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

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Assume a wave function of the form

where is the Bohr radius, is the variational parameter, and is the

spin state of the two electrons.

Perturbation Theory

5.38 Momentum Perturbation (Princeton)

A particle of mass moves in one dimension according to the Hamiltonian

All eigenfunctions and eigenvalues are known. Suppose we add

a term to the Hamiltonian, where and are constants and is the

momentumoperator:

Derive an expression for the eigenvalues and eigenstates of the new Hamil-

tonian H.

5.39 Ramp in Square Well (Colorado)

b) A small perturbation is added, Use perturbation

theory to calculate the change in the ground state energy to

5.40 Circle with Field (Colorado, Michigan State)

A particle with charge e and mass is confined to move on the circumfer-
ence of a circle ofradius Theonlyterm in theHamiltonian is the kinetic
energy, so the eigenfunctions and eigenvalues are

PROBLEMS

A particle of mass is bound in a squarewell where

a) What are theenergy andeigenfunction ofthe groundstate?