4.3 Proof of the Bloch-Theorem

For each vector R of the Bravais lattice we define a translation operator TR by

TRf(r) = f(r + R) (4.8)

for an arbitrary function f.
Since the Hamiltonian is periodic, we find

TRHψ(r) = H(r + R)ψ(r + R) = H(r)ψ(r + R) = HTRψ(r) (4.9)

Eq. (4.9) holds for all state functions ψ; so we can write

TRH = HTRor[TR,H] = 0. (4.10)

Successively applying the translation leads to

TRTRψ(r) = TRTRψ(r) = ψ r + R + R, (4.11)

i.e.

TRTR = TRTR = TR+R. (4.12)

Following Eq. (4.10) we can choose the Eigenvectorsystem of H to be simultaneously a Eigenvectorsystem of TR:

Hψ = Eψ TRψ = c(R)ψ (4.13)

Following Eq. (4.12) the Eigenvalues c(R) of the translation operator obey

TRTRψ(r) = c(R)T Rψ(r) = c(R)c(R)ψ(r). (4.14)

Thus for the Eigenvalues we find

c(R + R) = c(R)c(R). (4.15)

Let’s have a closer look on the primitive translation vectors of the Bravais lattice ai. For adequate variables xi we can write

c(ai) = ei2πxi . (4.16)

Thus for a general translation vector

R = n1a1 + n2a2 + n3a3 (4.17)

we find by successively applying Eq. (4.15)

c(R) = c(a1)n1 * c(a2)n2 * c(a3)n3 (4.18)

which may be rewritten as

c(R) = eikR (4.19)

using Eq. (4.16) and the following definitions:

k = x1b1 + x2b2 + x3b3 (4.20)

and

aibj = 2πδi,j. (4.21)

The above defined vectors b are the basic vectors of the reciprocal lattice.
Summing up, we have shown that for every vector of the real space the Eigenvectors ψ of H can be chosen to fulfill the following relation:

TRψ(r) = ψ(r + R) = c(R)ψ(r) = eikRψ(r). (4.22)

This is the definition of the Bloch theorem according to Eq. (4.5). Additional illustrative information you can find in the MaWi II script.


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© J. Carstensen (Quantum Mech.)