130_notes.dvi

The total number of states is always preserved. For example if I add twoℓ= 2 states together, I
get total angular momentum states withj= 0, 1 , 2 ,3 and 4. There are 25 product states since each
ℓ= 2 state has 5 different possiblems. Check that against the sum of the number of states we have
just listed.

5 ⊗5 = 9S⊕ (^7) A⊕ (^5) S⊕ (^3) A⊕ (^1) S
where the numbers are the number of states in the multiplet.
We will use addition of angular momentum to:

• Add the orbital angular momentum to the spin angular momentum foran electron in an atom
  J~=~L+S~;


• Add the orbital angular momenta together for two electrons in an atomL~=L~ 1 +L~ 2 ;


• Add the spins of two particles togetherS~=S~ 1 +S~ 2 ;


• Add the nuclear spin to the total atomic angular momentumF~=J~+I~;


• Add the total angular momenta of two electrons togetherJ~=J~ 1 +J~ 2 ;


• Add the total orbital angular momentum to the total spin angular momentum for a collection
  of electrons in an atomJ~=~L+~S;


• Write the product of spherical harmonics in terms of a sum of spherical harmonics.

1.29 Time Independent Perturbation Theory

Assume we have already solved and an energy eigenvalue problem andnow need to include an
additional term in the Hamiltonian. We can use time independent perturbation theory (See section
22) to calculate corrections to the energy eigenvalues and eigenstates. If the Schr ̈odinger equation
for thefull problemis
(H 0 +H 1 )ψn=Enψn

and we have already solved the eigenvalue problem forH 0 , we may use aperturbation series,to
expand both our energy eigenvalues and eigenstates in powers of the small perturbation.

En=En(0)+E(1)n +E(2)n +...

ψn=N

(

φn+

∑

k 6 =n

cnkφk

)

cnk=c(1)nk+c(2)nk+...

where the superscript (0), (1), (2) are the zeroth, first, and second order terms in the series.Nis
there to keep the wave function normalized but will not play an important role in our results.

By solving the Schr ̈odinger equation at each order of the perturbation series, wecompute the
corrections to the energies and eigenfunctions. (see section 22.4.1) We just give the first
few terms above.

En(1)=〈φn|H 1 |φn〉

c(1)nk=〈φk|H^1 |φn〉

En(0)−Ek(0)

En(2)=

∑

k 6 =n

|〈φk|H 1 |φn〉|^2

E(0)n−Ek(0)