A particle of charge and mass sits in a uniform field pointing along the -axis,

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Charged particles in electromagnetic fields

Problem

A particle of charge

q

and mass

m

sits in a uniform field pointing along the

x

-axis,

\vec B=B\hat x

, described in the symmetric gauge

\vec A=\tfrac{1}{2}\vec B\times\vec r

. ((a)) Write

\vec A

explicitly in Cartesian components. ((b)) Expand

\hat H=\frac{1}{2m}(\hat{\vec p}-q\vec A)^2

and show that it separates into the field-free kinetic energy, a term

-\frac{qB}{2m}\hat L_x

, and a term

\frac{q^2B^2}{8m}(\hat y^2+\hat z^2)