^1+k-γ24mq^1q^2, (1)

where k is the harmonic coefficients and γ is a damping coefficient.

The aim of this section is to derive the Green function G(x1,x2,x′1,x′2,t) of the Schrödinger equation by the method of integrals of the motion 7-9. The classical equation of motion for this system are

mq¨1+γq˙1+kq1=0, (2)

mq¨2-γq˙2+kq2=0. (3)

The classical paths in the phase space under the initial conditions q1(0) = q10, q2(0) = q20, p 1(0) = p10, and p2(0) = p20 are given by

q1t=q10e-γt/2mcosΩt+p20mΩe-γt/2msinΩt, (4)

q2t=q20eγt/2mcosΩt+p10mΩeγt/2msinΩt, (5)

p1t=p10eγt/2mcosΩt-q20mΩeγt/2msinΩt, (6)

p2t=p20e-γt/2mcosΩt-q10mΩe-γt/2msinΩt, (7)

Where Ω=k/m-γ2/4m2. Now we consider the system of Eqs. (4)-(7) as an algebraic system for unknown initial positions q10 and q20 and initial momentums p10 and p20.

The variables q1, q2, p1, p2, and t are taken as the parameters. The solution of this system can be written as the operator in Hilbert space as

q^10q^1,p^2,t=q^1eγt/2mcosΩt-p^2mΩeγt/2msinΩt, (8)

p^10q^2,p^1,t=p^1e-γt/2mcosΩt+q^2mΩe-γt/2msinΩt, (9)

p^20q^2,p^1,t=q^2e-γt/2mcosΩt-p^1mΩe-γt/2msinΩt, (10)

p^20q^1,p^2,t=p^2eγt/2mcosΩt+q^1mΩeγt/2msinΩt. (11)

The operators q^10,q^20,p^10, and p^20 are the integrals of the motion because theirs satisfy equation of

dI^dt=I^t+i[H^,I^]=0, (12)

where Î maybe q^10,q^20,p^10, and p^20. Then these operators must satisfy equations for the Green function G(x 1 , x2 , x′1 , x′2 , t) 7-9,

q^10x1Gx1,x2,x1',x2',t=q^1x1'×Gx1,x2,x1',x2',t, (13)

p^10x1Gx1,x2,x1',x2',t=-p^1x1'×Gx1,x2,x1',x2',t, (14)

q^20x2Gx1,x2,x1',x2',t=q^2x2'×Gx1,x2,x1',x2',t, (15)

p^20x2Gx1,x2,x1',x2',t=-p^2x2'×Gx1,x2,x1',x2',t, (16)

where the operators on the left-hand sides of the equations act on variables x1 and x2, and on the right-hand sides, on x′1 and x′2. Now we write Eqs. (13)-(16) explicitly,

x1eγt/2mcosΩt+imΩeγt/2msinΩtx2×Gx1,x2,x1',x2',t=x1'Gx1,x2,x1',x2',t, (17)

-ie-γt/2mcosΩtx1+x2 mΩe-γt/2msinΩt×Gx1,x2,x1',x2',t=iGx1,x2,x1',x2',tx1', (18)

x2e-γt/2mcosΩt+imΩe-γt/2msinΩtx1×Gx1,x2,x1',x2',t=X2'Gx1,x2,x1',x2',t, (19)

x1mΩeγt/2msinΩt-ieγt/2mcosΩtx2×Gx1,x2,x1',x2',t=iGx1,x2,x1',x2',tx2' . (20)

By modifying Eqs. (17)-(20), the system of equation for deriving the Green function G(x1 , x2 , x′1 , x′2 , t) are

Gx1,x2,x1',x2',tx2=-imΩe-γt/2mcscΩtx1'-mΩcotΩtx1Gx1,x2,x1',x2',t, (21)

Gx1,x2,x1',x2',tx1=-imΩe-γt/2mcscΩtx2'-mΩcotΩtx2Gx1,x2,x1',x2',t, (22)

Gx1,x2,x1',x2',tx1=imΩcotΩtx2'-mΩe-γt/2mcscΩtx2Gx1,x2,x1',x2',t, (23)

Gx1,x2,x1',x2',tx2=imΩcotΩtx1'-mΩeγt/2mcscΩtx1Gx1,x2,x1',x2',t, (24)

Now one can integrate Eq. (21) with respect to the variable x 2 to obtain

Gx1,x2,x1',x2',t=Cx1,x1',x2',texpimΩ×cotΩtx1x2-mΩe-γt/2mcscΩtx1'x2, (25)

where C(x1 , x′1 , x′2 , t) is the function of x1 , x′1 , x′2 , and t. Substituting Eq. (25) into Eq. (22) to obtain C (x1 , x′ 1 , x′2 , t), the result is

Cx1,x2,x1',x2',t=Cx1',x2',t×exp-imΩeγt/2m cscΩtx1x2'. (26)

So, the Green in Eq. (25) can be written as

Gx1,x2,x1',x2',t=Cx1',x2',texpimΩcotΩtx1x2-mΩcscΩteγt/2mx1x2'+e-γt/2mx1'x2 (27)

The next step is substituting the Green function in Eq. (27) into Eq. (23) to obtain C(x′1 , x′2 , t) as

Cx1',x2',t=Cx2',texpimΩcotΩtx1'x2'. (28)

Thus, the Green function in Eq. (27) becomes

Gx1,x2,x1',x2',t=Cx2',texpimΩcotΩt ×x1,x2+x1'x2'-mΩcscΩt eγt/2mx1x2'+e-γt/2mx1'x2. (29)

Substituting the Green function in Eq. (29) into Eq. (24) to obtain C(x′2 , t), the result is

Cx2',tx2'=0. (30)

So, it can be implied that C(x′2 ,t) = C(t). The Green function in Eq. (29) can be expressed as

Gx1,x2,x1',x2',t=Ctexpi×mΩcotΩtx1x2+x1'x21-mΩcscΩteγt/2mx1x2'+e-γt/2mx1'x2. (31)

To find C(t), we must substitute the Green function of Eq. (31) into the Schrodinger equation

iGx1,x2,x1',x2',tt=-2m2Gx1,x2,x1',x2',tx1x2-iγ2mx1Gx1,x2,x1',x2',tx2+iγ2mx1Gx1,x2,x1',x2',tx1+k-γ24mx1x2Gx1,x2,x1',x2',t. (32)

After some algebra, we obtain an equation

dC(t)dt=-CtΩcotΩt. (33)

Equation (33) can be simply integrated with respect to time t, and one obtains

Ct=CsinΩt, (34)

where C is a constant. Substituting Eq. (34) into Eq. (31) and applying the initial condition

limt0+Gx1,x2,x1',x2',t=δx1-x1'δx2-x2', (35)

we obtain

C=mΩ2πi . (36)

So, the Green function for a dual damped oscillator is

Gx1,x2,x1',x2',t=mΩ2πisinΩt×expimΩcotΩt x1x2+x1'x2'-mΩcscΩteγt/2mx1x2'+e-γt/2mx1'x2. (38)

which is the same form as the result of S. Pepore and B. Suk-bot calculated by the Schwinger method 5.

3. The Green function for the coupled harmonic oscillators

Considering a system of two harmonic oscillators which are coupled together by another spring. Assuming that the masses of the oscillators and three spring constants are all the same. Let their displacements be q1 and q2. The Hamiltonian operator for the coupled harmonic oscillator can be written as 6.

H^t=p^122m+p^222m+mω2q^12-q^1q^2+q^22, (38)

where ω is the constant frequency. The classical equations of motion determining the oscillator positions and momentums are

q¨1+ω22q1-q2=0, (39)

q¨2+ω22q1-q2=0. (40)

The classical paths in the phase space under the initial conditions q1 (0) = q10, q2 (0) = q20 , p1 (0) = p10, and p2 (0) = p20 are

q1t=q10cosωt+cos3ωt2+q20cosωt+cos3ωt2+p103sinωt+3sin3ωt6mω+p203sinωt-3sin3ωt6mω, (41)

q2t=q10cosωt-cos3ωt2+q20cosωt+cos3ωt2+p103sinωt-3sin3ωt6mω+p203sinωt+3sin3ωt6mω, (42)

p1t=p10cosωt+cos3ωt2+p20cosωt-cos3ωt2-q10mωsinωt+3mωsin3ωt2-q20mωsinωt- 3mωsin3ωt2, (43)

p2t=p10cosωt-cos3ωt2+p20cosωt+cos3ωt2-q10mωsinωt-3mωsin3ωt2-q20mωsinωt+ 3mωsin3ωt2. (44)

Now we consider the system of Eqs. (41)-(44) as an algebraic system for unknown initial positions q10 and q20 and initial momentums p10 and p20. The variables q1 , q2 , p1 , p2, and t are taken as the parameters. The solution of this system can be written as the operator in Hilbert space as

q^10q^1,q^2,p^1,p^2,t=q^1cosωt+cos3ωt2+q^2cosωt-cos3ωt2-p^13sin3ωt+3sinωt6mω+p^23sin3ωt-3sinωt6mω, (45)

q^20q^1,q^2,p^1,p^2,t=q^1cosωt-cos3ωt2+q^2cosωt+cos3ωt2+p^13sin3ωt-3sinωt2+q^2cosωt+cos3ωt2, (46)

p^10q^1,q^2,p^1,p^2,t=q^1mωsinωt+3mωsin3ωt2+q^2mωsinωt-3mωsin3ωt2+p^1cosωt+cos3ωt2+p^2cosωt-cos3ωt2, (47)

p^20q^1,q^2,p^1,p^2,t=q^1mωsinωt-3mωsin3ωt2+q^2mωsinωt+3mωsin3ωt2+p^1cosωt-cos3ωt2+p^2cosωt+cos3ωt2. (48)

The operators q^10,q^20,p^10,p^20, are the integrals of the motion because their satisfy Eq. (12). Then these operators must satisfy Eqs. (13)-(16) and can be explicitly written as

x1cosωt+cos3ωt2+x2cosωt-cos3ωt2+i3sin3ωt+3sinωt6mωx1-i3sin3ωtsinωt6mωx2Gx1,x2,x1',x2',t=x1'Gx1,x2,x1',x2',t, (49)

x1mωsinωt+3mωsin3ωt2+x2mωsinωt-3mωsin3ωt2-icosωt+cos3ωt2x1-icosωt-cos3ωt2x2Gx1,x2,x1',x2',t=iGx1,x2,x1',x2',txa', (50)

x1cosωt-cos3ωt2+x2cosωt+cos3ωt2-i3sin3ωt-3cin ωt6mωx1+i3sin3ωt+3sinωt6mωx2Gx1,x2,x1',x2',t=x2'Gx1,x2,x1',x2',t, (51)

x1mωsinωt-3mωsin3ωt2+x2mωsinωt+3mωsin3ωt2-icosωt-cos3ωt2-icosωt+cos3ωt2x2Gx1,x2,x1',x2',t=iGx1,x2,x1',x2',tx2'. (52)

By modifying Eqs. (49)-(52), the system of equations for deriving the Green function G(x1 , x2 , x′1 , x′2 , t) are

Gx1,x2,x1',x2',tx1=ix1mω2cotωt+32mωcot3ωt+x2mω2cotωt-32mωcot3ωt-x1'mω2cscωt+32mωcsc3ωt-x2'mω2cscωt-32mωcsc3ωtGx1,x2,x1',x2',t, (53)

Gx1,x2,x1',x2',tx2=ix1mω2cotωt-32mωcot3ωt+x2mω2cotωt+32mωcot3ωt-x1'mω2cscωt-32mωcsc3ωt-x2'mω2cscωt+32mωcsc3ωtGx1,x2,x1',x2',t, (54)

Gx1,x2,x1',x2',tx1=ix1mω2cscωt+32mωcsc3ωt+x2mω2cscωt-32mωcot3ωt-x1'mω2cotωt+32mωcot3ωt-x2'mω2cotωt-32mωcot3ωtGx1,x2,x1',x2',t, (55)

Gx1,x2,x1',x2',tx1=ix1mω2cscωt-32mωcsc3ωt+x2mω2cscωt+32mωcsc3ωt-x1'mω2cotωt-32mωcot3ωt-x2'mω2cotωt+32mωcot3ωtGx1,x2,x1',x2',t, (56)

Now we can integrate Eq. (53) with respect to the variable x1 to obtain

Gx1,x2,x1',x2',t=Cx1',x2,x2',texpix12mω4cotωt+34mωcot3ωt+x1x2mω2cotωt-32mωcot3ωt-x1x1'mω2cscωt+32mωcsc3ωt-x1x2'mω2cscωt-32mωcsc3ωt, (57)

where C(x′1 , x2 , x′2 , t) is the function of x′ 1 , x 2 , x′ 2 , and t. Substituting Eq. (57) into Eq. (54) to find C(x′ 1 , x 2 , x′ 2 , t), we get

Cx1',x2,x2',t=Cx1',x2',texpix22mω4cotωt+34mωcot3ωt-x2x1'mω2cscωt-32mωcsc3ωt-x2x2'mω2cscωt+32mωcsc3ωt, (58)

So, the Green function in Eq. (57) can be written as

Gx1,x2,x1',x2',t=Cx1',x2',texpix12+x22mω4cotωt+34mωcot3ωt+x1x2mω2cotωt-32mωcot3ωt-x1x1'+x2x2'mω2cscωt+32mωcsc3ωt-x1x2'+x2x1'mω2cscωt-32mωcsc3ωt, (59)

Substituting Eq. (59) into Eq. (55), we obtain

Cx1',x2',t=Cx2',texpix'12mω4cscωt+34 mωcot3ωt+x1'x2'mω2cotωt-32mωcot3ωt, (60)

Thus, the Green function of Eq. (59) becomes

Gx1,x2,x1',x2',t=Cx2',t=expix12+x22+x'12mω4cotωt+34mωcot3ωt+x1x2+x1'x2'mω2cotωt-32mωcot3ωt-x1x2'+x2x1'mω2cscωt-32 mωcsc3ωt-x1x2'+x2x1'mω2cscωt-32mωcsc3ωt, (61)

Substituting Eq. (61) into Eq. (56), we get

Cx2',t=Ctexpix'22mω4cotωt+34mωcot3ωt, (62)

So, the Green function in Eq. (61) can be written as

Gx1,x2,x1',x2',t=Ctexpix12+x22+x'12+x'22mω4cotωt+34mωcot3ωt+x1x2+x1'x2'mω2cotωt-32mωcot3ωt-x1x1'+x2x2'mω2cscωt+32mωcsc3ωt-x1x2'+x2x1'mω2cscωt-32mωcsc3ωt, (63)

To find C(t), we must substitute the Green function of Eq. (63) into the Schrödinger equation

iGx1,x2,x1',x2',tt=-22m2Gx1,x2,x1',x2',tx12-22m2Gx1,x2,x1',x2',tx22+mω2x12-x1x2+x22Gx1,x2,x1',x2',t. (64)

After some algebra, we obtain an equation

dC(t)dt=-Ctω2cotωt+32ωcot3ωt. (65)

Integrating Eq. (65) with respect to time, we obtain

Ct=csinωtsin3ωt1/2, (66)

where C is a constant.

Substituting Eq. (66) into Eq. (63) and applying the initial condition in Eq. (35), the constant C is

C=31/4mω2πi. (67)

So, the Green function for a coupled harmonic oscillator can be expressed as

Gx1,x2,x1',x2',t=mω2πi3sinωtsin3ωt1/2expix12+x22+x'12+x'22mω4cotωt+34 mωcot3ωt+x1x2+x1'x2'mω2cotωt-32mωcot3ωt-x1x1'+x2x2'mω2cscωt+32 mωcsc3ωt-x1x2'+x2x1'mω2cscωt-32mωcsc3ωt, (68)

which is the same form as the calculation of S. Pepore and B. Sukbot by the Schwinger method 6.

4. Conclusion

The method in calculating the Green functions with the aid of integrals of the motion presented in this article can be successfully applied in solving the dual damped oscillator and the coupled harmonic oscillator problems. This method has the crucial steps in deriving the integrals of the motions q^10,q^20,p^10, and p^20 and implying that the Green functions G(x 1, x 2, x'1, x'2, t) is the eigenfunctions of the operators q^10,q^20,p^10 and p^20.

In fact, this method has many common features with the Schwinger method 3-6, but the Schwinger method uses the operator q^(t) and p^(t) in calculating the matrix element of Hamiltonian operator in deriving the Green function

Gx,x',t=C(x,x')×exp-i0txtH^x^t,x^(0)x'0xtx'0. (69)

In the Feynman path integral 1, the pre-exponential function C(t) comes from sum over all fluctuating paths that depend on calculation of the functional integration while in the integrals of the motion method this term appears from solving the Schrodinger equation of Green function. In the Schwinger formalism 2-6, the pre-exponential function C(t) arises from the commutation relation of x^t,x^(0). These different points of view may show the connection between classical mechanics and quantum mechanics. It can be conclude here that the integrals of the motion method in this paper seems to be more simple from the viewpoint of calculation.

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Received: August 31, 2017; Accepted: November 09, 2017

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