A negative muon (charge

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Quantum Mechanics · The formalism of quantum mechanics

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

Problem

q=-e

, mass

m_\mu=1.884\times10^{-28}\,\mathrm{kg}

) is studied in two separate experiments. Ignore spin and treat the motion as non-relativistic throughout. ((a)) In a static, uniform field

\vec B=B\hat z

with

B=0.8\,\mathrm{T}

, described equally by the Landau gauge

\vec A_L=(0,Bx,0)

and the symmetric gauge

\vec A_S=\tfrac12(-By,Bx,0)

, find the gauge function

\Lambda(x,y)

with

\vec A_S=\vec A_L+\nabla\Lambda

and

\Lambda(0,0)=0

, write down the phase relating the state in the two gauges, and evaluate that phase angle at

(x_0,y_0)=(-8\,\mathrm{nm},12\,\mathrm{nm})

. ((b)) Still in this field, use only the commutator

[\hat\pi_x,\hat\pi_y]=i\hbar qB

to build ladder operators for the motion transverse to

\vec B

and obtain the Landau spectrum. Give

\hbar\omega_c

\mu\mathrm{eV}

\ell_B

in nm, and the energy

E_1

of the

n=1

level, in

\mu\mathrm{eV}

. ((c)) In the second experiment the same muon passes through an interferometer whose two arms encircle a long, thin, shielded solenoid carrying flux

\Phi=1.75\,h/e

, with

h/e=4.136\times10^{-15}\,\mathrm{Wb}

, outside which the field is exactly zero. Derive

\delta

in terms of

\Phi

and the muon's flux quantum

\Phi_0=h/q

, and give its value modulo

2\pi

in rad and the fringe shift, with its sign, as a fraction of one fringe.