## Indicators

• Similars in SciELO

## Print version ISSN 0035-001X

### Rev. mex. fis. vol.63 n.5 México Sep./Oct. 2017

Research

The initial value problem method for time-dependent harmonic oscillator

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 initial value problem method is formulated to calculate the propagator for time- dependent harmonic oscillators. The method is based on finding the initial position operator from Heisenberg equations. The investigated models in this paper are the damped harmonic oscillator, the harmonic oscillator with strongly pulsating mass, and the harmonic oscillator with mass growing with time. The comparison of the initial value problem method with Feynman path integral and Schwinger method is also described.

Keywords: The initial value problem method; propagator; time-dependent harmonic oscillators

PACS: 03.65.-w

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

k(x,t;x0,0)=ϕ(t)expihA(x,t;x0,0), (1)

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),

idx^(t)dt=x^(t),H^,idp^(t)dt=p^(t),H^, (2)

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

x^(0)k(x,t;x0,0)=x0k(x,t;x0,0). (3)

(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

ik(x,t;x0,0)t=H^(t)k(x,t;x0,0). (4)

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

H(t)=p22m(t)+12m(t)ω2x2, (5)

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

m(t)=mert, (6)

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

m(t)=mcos2vt, (7)

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

m(t)=m(1+αt)2, (8)

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

H^(t)=e-rtp^22m+12mω2ertx^2. (9)

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

x^t=e-rtcos Ωtx^0+re-rt2Ωsin Ωtx^0 +e-rtmΩ sin Ωtp^0 (10)

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

p^t=-mΩert2 sin Ωtx^0-mr24Ωert2sinΩtx^0 +ert2cosΩtp^0 - r2Ωert2sinΩtp^0 (11)

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

x^t=ert2cosΩtx^t-rert22ΩsinΩtx^t -sinΩtmΩe-rt2p^t (12)

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

xert2cosΩt-xr2Ωert2sinΩt+isinΩte-rt2mΩ ×x Kx, t; x0, 0=x0Kx, t, x0, 0 (13)

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

ert2cosΩt - r2Ωert2sinΩt x-sinΩte-rt2 mΩ Ax,  t; x0, 0x=x0 (14)

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

Ax, t; x0, 0=12mΩcotΩtertx2-14mrertx2-mΩcscΩtert2 xx0 (15)

So, the propagator can be written as

Kx, t; x0, 0 = ϕtexpi2mΩcotΩtertx2-12mrertx2-2mΩcscΩtert2xx0 (16)

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

iKx, t; x0, 0t=-e-rt22m 2Kx, t; x0, 0x2 + 12mω2x2ertkx, t; x0, 0 (17)

to get the differential equation for ϕ(t) as

iϕtt=12mΩ2 csc2 Ωtx02ϕt-iΩ2cotΩtϕt + ir4ϕt (18)

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

ϕ(t)=Cert4sinΩtexpimΩ2cot tΩtx02, (19)

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

Kx, t; x0, 0=Cert2sinΩt12expimΩ2 × ertx2+x02cotΩt-2xx0sinΩtert2 ×exp-imr4ertx2 (20)

After applying the initial condition

limt0+K(x,t;x0,0)=δ(x-x0),

the constant C is

C=mΩ2πi. (21)

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

Kx, t; x0, 0=mΩert22πisinΩt12 ×expimΩ2sinΩtertx2+x02cosΩt -2xx0ert2,  ×exp-imr4ertx2 (22)

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

H^(t)=p^22mcos2vt+12cos2vtω2x^2. (23)

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

x^t=sec vt cos Ωtx^(0)+sec vtmΩsin Ωtp^(0). (24)

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

p^t=-mΩcosvtsinΩtx^0 +mvsinvtcosΩtx^0 +cosvtcosΩtp^0 +vΩsinvtsinΩtp^0 (25)

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

x^0=cosvtcosΩtx^t + vΩsinvtsinΩtx^t-secvtmΩsinΩtp^t (26)

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

cosvtcosΩt + vΩsinvtsinΩtx +imΩsecvtsinΩtxKx, t; x0, 0 = x0Kx, t; x0, 0 (27)

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

cosvtcosΩt + vΩsinvtsinΩtx - secvtmΩsinΩt × Ax, t; x0, 0x = x0 (28)

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

Ax, t; x0, 0 = 12mΩ cos2 vtcotΩtx2+12mvsinvtcosvtx2 -mΩcosvtcscΩtxx0 (29)

So, the propagator becomes

Kx, t; x0, 0=ϕtexpi2mΩ cos2 vtcotΩt +mvsinvtcosvtx2-2mΩcosvtcscΩtxx0 (30)

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

iKx, t; x0, 0t = -sec2 vt22m 2Kx, t; x0, 0x2 + 12m cos2 vtω2 x2 K x, t; x0, 0 (31)

it can be shown that

i ϕtt=12mΩ2csc2Ωtx02ϕt - i2ΩcotΩt +vtanvtϕt (32)

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

ϕ(t)=CcosvtsinΩtexpimΩ2cot Ωtx02, (33)

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

K x, t; x0, 0 =CcosvtsinΩt ×expi2mΩcotΩt cos2 vtx2 + x02 -2mΩcosvtcscΩtxx0 ×expimv2sinvtcosvtx2 (34)

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

limt0+K(x,t;x0,0)=δ(x-x0). (35)

The constant C becomes

C=mΩ2πi. (36)

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

K x, t; x0, 0 =mΩcosvt2π isinΩtexpi2 × mΩcotΩtcos2 vtx2 + x02 -2mΩcosvtcscΩtxx0 ×expimv2sinvtcosvtx2 (37)

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

H^(t)=p^22m(1+αt)2+12m(1+αt)2ω2x^2. (38)

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

x^t = cosωt1+αtx^0 + asinωtω 1+αtx^0 + sinωtmω 1+αtp^0 (39)

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

p^t=mαcosωtx^0-mω1+αtsinωtx^ 0 + mα 1+αt cosωtx^0 - mα2ωsinωtx^ 0 + 1+αtcosωtp^ 0 - αωsinωtp ^0 (40)

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

x^0=1+αtcosωt -αωsinωt x^t - sinωtmω 1+αt p^t (41)

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

x1+αtcosωt -αωsinωt+isinωtmω1+αt x ×K x, t; x0, 0 = x0Kx, t; x0, 0 (42)

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

x1+αtcosωt -αωsinωt - sinωtmω1+αt × Ax, t; x0, 0x=x0 (43)

By solving Eq. (43), the answer is

Ax, t; x0, 0 = 12 m 1+αtω 1+αtcotωt-αx2-mωcscωt 1+αtxx0 (44)

So, the propagator becomes

Kx, t; x0, 0=ϕtexpi2m1+αtx2 × ω1+αtcotωt -α -2mωcscωt1+αtxx0 (45)

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

iKx, t; x0, 0t=-22m1+αt 2Kx, t; x0, 0x2+12m 1+αt2ω2x2Kx, t; x0, 0 (46)

to obtain the differential equation for ϕ(t) as

iϕtt=12mω2csc2ωtx02ϕt-i21+αtω1+αtcotωt -αϕt (47)

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

ϕ(t)=C1+αtsinωtexpimω2cotωtx02, (48)

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

Kx, t; x0, 0=C1+αtsinωtexpi2mω1+αt2 ×cotωt -mα1+αtx2 +mωcotωtx02 -2mωcscwt1+αtxx0 (49)

The last step is finding the constant C by using

limt0+K(x,t;x0,0)=δ(x-x0), (50)

to get

C=mω2πi. (51)

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

Kx, t; x0, 0 = mω1+αt2πisinωt12expi2 × mω1+αt2cotωt-mα1+αtx2+mωcotωtx02-2mωcscωt 1+αtxx0 (52)

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.

References

1. D.C. Khandekar, S.V. Lawande, and K.V. Bhagwat, Path Integral Method and Their Applications, (Woorld Scientific, Singapore, 1993). [ Links ]

2. R.P. Feynman and A.R Hibbs, Quantum Mechanics and Path Integral, (McGraw-Hill, New York, 1965). [ Links ]

3. S. Pepore and B. Sukbot, Chinese. J. Phys. 47 (2009) 753. [ Links ]

4. S. Pepore and B. Sukbot , Chinese. J. Phys. 53 (2015) 060004. [ Links ]

5. S. Pepore and B. Sukbot , Chinese. J. Phys. 53 (2015) 100002. [ Links ]

6. S. Pepore and B. Sukbot , Chinese. J. Phys. 53 (2015) 120004. [ Links ]

7. S. Pepore , P. Winotai, T. Osotchan, and U. Robkob, Science Asia. 32 (2006) 173. [ Links ]

8. P. Caldirola, Nuovo Cim. 18 (1941) 393. [ Links ]

9. E. Kanai, Prog. Theor. Phys. 3 (1948) 440. [ Links ]

10. R.K. Colegrave and M.S. Abdalla, J. Phys. A. 19 (1982) 1549. [ Links ]

11. M. Sabir and S. Rajagopalan, J. Phys. A. 37 (1991) 253. [ Links ]

Received: May 16, 2017; Accepted: June 30, 2017