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

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

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

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

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

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,

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,

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

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

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

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

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

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

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

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

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

After some algebra, we obtain an equation

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

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

we obtain

So, the Green function for a dual damped oscillator is

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

After some algebra, we obtain an equation

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

where C is a constant.

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

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

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