Thermodynamics, Statistical Physics, and Quantum Mechanics

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5.24 Triplet Square Well (Stony Brook)

Consider atwo-electronsystem inonedimension, whereboth electrons have

spins aligned inthe samedirection(say,up). They interact onlythrough

the attractivesquarewell in relativecoordinates

What is the lowestenergy of the two-electron state? Assume the total

momentum is zero.

5.25 Dipolar Interactions (Stony Brook)

Two spin-1/2 particles are separated bya distance and interact only

through themagnetic dipoleenergy

where is the magnetic moment of spin Thesystem of two spins

consists ofeigenstates of thetotalspin andtotal

a) Write the Hamiltonian in terms of spin operators.

b) Write the Hamiltonian in terms of and

c) Give theeigenvalues for all states.

5.26 Spin-Dependent Potential (MIT)

Consider twoidentical particles ofmass andspin 1/2. They interact only
through thepotential

where and are Pauli spin matrices which operate on the spin of

particle

a) Construct thespineigenfunctions for the two particle states. What

is the expectation value of Vfor each ofthese states?

b) Give the eigenvalues of all of theboundstates.

Show that commuteswith

Derive the spectrum of from thecommutation relations.

a)

b)

QUANTUM MECHANICS