130_notes.dvi

The full Schr ̈odinger equation is.

(H 0 +λH 1 )





∑

i∈N

αiφ(i)+

∑

k6∈N

cnk(λ)φk



= (E(0)n +λE(1)+...)





∑

i∈N

αiφ(i)+

∑

k6∈N

cnk(λ)φk





If we keep the zeroth and first order terms, we have

(H 0 +λH 1 )

∑

i∈N

αiφ(i)+H 0

∑

k6∈N

λc(1)nkφk= (E(0)n +λE(1))

∑

i∈N

αiφ(i)+En(0)

∑

k6∈N

λc(1)nkφk.

Projecting this onto one of the degenerate statesφ(j), we get
∑

i∈N

〈φ(j)|H 0 +λH 1 |φ(i)〉αi= (E(0)n +λE(1))αj.

By putting both terms together, our calculation gives us the full energy to first order, not just the
correction. It is useful both for degenerate states and for nearly degenerate states. The result may
be simplified to ∑

i∈N

〈φ(j)|H|φ(i)〉αi=Eαj.

This is just the standard eigenvalue problem for the full Hamiltonian inthesubspace of (nearly)
degenerate states.

22.5 Homework Problems

1. An electron is bound in a harmonic oscillator potentialV 0 =^12 mω^2 x^2. Small electric fields in
   thexdirection are applied to the system. Find the lowest order nonzero shifts in the energies
   of the ground state and the first excited state if a constant fieldE 1 is applied. Find the same
   shifts if a fieldE 1 x^3 is applied.


2. A particle is in a box from−atoain one dimension. A small additional potentialV 1 =
   λcos(πx 2 b) is applied. Calculate the energies of the first and second excited states in this new
   potential.


3. The proton in the hydrogen nucleus is not really a point particle like the electron is. It has
   a complicated structure, but, a good approximation to its charge distribution is a uniform
   charge density over a sphere of radius 0.5 fermis. Calculate the effect of this potential change
   for the energy of the ground state of hydrogen. Calculate the effect for then= 2 states.


4. Consider a two dimensional harmonic oscillator problem described by the HamiltonianH 0 =
   p^2 x+p^2 y
   2 m +

1

2 mω

(^2) (x (^2) +y (^2) ). Calculate the energy shifts of the ground state and the degenerate
first excited states, to first order, if the additional potentialV= 2λxyis applied. Now solve
the problem exactly. Compare the exact result for the ground state to that from second order
perturbation theory.

5. Prove that

∑

n

(En−Ea)|〈n|x|a〉|^2 = ̄h

2

2 mby starting from the expectation value of the commu-

tator [p,x] in the stateaand summing over all energy eigenstates. Assumep=mdxdtand write

dx

dtin terms of the commutator [H,x] to get the result.

6. If the general form of the spin-orbit coupling for a particle of massmand spinS~moving in a
   potentialV(r) isHSO= 2 m^12 c 2 L~·S~^1 rdVdr(r), what is the effect of that coupling on the spectrum
   of a three dimensional harmonic oscillator? Compute the relativistic correction for the ground
   state of the three dimensional harmonic oscillator.