A Wave Packet in Simple Harmonic Motion: Coherent State of Simple Harmonic Oscillator

 

Background: Without the general tools for solving the Time Dependent Schrödinger Equation (TDSE) which we will learn in the coming lectures, direct solutions to the TDSE are difficult but not impossible to find. In this problem, you will consider one such solution, the ``Coherent States'' of a Simple Harmonic Oscillator (SHO) of frequency . We will use the solution to this problem to illustrate the general principles of the Correspondence Principle and the Conservation of Probability discussed in lecture.

Consider a particle of mass m in a one-dimension simple Harmonic oscillator of frequency . The Hamiltonian for this system is then given by

We begin with trial solutions of the form

 

Assignment:

• Effective equations for A(t), B(t) and C(t)
• Solutions of constant width
• Rigid shape of the probability distribution
• Conservation of probability
• Correspondence Principle/Ehrenfest's Theorem
• Connection to Problem 2