Solid State 2 – Exercise 7
1. Using the Wigner-Eckart theorem:
ˆ
ˆ
ˆ
 JLSJ z | ( L  g0 S ) | JLSJ ' z  g ( JLS )  JLSJ z | J | JLSJ ' z 
Show that the value for the Lande g-factor is
g ( JLS ) 
3 1  S ( S  1)  L( L  1) 
 

2 2
J ( J  1)

Hint: See Appendix P, p. 797, in Ashcroft & Mermin.
2. Consider the spin-orbit Hamiltonian:
HSO   ( Lˆ Sˆ )
a) Show that the splitting of an LS-multiplet due to spin-orbit interation is:
EJ max  EJ min   S (2 L  1),
L>S
  L(2S  1),
S<L
b) Show that the splitting between two successive J multiplets within the
same LS-multiplet is:
EJ 1  EJ   ( J  1)
3. Consider the following Hamiltonian with both spin-spin interaction and a
Zeeman term (with magnetic field in the z direction):
ˆ ˆ
H  J  Si S j  H  Sˆi , z
i , j 
i
Where <i,j> is a sum of nearest neighbours.
Solve the question for the (simple) case of a 1D chain with only 3 spins.
(Each of spin ½).
1
3
2
(Each site can have on spin either up (+) or down (-) ).
a)
Using the fact that:
Sˆ  Sˆx  iSˆ y
Show that:


ˆ ˆ 1
Si S j  Sˆi  Sˆ j   Sˆi  Sˆ j   Si , z S j , z
2
b)
Now use the fact that:
Sˆ | S z   (S SZ )(S  1  S z ) | S z  1 
To write the matrix representation of the Hamiltonian in the basis where
|1> = | +++ > |2> = | ++- > |3> = | +-+ > |4> = | -++ >
|5> = | +-- >
|6> = | -+- > |7> = | --+ > |8> = | --- >
c)
What are the eigen-energies ?
d)
Diagonalize the Hamiltonian.
e)
The system is prepared in state |2> at time t=0. Find the
probability of the system to be in state |2> and state |3> as a
function of t.