A particle of mass falls in a uniform gravitational field

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Quantum Mechanics · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch The path integral

Problem

A particle of mass

m

falls in a uniform gravitational field, with potential

V(x)=-mgx

and

x

measured downward. (a) Write the short-time factor

K(x_j,\epsilon;x_{j-1},0)

for this potential, and identify the Lagrangian in its exponent. (b) Put

x=x_{\rm cl}+y

with

y(0)=y(t)=0

. Show that the action is exactly

S_{\rm cl}+\tfrac12\int_0^tm\dot y^2\,dt'

, and hence that the fluctuation factor is the free particle's. (c) Find

S_{\rm cl}

and write

K(x,t;x',0)

. Check that it reduces to the free propagator as

g\to0