Thermodynamics, Statistical Physics, and Quantum Mechanics

69

What are the eigenvalues of thegroundstate and the lowestexcited

states ofthe three-dimensionalharmonicoscillator?

Use perturbationtheory to estimate howtheseeigenvalues arealtered

by the spin–orbitinteraction.

a)

b)

5.47 Interacting Electrons (MIT)

Consider two electronsbound to a proton byCoulombinteraction.Neglect

the Coulomb repulsion betweenthe two electrons.

a) What are the ground state energy andwavefunctionfor thissystem?

b) Considerthat aweak potentialexists between the twoelectrons of

the form

where is a constant and isthe spin operator for electron (ne-

glect the spin–orbitinteraction). Usefirst-orderperturbationtheory

to estimate howthis potentialalters thegroundstate energy.

5.48 Stark Effect in Hydrogen (Tennessee)

Consider a single electron in the state of the hydrogen atom. We ig-
nore relativisticcorrections, so the and states areinitiallydegenerate.
Then we impose a small static electric field Use perturbation
theory to derivehow the energy levelsare changed tolowestorder in
powers of

5.49 Hydrogen with Electric and Magnetic

Fields (MIT)

Consider an electron in the state of thehydrogen atom. We ignore
relativistic corrections, so the and states areinitially degenerate.

QUANTUM MECHANICS