A Guide to Physics Problems. Part 2. Thermodynamics, Statistical Physics, and Quantum Mechanics

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5.74 Electron in Magnetic Field (Stony Brook,

Moscow Phys-Tech)

An electron is in freespaceexcept for aconstant magnetic field Bin the

a)

b)

Showthat themagnetic field can berepresented bythe vectorpoten-

tial

Use thisvectorpotential toderive the exacteigenfunctions andeigen-

values for theelectron.

5.75 Electric and Magnetic Fields (Princeton)

Consider aparticle of massmand chargeewhich is inperpendicularelectric

and magnetic fields:

Write theHamiltonian, using a convenient gauge for the vector po-

tential.

Find theeigenfunctions andeigenvalues.

Find theaveragevelocity in the for any eigenstate.

a)

b)

c)

5.76 Josephson Junction (Boston)

Considersuperconductingmetals I and II separated by a verythin insulat-

ing layer,such thatthatelectron wavefunctions canoverlap between the

metals (Josephsonjunction). AbatteryVis connectedacross thejunction

to ensure an averagechargeneutrality(seeFigureP.5.76). This situation

can be described by means of the coupled Schrödinger equations:

Here and are theprobability amplitudes for anelectron in I and
II, and are the electric potential energies in I and II, K is the cou-
pling constant dueto the insulatinglayer, and and
describe thebattery as asource ofelectrons.

a) Showthat areconstant intime.

QUANTUM MECHANICS