A particle of mass moves in three dimensions in a potential energy that depends only…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Symmetries, generators and conservation laws

Problem

A particle of mass

m

moves in three dimensions in a potential energy

V(y,z)

that depends only on

y

and

z

, not on

x

(for instance a charged particle between two infinite charged plates whose separation lies along

y

, the plates extending without limit along

x

and

z

, so the electrostatic potential energy does not vary along

x

). Let

\hat T(a) = e^{-i\hat p_x a/\hbar}

be translation by

a

along

x

, and

\hat H = \dfrac{\hat p_x^2+\hat p_y^2+\hat p_z^2}{2m}+V(\hat y,\hat z)

. ((a)) Show that

\hat T(a)

is a symmetry of

\hat H

for every real

a

\hat T^\dagger(a)\hat H\hat T(a)=\hat H

. ((b)) Expand

\hat T(a)

to first order in

a

and deduce

[\hat H,\hat p_x]=0

. ((c)) Use the Heisenberg equation of motion to show that

\langle\hat p_x\rangle

is constant in time for every state of this particle.