Principles of Quantum Mechanics · Problem 5.4.3

The Harmonic Oscillator — R. Shankar

Physics — Quantum Mechanics · propagator-constant-force

Problem

Consider a particle subject to a constant force

f

in one dimension. Solve for the propagator in momentum space and get

U(p, t; p', 0) = \delta(p - p' - ft)\, e^{i(p'^3 - p^3)/6m\hbar f}

Transform back to coordinate space and obtain

U(x, t; x', 0) = \left(\frac{m}{2\pi\hbar it}\right)^{1/2} \exp\left\{\frac{i}{\hbar}\left[\frac{m(x-x')^2}{2t} + \frac{1}{2}f(x+x') - \frac{f^2 t^3}{24m}\right]\right\}

[Hint: Normalize

\psi_E(p)

such that

\langle E|E'\rangle = \delta(E-E')

. Note that

E

is not restricted to be positive.]