**I. Path Integrals
for Quantum Amplitudes**

- For a classical particle moving under conservative forces show
that the momentum of a particle at the endpoint x
_{b}is given by p = ¶S_{class}/¶x_{b}where S = ò_{ta}^{tb}dt L(x,[x\dot];t) is the action functional. Also show that the energy of the particle is given by E = ¶S_{class}/¶t_{b}. Recall that the energy of a classical particle is given by E = L-p[x\dot]. - The path integral for the amplitude a particle to be at x
_{b}at time t_{b}given the particle was at x_{a}at t_{a}isf(x _{a},t_{a};x_{b},t_{b}) =ó

õb

a*D*[x(t)] e^{[i/((h/2p) )]òtatb L(x,[x\dot];t) dt}.(1) - Evaluate the Gaussian integral
ò
_{-¥}^{¥}e^{iax2}dx for Re a > 0. Hint: Change variables to t = -iax^{2}and consider the closed contour in the complex t-plane: {t| 0® R,Rexp-ij [0 < j < p/2], -iR® 0} and R® ¥. Consider amplitude for a particle whose motion is derived from a Lagrangian of the form L = a(t)[x\dot]^{2}+b(t)x^{2}+c(t)x+d(t).- Show that f(x
_{a},t_{a};x_{b},t_{b}) =*A*(t_{a},t_{b}) e^{iSclass[a,b]/(h/2p)}for the general case. Hint: re-write the path integral in terms of the deviations from the classical path, [`x](t). - For a free particle of mass m show that
*A*(t_{a},t_{b}) =*A*(t_{b}-t_{a}) = Ö{[m/(2pi(^{h}/_{2p}) (t_{b}-t_{a}))]}.

- Show that f(x
- The amplitude for a free particle to be at x at time t given
the particle was at x = 0 at t = 0 is
f(x,t;0,0) = Ö{[m/(2pi(
^{h}/_{2p}) t)]}e^{imx2/2(h/2p) t}.- Consider the spatial variation of f(x,t;0,0) at fixed t and large x. Show that the amplitude has a wavelength l given by the de Broglie relation, p = h/l, for a particle of momentum p.
- Similarly, consider the temporal variation of f at
fixed x and show that the amplitude has a frequency w given by
the Einstein relation E = (
^{h}/_{2p})w for a particle with energy E.

- Calculate the amplitude for a particle of mass m moving
in a harmonic potential with natural frequency w. Express the
result in terms of the initial and final positions, x
_{a}and x_{b}, and the time interval t = t_{b}-t_{a}.

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On 17 Jan 2000, 16:10.