Question

A Wave Packet in Simple Harmonic Motion: Coherent State of Simple Harmonic Oscillator 2 Background: Without the general tools for solving the Time Dependent Schrödinger Equation DSwhich we will lear in ciect ssoltions io the TDSEi are diflieli 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 w. We will use the solution to this problem to illustrate the general principles of the Correspondence Principle and the Conservation ndnce Prineiple and the e tihis of Probability discussed in lecture.

Answer 1

H^- = p^2/2 m + m w^2/2 x^2 & Psi (x, t) = exp (-A (t) x^2 + B (t) x + C (t)) We know that Schrodinger wave equation -h^2/2 m d^2 Psi (x, t)/d x^2 + 1/2 m w^2 x^2 Psi (x, t) = i h d/dt (Psi (x, t)) So -h^2 [(B (t) - 2 A x)^2 - 2 A (t)] Psi (x, t)] + 2/2 m^2 w^2 x^2 Psi (x, t) = 2 m i h [-A' (t) x^2 + B' (t) x + C' (t)] Psi (x, t) rightarrow -h^2 [B^2 (t) + 4 (A (t)^2 x^2 - 4 A (t) B (t) x - 2 A (t)] + m^2 w^2 x^2 = 2 m i h [-A' (t) x^2 + B' (t) x + C' (t)] Now comparing x^2, x & x^0 coefficient in the term, we get -4 h^2 (A (t))^2 + m^2 w^2 = -2 m i h A' (t) rightarrow A' (t) = i/2 h m [m^2 w^2 - 4 h^2 (A (t))^2]