Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

We will need to find a number of single-particle states equal to at least half the
number of electrons in the system, assuming that the many-body wave function
is build as an anti-symmetrized product of single-electron states taken as spinup
and spin-down pairs (as in the case of He and H 2 ). Of course the resulting
state will have zero magnetization (S = 0). The exact number of electrons in a
crystal depends upon its atomic composition. Even if we assume the minimal
case of one electron per atom, we still have N electrons and we need to calculate
N/2 states, with N → ∞. How can we deal with such a macroscopic number
of states?
9.1.2 Bloch Theorem
At this point symmetry theory comes to the rescue under the form of the Bloch
theorem. Let us indicate with T the discrete translation operator: T ψ(x) =
ψ(x + a). What is the form of the eigenvalues and eigenvectors of T ? It can be
easily verified (and rigorously proven) that T ψ(x) = λψ(x) admits as solution
ψ k (x) = exp(ikx)u k (x), where k is a real number, u k (x) is a periodic function
of period a: u k (x + a) = u k (x). This result is easily generalized to three
dimensions, where k is a vector: the Bloch vector. States ψ k are called Bloch
states. It is easy to verify that for Bloch states the following relation hold:
ψ k (x + a) = ψ k (x)e ika . (9.1)
Let us classify our solution using the Bloch vector k (in our case, a onedimensional
vector, i.e. a number). The Bloch vector is related to the eigenvalue
of the translation operator (we remind that H and T are commuting operators).
Eq.(9.1) suggests that all k differing by a multiple of 2π/a are equivalent
(i.e. they correspond to the same eigenvalue of T ). It is thus convenient to
restrict to the following interval of values for k: k: −π/a < k ≤ π/a. Values
of k outside such interval are brought back into the interval by a translation
G n = 2πn/a.
We must moreover verify that our Bloch states are compatible with PBC.
It is immediate to verify that only values of k such that exp(ikL) = 1 are
compatible with PBC, that is, k must be an integer multiple of 2π/L. As
a consequence, for a finite number N of atoms (i.e. for a finite dimension
L = Na of the box), there are N admissible values of k: k n = 2πn/L, con
n = −N/2, ..., N/2 (note that k −N/2 = −π/a is equivalent to k N/2 = π/a). In
the thermodynamical limit, N → ∞, these N Bloch vectors will form a dense
set between −π/a and π/a, in practice a continuum.
9.1.3 The empty potential
Before moving towards the solution, let us consider the case of the simplest
potential one can think of: the non-existent potential, V (x) = 0. Our system
will have plane waves as solutions: ψ k (x) = (1/ √ L)exp(ikx), where the factor
ensure the normalization. k may take any value, as long as it is compatible
with PBC, that is, k = 2πn/L, with n any integer. The energy of the solution
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