A hydrogen atom ( ) is prepared in the level

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Approximation methods

The Structure of Quantum Mechanics, Ch Hydrogen, and the symmetry that explains it

Problem

A hydrogen atom (

Z=1

) is prepared in the

n=3

level. The electron's spin is neglected throughout this problem. A weak uniform magnetic field

\vec B = B\hat z

is switched on, adding the term

\hat H'=\dfrac{\mu_B}{\hbar}B\hat L_z

to the Hamiltonian, where

\mu_B

is the Bohr magneton. ((a)) Using the Casimir relation, state

E_3

(in eV, to four significant figures), the total number of degenerate states, and the allowed orbital quantum numbers

l

with their degeneracies, with no radial equation solved. ((b)) State which of

\hat L^2

\hat L_z

, and the operator

\hat{\vec K}

of the

SO(4)

algebra commute with

\hat H'

, and hence say which quantum numbers among

l

m

remain good once the field is on. ((c)) The

9

degenerate

n=3

states separate according to

m

. For each

m=-2,\dots,2

, state how many of the original

9

states carry that value of

m

(the degeneracy of the resulting level), and explain why states of the same

m

but different

l

remain exactly degenerate under this perturbation. Do not compute the numerical size of the splitting.