Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.17

Particle in a Gravitational Field

Physics — Landau & Lifshitz · particle-in-gravitational-field

Problem

The Hamiltonian for a charged particle of mass

m

and charge

q

in a uniform electric field

\mathcal{E}

pointing in the

-z

direction is

H = \frac{p^2}{2m} - q\mathcal{E}z

(a) By analogy with the gravitational problem, write down the energy eigenvalues and eigenfunctions (for a particle confined to

z > 0

with

\psi(0) = 0

). (b) Show that the energy scale and length scale are

E_0 = \left(\frac{\hbar^2 q^2 \mathcal{E}^2}{2m}\right)^{1/3}, \quad z_0 = \left(\frac{\hbar^2}{2mq\mathcal{E}}\right)^{1/3}

\mathcal{E} = 10^6

V/m, calculate

E_0

and

z_0