Problem 2. Spin-orbit coupling (10 points).
Inside atoms, besides the Column interaction there exists a term called spin-orbit (SO)
coupling. The origin of SO coupling arises from relativistic Dirac equation, which is the
first order relativistic correction to the non-relativistic Schrödinger Eq. (The zero-th order
approximation is called Pauli Eq. that you have already seen in the midterm.) The SO
coupling can be written as
Hso = f (r)~σ ·~L,
(4)
where ~L is the orbital angular momentum, and f (r) is a scalar wavefunction that is proportional to the magnitude of radial electric field. We consider a simplified version of Eq. 4
H = ω~σ ·~L
(5)
We will solve its eigenvalues and eigenstates.
We consider the spheric harmonics with orbital angular momentum l and couple with
spin- 21 . we start from the basis |lm; ssz i defined
11
1 1
Ylm (θ, φ)
0
|lm;
i=
, |lm; − i =
.
(6)
0
Ylm (θ, φ)
22
2 2
1) Prove that there are two sets of different eigenvalues E+ > 0 and E− < 0. What the
values of E+ and E− ?
(Hint: you may use the operator identity ~σ ·~L = J~2 −~L2 − ~S2 where ~S = h̄2~σ and J~ = ~L + ~S.
Please note that ~L, ~S and J~ are all operators.)
For the E± -sectors, the normalized eigenstates are denoted as Y j+ , jz and Y j− , jz , respectively, where j± = l ± 21 . They are eigenstates of total angular momentum J~2 and Jz with
eigenvalue j± ( j± + 1) and jz , respectively. These two sectors are called sectors with positive
and negative helicitity, respectively.
2) Express Y j± jz as
Y j± , jz (θ, φ) =
a j± jz Yl,m (θ, φ)
b j± , jz Yl,m+1 (θ, φ)
,
(7)
where j± = l ± 12 and jz = m + 21 . Derive the expressions for a j± , jz and b j± , jz . Please note
that Y j± jz need to be normalized.
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