William Angerer - Department of Physics and Astronomy - University ...

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equation (5.2), is represented using matrix elements of the form (mlrili) as
(5.2)
+ (1Ir,ln)(nlrilm)(mlrkll)
[(W-Wmn )+hmn J[(Wo-rWnl) -hnl J
+ (1Ir,ln){n1r,lm)(m1rkll)..
[(W+Wnm )-l1'nm J[(Wo-Wml )+17ml!
+ (1Ir,ln)(nlrk Im)(mlr.ll) )
[(w+wmil-hmd[(wo+wnil+I"Ynd
•
In equation (5.2), Ii), 1m), and In) are the quantum mechanical states of the system:
pf?) is the unperturbed density of the state Ii); -eri is the dipole moment operator
along the direction i; hW nm is the energy difference between states In), and 1m):
and ~/nm
is a dephasing term. Determination of nonzero matrix elements is greatly
simplified through group theory. Group theory assigns symmetries to states based on
the underlying symmetry of the crystal.
The power of group theory to determine band symmetries becomes clear when
we examine the relationship between the Hamiltonian, H, the symmetry operations
of the crystaL P(Xd, and the eigenstates of the Hamiltonian. wnJo
Note the index
n corresponds to the an eigenstate of energy En and j labels the I-fold degenerate
states. In general, P(Xd are all of the symmetry operations of the crystal including
translations and point operations of the crystaL vVe will state without proof the
following theorem.
If H is invariant under the group G of P(Xd then the eigenfunctions belonging to the
same energy level, l/JnJ' form a basis of the representation of G.
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