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PathIntegralSHO - Finding the Prefactor in the Simple...

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Finding the Prefactor in the Simple Harmonic Oscillator Propagator Michael Fowler 10/24/07 Hand-waving Argument Recall that the free particle propagator has the form ( ) ( ) 2 , ; ,0 exp . 2 2 im x x m U x T x iT T π = = = From a classical mechanical evaluation of the action, we can show the simple harmonic oscillator propagator has the form ( ) ( ) ( ) 2 2 , ; ,0 exp cos 2 2 sin im U x T x A T x x T xx T ω ω ω = + = where A ( T ) is a so far unknown function of time. For sufficiently small T , though, much less than the period of the oscillator, the potential will have no significant effect, so the propagator must tend to the free particle propagator. That is, for T tending to zero, ( ) / 2 . A T m iT π = = But we also know that at time 2 / t π ω = , all simple harmonic wavefunctions return to their t = 0 values, so a particle localized at x will be localized at x again after one period. In the exponent we do indeed see this cyclic behavior. In the prefactor, we must replace T by ( ) sin / t ω ω , that is, ( ) . 2 sin m A T i T ω π ω = = Note this expression is also consistent with the free particle propagator in the limit ω going to zero, in other words, the vanishing of the simple harmonic oscillator potential. Approximating Integrals by Stationary Phase Techniques Actually, this result can be derived from the integral over the fluctuations about the classical path. The argument is closely analogous to that for the free particle, and the following equation is a straightforward generalization of that case (discussed in the previous lecture):
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