1. Introduction

The propagators have application in many areas of physics such as quantum statistical mechanics, condensed matter physics, polymer physics, and economics ¹. In non-relativistic quantum mechanics, the propagator describe the transition probability amplitude for a particle to travel from initial space-time configuration to final space-time configuration. The most popular methods to calculate the propagator are the Feynman path integral ² and the Schwinger method ^3,4,5,6. However, both methods have some mathematical difficulties. The aim of this paper is to present the simple method called the initial value problem method to calculate the non-relativistic propagator.

The initial value problem method begins with the assumption that the propagator for the quadratic potentials can be written as ²

where the pre-exponential factor ϕ(t) is the pure function of time and A(x, t; x ₀, 0) is the two-point characteristic function.

The main idea of the initial value problem method consists in the following steps.

(1) The first step is solving the Heisenberg equations for x^(t) and p^(t),

and writing the solution for x^(0) only in terms of the operators x^(t) and p^(t).

(2) Next, we substitute the propagator in Eq. (1) into an eigenvalue equation of

(3) Solving the differential equation for A(x, t; x ₀, 0), we obtain the two-point characteristic function.

(4) The final step is finding ϕ(t) by substituting the obtained propagator form step (3) into the Schrödinger equation

The problem that use to demonstrate the application of the initial value problem method is the time-dependent harmonic oscillator described by the Hamiltonian ⁷

where m(t) is the time-dependent mass. The time-dependent mass m(t) of this paper can be divided to the three system.

The first system is the damped harmonic oscillator or the Caldirola-Kanai oscillator, ^7,8,9 which the time-dependent mass can be written as

where r is the damping constant coefficient.

The second system is the harmonic oscillator with strongly pulsating mass, ^7,10 which the time-dependent mass can be described by

where v is the frequency of mass.

The third system is the harmonic oscillator with mass growing with time, ^4,11 which the time-dependent mass has the law as

where α is a constant parameter.

In Sec. 2, the propagator for a damped harmonic oscillator is derived. The calculation of the propagator for a harmonic oscillator with strongly pulsating mass is shown in Sec. 3. In Sec. 4, the evaluation of the propagator for a harmonic oscillator with mass growing with time is illustrated. Finally, the conclusion and discussion are described in Sec. 5.

2. The initial value problem method for a damped harminic oscillator

This section is the calculation of the propagator for a damped harmonic oscillator or the Caldirola-Kanai oscillator ^7,8,9 described by the Hamiltonian operator

By applying the Heisenberg equation in Eq. (2) to the Hamiltonian operator in Eq. (9), the position operator x^(t) can be written as

where x^(0) and p^(0) are the position and momentum operators respectively at t = 0, and Ω2=ω2-(r2/4)

The momentum operator p^(0)=mertx^̇(t) can be written by using Eq. (10) as

The next step is expressing x^(0) only in terms of x^(t) and p^(t) by eliminating p^(0) from Eq. (11) with the using of Eq. (10) to obtain

The eigenvalue equation for the propagator in Eq. (3) can be written in coordinate representation as

Substituting propagator assumed in Eq. (1) into Eq. (13), the result is

Solving Eq. (14) to find A(x, t; x ₀, 0), the result is

So, the propagator can be written as

The next step is substituting the propagator in Eq. (16) into the Schrodinger equation for a damped harmonic oscillator

to get the differential equation for ϕ(t) as

The next step is solving Eq. (18) to obtain the pre-exponential function ϕ(t) as

where C is a constant. Substituting Eq. (19) into Eq. (16), the propagator becomes

After applying the initial condition

the constant C is

So, the propagator for a damped harmonic oscillator can be written as

which agree with the result of S. Pepore and et. al.⁷ calculated by Feynman path integral.

3. The initial value problem method for a harminic oscillator with strongly pulsating mass

The Hamiltonian operator for a harmonic oscillator with strongly pulsating mass can be written as ⁷

By solving Heisenberg equation, the position operator x^(t) becomes

The momentum operator p^(t)=m cos2vtx^̇(t) can be expressed as

By eliminating p^(0) from Eq. (25), the initial position operator x^(0) is

So, we can write the eigenvalue equation for the propagator as

Substituting the propagator in Eq. (1) into Eq. (27), the differential equation for A (x, t; x ₀, 0) can be written as

Solving Eq. (28) to find A (x, t; x ₀, 0), it can be shown that

So, the propagator becomes

Substituting the propagator in Eq. (30) into the Schrodinger equation for a harmonic oscillator with strongly pulsating mass

it can be shown that

After solving Eq. (32), the pre-exponential function ϕ(t) can be written as

where C is a constant. Combining Eq. (33) with Eq. (30), the propagator can be written as

To find the constant C, we apply the initial condition of the propagator

The constant C becomes

So, the propagator for a harmonic oscillator with strongly pulsating mass is

which is the same result as deriving by Feynman path integral ⁷.

4. The initial value problem method for a harminic oscillator with mass growing with time

The last system to investigate the initial value problem method is the harmonic oscillator with mass growing with time expressed by the Hamiltonian operator

By solving the Heisenberg equation, the operator x^(t) can be written as

The momentum operator p^(t)=m(1+αt)2x^̇(t) can be expressed as

Eliminating p^(t) from Eq. (40) by using Eq. (39), the initial position operator is

The eigenvalue equation for the propagator in Eq. (3) can be shown that

The next step is substituting the propagator in Eq. (1) into Eq. (42) to obtain

By solving Eq. (43), the answer is

So, the propagator becomes

The next step is substituting the propagator in Eq. (45) into the Schrodinger equation for a harmonic oscillator with mass growing with time

to obtain the differential equation for ϕ(t) as

The next step is solving Eq. (47) for ϕ(t) as

where C is a constant. Substituting Eq. (48) into Eq. (45), the result is

The last step is finding the constant C by using

to get

Finally, the propagator for a harmonic oscillator with mass growing with time can be written as

which agree with the calculation of S. Pepore and B. Sukbot by Schwinger method and Feynman path integral ⁴.

5. Conclusions

We have successfully calculated the exactly propagator for three systems of time-dependent harmonics oscillators. The method in this paper is simple. It requires only solving the Heisenberg equation for x^(t). This method reduce the solving second order differential equation of Schrodinger equation to solve the first order differential equation of the pre-exponential factor ϕ(t).

The initial value problem method in this paper have some similarity with the Schwinger method ^3,4,5,6 in solving Heisenberg equation but the Schwinger methods requires the knowledge of commutator algebra for x^(0),x^(t). The calculation of propagator for Feynman path integral have some mathematical difficulties in deriving the classical action and in time-slicing process ².

We can conclude here that the initial value problem method may be the new techniques to calculate the non-relativistic propagator for the quadratic potentials.