Quantum Mechanics Homework: 1D Harmonic Oscillator & Free Particle - Prof. Thomas D. Cohen, Assignments of Quantum Mechanics

Download Quantum Mechanics Homework: 1D Harmonic Oscillator & Free Particle - Prof. Thomas D. Cohen and more Assignments Quantum Mechanics in PDF only on Docsity! Homework 6: Due October 20 Please use units with 1=h 1. Verify that the causal propagator for the one dimensional Harmonic oscillator is given by ( )( ) .)'( )'sin(2 '2)'cos(' exp )'sin(2 )'()'(ˆ')0,;','( 22 t t xxtxxmi ti m txtUxxtxK C θ ω ωω ωπ ω θ       −+ == That is you should show that )'()'()0,;','( '2 ' '2 1 22 2 2 txxixtxK t i xm xm C δδ ω −−=      ∂ ∂ −+ ∂ ∂− . The book has various suggestions as to how you might prove this. 2. In class it was argued that the time-evolution operator at imaginary time is related to the partition function. This has important implications for the propagator In particular one expects that ∑∫ −=− n Ec nexixKdx ββ )0,;,( where ( )∑ −≡ n nEZ ββ exp)( is the partition function in statistical mechanics (for which kT 1 =β where k is Boltzmann’s constant) and )0,\;,( xixK c hβ− is understood in the sense of an analytic continuation from the real function. In this problem, we focus on the case of the one dimensional harmonic oscillator. a. By directly summing using the harmonic oscillator energy levels ( ) 2 1+= nEn ω show that ( ) )2/exp()2/exp( 1 exp βωωβ β −− =−∑ n nE . b. Starting from the propagator for the harmonic oscillator derived in last week’s homework ( )( ) . )'sin(2 '2)'cos(' exp )'sin(2 )0,;','( 22       −+ = t xxtxxmi ti m xtxK C ω ωω ωπ ω Evaluate )0,;,( xixKdx c β−∫ and show that you reproduce the partition function from part a. 3. Equation 2.5.16 gives the propagator for a free particle (i.e. one for which the potential is zero.) Use this propagator to solve the following problem: Suppose that one has a system which is in the ground state of a harmonic oscillator for a particle of mass m and frequency ω. Suppose, further at t=0 potential is instantly tuned off leaving the system as a free particle..