Introduction to Solid State Theory

JProf. Dr. Lorenz Bartosch Winter 2013/14
M. Sc. Simon Streib
Introduction to Solid State Theory
Problem set 6
Exercise 13 (Born-Oppenheimer Approximation for two coupled oscillators):
Consider the Hamiltonian for two coupled one-dimensional harmonic oscillators,
Ĥ = ˆp2 1
2M + ˆp2 2
2m + Kˆx2 1
+ kˆx2 2
2 2 + λˆx 1ˆx 2 ,
with [ˆx n , ˆp m ] = iδ m,n and M ≫ m. (ˆx 1 , ˆp 1 ) shall describe the movement of a ’nucleus’
with mass M and (ˆx 2 , ˆp 2 ) the movement of an ’electron’ with mass m.
(a) Show that the exact eigenvalues of Ĥ are given by
with n 1 , n 2 = 0, 1, 2, . . . and
(
E n1 ,n 2
= ω + n 1 + 1 ) (
+ω − n 2 + 1 )
,
2
2
ω± 2 = 1 ( k
2 m + K )
√
±
1 ( k
M 4 m − K M
) 2
+ λ2
mM .
Hint: You need to decouple the oscillators. Introduce a coordinate transformation
ˆx 1 = √ ˜x 1
M
, ˆx 2 = √ ˜x 2
m
, ˆp 1 = √ M ˜p 1 , ˆp 2 = √ m˜p 2 , so that the Hamiltonian can
be written as
Ĥ = 1 (
˜p 2 + ˜x T D ˜x ) ,
2
with ˜x = (˜x 1 , ˜x 2 ) T , ˜p = (˜p 1 , ˜p 2 ) T . By diagonalization of D, one can describe Ĥ
by two independent oscillators.
(b) Calculate E n1 ,n 2
within the Born-Oppenheimer approximation.
Hint: In the first step, neglect the kinetic energy of the ’nucleus’ and solve the
’electronic’ problem for fixed x 1 , obtaining energies E n (x 1 ). Then, solve the
’atomic’ Hamiltonian ˆp2 1
2M + E n(x 1 ).
(c) Compare the two solutions from (a) and (b) for small m M .
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Exercise 14 (Primitive Vectors of the Reciprocal Lattice):
(a) Prove that the reciprocal lattice primitive vectors defined by b i = 2π
where (i, j, k) = (1, 2, 3) and cyclic permutations thereof, satisfy
b 1 · (b 2 ×b 3 ) =
(2π) 3
a 1 · (a 2 ×a 3 ) .
a j×a k
a i·(a j ×a k ) ,
Recall that v = |a 1 · (a 2 ×a 3 )| is the volume of the primitive unit cell and argue
that the volume of the reciprocal lattice primitive cell is (2π) 3 /v.
Hint: Write b 1 (but not b 2 and b 3 ) in terms of the a i and use the orthogonality
relations b i ·a j = 2πδ ij .
(b) Suppose that the primitive vectors are constructed from the b i in the same
manner as the b i are constructed from the a i . Prove that these vectors are just
the a i themselves; i.e., show that
2π
b 2 ×b 3
b 1 · (b 2 ×b 3 ) = a 1,
etc.
Hint: Write b 3 in the numerator (but not b 2 ) in terms of the a i , use the vector
identity A × (B × C) = B(A · C) − C(A · B) and appeal to the orthogonality
relations b i ·a j = 2πδ ij and the result in (a).
Exercise 15 (Reciprocal Lattice of a Hexagonal and Trigonal Lattice):
(a) Using the primitive vectors given by a 1 = aˆx, a 2 = (a/2)ˆx + ( √ 3a/2)ŷ, and
a 3 = cẑ, show that the reciprocal of the simple hexagonal Bravais lattice is also
simple hexagonal, with lattice constants 2π/c and 4π/ √ 3a, rotated through 30 ◦
about the c-axis with respect to the direct lattice.
(b) For what value of c/a does the ratio have the same value in both the direct and
the reciprocal lattices?
(c) The Bravais lattice generated by three primitive vectors of equal length a, making
equal angles θ with one another, is known as a trigonal Bravais lattice. Show
that the reciprocal of a trigonal Bravais lattice is also trigonal, with an angle θ ∗
given by − cos θ ∗ = cos θ/[1 + cos θ] and a primitive vector length a ∗ given by
a ∗ = (2π/a)(1 + 2 cos θ cos θ ∗ ) −1/2 .
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