text new page (beta)
English (pdf)
Article in xml format
Article references
Automatic translation
Send this article by e-mail
Cited by SciELO 
Similars in
SciELO 
Permalink
PACS: 03.65.-w
1. Introduction
In non-relativistic quantum mechanics, the propagator is represented as the transition probability amplitude for a particle to motion from initial space-time configuration to final space-time configuration. The Feynman path integral 1 and the Schwinger action principle 2 are the well-known methods in calculating the propagator. The aim of this paper is to present the connection between the integrals of the motion of a quantum system and its Green function or propagator.
As reveal by V.V. Dodonov et al. 3 that the Green function is the eigenfunction of the integrals of the motion describing initial points of the system trajectory in the phase space. D.B. Lemeshevskiy and V.I. Man’ko 4 constructed the Green functions for the driven harmonic oscillator with the aid of integrals of the motion. In the present paper we want to calculate the Green functions or propagators for the damped harmonic oscillator 5,6,7, the harmonic oscillator with strongly pulsating mass, 8 and the harmonic oscillator with mass growing with time 9 by the method developed by V.V. Dodonov et al. 3
This paper is organized as follows. In Sec. 2, the Green function for the damped harmonic oscillator is derived. In Section 3, the calculation of the Green function for the harmonic oscillator with strongly pulsating mass is presented. The Green function for the harmonic oscillator with mass growing with time is evaluated in Sec. 4. Finally, the conclusion is given in Sec. 5.
2. The Green function for a damped harmonic oscillator
The Hamiltonian operator for a damped harmonic oscillator is described by 5,6,7
where r is the damping constant coefficient.
The aim of this section is to drive the Green function G(x, x', t) of the Schrodinger equation by the method of integrals of motion 3,4. The classical correspondence of the Hamiltonian operator in Eq. (1) is
The Hamilton equation of motion for position and momentum are 10
The classical paths in the phase space under the initial conditions q(0) = q 0 and p(0) = p 0 are given by
where
We define operators acting in the Hilbert space as follows
Calculating the total derivative of the operator
Similarly, the total time-derivative of the operator
Thus, operators in Eqs. (8) and (9) are integrals of the motion and correspond to the initial position and momentum. Then these operators must satisfy equations for the Green function G(x, x', t), 3,4
where the operators on the left-hand sides of the equations act on variable x, and on the right- hand sides, on x'. Now we write Eqs. (12) and (13) explicitly,
By modifying Eqs. (14) and (15), the system of equations for deriving the Green function G(x, x', t) are
Now one can integrate Eq. (16) with respect to the variable x to obtain
Where C(x', t) is the function of x' and t.
Substituting Eq. (18) into Eq. (17), we obtain the differential equation for C(x', t) as
Solving Eq. (19), the function C(x', t) can be expressed as
where C(t) is the pure function of time.
So, the Green function in Eq. (18) can be written as
To find C(t), we must substitute the Green function of Eq. (21) into the Schrodinger equation
After some algebra, we obtain an equation that does not contain the variables x and x',
Eq. (23) can be simply integrated with respect to time t, and one obtains
where C is a constant.
Substituting Eq. (24) into Eq. (21) and applying the initial condition
we get
So, the Green function or propagator for a damped harmonic oscillator can be written as
which is the same form as the result of S. Pepore et al. 5 calculating from Feynman path integral.
3. The Green function for a harmonic oscillator with strongly pulsating mass
The Hamiltonian operator for a harmonic oscillator with strongly pulsating mass can be expressed as 9
where v is the frequency of mass. The classical analog of the Hamiltonian operator in Eq. (28) is
The classical equations of motion determining the oscillator position and momentum are
The classical trajectories in the phase space under the initial conditions q(0) = q 0 and p(0) = p 0 can be written as
where
By eliminating p 0 in Eq. (31) and q 0 in Eq. (32), the solutions are
The Hilbert space operators of q 0 and p 0 are
We can determine that
Then these operators must satisfy the equations for the Green function G(x, x', t) 3,4
Now we write Eqs. (39) and (40) explicitly,
The system of equations for defining the Green function G(x, x', t) are
Now we can integrate Eq. (43) with respect to the variable x to get
Substituting Eq. (45) into Eq. (44), we obtain the differential equation for C(x', t) as
Solving Eq.(46), we obtain
After substituting Eq. (47) into Eq. (45), we arrive at
To find C(t), we must substitute the Green function of Eq. (48) into the Schrödinger equation
After some algebra, we get
Integrating Eq. (50) with respect to time, we obtain
Substituting Eq. (51) into Eq. (48) and applying the initial condition in Eq. (25), the constant C is
Thus, the Green function for a harmonic oscillator with strongly pulsating mass can be expressed as
which is the same result as M. Sabir and S. Rajagopalan 9 by Feynman path integral method.
4. The Green function for a harmonic oscillator with mass growing with time
The Hamiltonian operator for a harmonic oscillator with mass growing with time can be written as
where
The equation of motion for this oscillator is
The classical paths in the space under the initial conditions q(0) = q0 and p(0) = p0 can be expressed as
We can express q0 and p0 in terms of q, p, and t by
We define operators acting in the Hilbert space as follows
Calculating the total derivatives of the operators
Hence, operators in Eqs. (61) and (62) are integrals of motion and correspond to the initial position and momentum. Then these operators must satisfy the equations for the Green function G(x, x', t)3,4
Writing Eqs. (65) and (66) explicitly, it can be shown that
The system of equations for calculating the Green function G(x, x', t) are
Now we can integrate Eq. (70) with respect to the variable x to obtain
Substituting Eq. (71) into Eq. (70), we obtain the differential equation for C(x', t) as
Solving Eq. (72), we obtain
After substituting Eq. (73) into Eq. (71), we obtain
To get C(t), we must substitute the Green function of Eq. (74) into the Schrödinger equation
After some algebra, we obtain
Integrating Eq. (76) with respect to time, we get
Substituting Eq. (77) into Eq. (74) and employing the initial condition in Eq. (25), the constant C becomes
So, the Green function for a harmonic oscillator with mass growing with time can be written as
which is agreement with the result of S. Pepore and B. Sukbot 11 calculating by Schwinger method.
5. Conclusion
The method for deriving the Green functions with the helping of integrals of the motion
presented in this paper can be successfully applied in solving time-dependent mass
harmonic oscillator problems. This method has the important steps in finding the
constant of motions q0 and
p0 and implying that the Green functions
G(x, x', t)
is the eigenfunctions of the operators
In fact, this method has many common features with the Schwinger method, 11,12,13,14 but the Schwinger method requires the operator
In Feynman path integral, the pre-exponential function C(t) comes from sum over all historical paths that depend on the calculation of functional integration while in the integrals of motion method this term appears in solving Schrodinger equation of Green function. In my opinion the method in this article seems to be more simple from the viewpoint of calculation.
