ScienceAsia 32 (2006): 173-179 Path Integral for a Harmonic Oscillator with Time-Dependent Mass and Frequency Surarit Pepore a* , Pongtip Winotai a , Tanakorn Osotchan b and Udom Robkob b a Department of Chemistry, Faculty of Science, Mahidol University, Rama VI Road, Bangkok 10400, Thailand. b Department of Physics, Faculty of Science, Mahidol University, Rama VI Road, Bangkok 10400, Thailand. * Corresponding author, E-mail: g4437430@student.mahidol.ac.th Received 18 Oct 2004 Accepted 6 Feb 2006 ABSTRACT: The exact solutions to the time-dependent Schrodinger equation for a harmonic oscillator with time-dependent mass and frequency were derived in a general form. The quantum mechanical propagator was calculated by the Feynman path integral method, while the wave function was derived from the spectral representation of the obtained propagator. It was shown that the propagator and the wave function depended on the s solution of a classical oscillator, in which the amplitude and phase satisfied the auxiliary equations. To demonstrate the derivation of the solution from our auxiliary equations, exponential and periodic functions of mass with constant frequency were imposed to evaluate the propagator and wave function for the Caldirola-Kanai and pulsating mass oscillators, respectively. KEYWORDS: Path integral, propagator, wave function, a harmonic oscillator with time-dependent mass and frequency. doi: 10.2306/scienceasia1513-1874.2006.32.173 INTRODUCTION In recent years, the study of Hamiltonian with explicitly time-dependent coefficients becomes very popular. 1-11 The mathematical challenge and important applications in various areas of physics, such as quantum optics, 12 cosmology, 13 and nanotechnology, 14 are the main reasons for intensive studies. The most common problem in this area is the harmonic oscillator with time-dependent frequency and/or mass. The harmonic oscillator with time-dependent frequency is the first exactly solved problem. 15 The standard method for solving the time-dependent problems is the Lewis- Riesenfeld (LR) invariant operator method. 2,3,5,11,15 This method is based on constructing an invariant operator and writing Schrodinger’s wave function in terms of invariant operator eigenstates connecting with time- dependent phase factor. However, in the case of a harmonic oscillator with time-dependent frequency and mass the LR- invariant operator method has some difficulty. 16-17 In 1992, Dantas and et al. 16 constructed an invariant operator from the canonical transformation variables and transformed the invariant operator to a simple harmonic oscillator operator by unitary transformation. However, their wave functions satisfied the Schrodinger’s equation only in the case of constants mass, but were not applicable in general case of time- dependent mass. Later in 1997, Pedrosa 17 revised this problem by modifying the invariant operator and used another unitary operator to include the time-dependent mass parameter. His result presented the first wave function for the harmonic oscillator with time- dependent mass and frequency. Finally, Ciftja 18 proposed an alternative method by assuming the Schrodinger’s wave function in terms of the Gaussian function with time-dependent coefficients and using the space-time transformation to reduced the problem to a simple harmonic oscillator. He suggested that there should be some attempt to develop an easier method than the LR-invariant operator method to tackle the time-dependent problem. The aim of this paper is to derive the propagator and wave function for a harmonic oscillator with time- dependent mass and frequency in any function form as described by the Hamiltonian 2 2 2 1 () () () , 2 () 2 p Ht mt tx mt ω = + (1) where () mt and () t ω are the time-dependent mass and frequency, respectively. Our developed method is not based on the Hamiltonian and solving the differential equations as described in previously reported articles, 16-18 but base on the Lagrangian and solving the integral equation by the Feynman path integral approach. 19 In this formulation the time-dependent Schrodinger’s equation is replaced by integral equation