2 Schrödinger Equation
The framework of most solid state physics theory is theSchrödinger equationof non relativistic
quantum mechanics:
i~∂Ψ
∂t=
−~^2
2 m
∇^2 Ψ +V(r,t)Ψ (1)The most remarkable thing is the great variety ofqualitativly different solutionsto Schrödinger’s
equation that can arise. In solid state physics you can calculate all properties with the Schrödinger
equation, but the equation is intractable and can be only solved with approximations.
For solving the equation numerically for a given system, the system and therewithΨare discretized
into cubes of the relevant space. To solve it for an electron, about 106 elements are needed. This is
not easy, but possible. But let’s have a look at an other example:
For a gold atom with 79 electrons there are (in three dimensions) 3 · 79 terms for the kinetic energy in
the Schrödinger equation. There are also 79 terms for the potential energy between the electrons and
the nucleus. With the interaction of the electrons there are additionally^792 ·^78 terms. So the solution for
Ψwould be a complex function in 237 (!) dimensions. For a numerically solution we have to discretize
each of these 237 axis, let’s say in 100 divisions. This would give 100237 = 10^474 hyper-cubes, where
a solution is needed. That’s a lot, because there are just about 1068 atoms in the Milky Way galaxy!
Out of this it is possible to say:
The Schrödinger equation explains everything but can explain nothing.
Out of desperation: The model for a solid is simplified until the Schrödinger equation can be solved.
Often this involves neglecting the electron-electron interactions. Back to the example of the gold atom
this means that the resulting wavefunction is considered as a product of hydrogen wavefunctions.
Because of this it is possible to solve the new equation exactly. The total wavefunction for the
electrons must obey the Pauli exclusion principle. The sign of the wavefunction must change when
two electrons are exchanged. The antisymmetricNelectron wavefunctioncan can be written as a
Slater determinant:
Ψ(r 1 ,r 2 ,...,rN) =1
√
N!
∣∣
∣∣
∣∣
∣∣
∣∣Ψ 1 (r 1 ) Ψ 1 (r 2 ) ... Ψ 1 (rN)
Ψ 2 (r 1 ) Ψ 2 (r 2 ) ... Ψ 2 (rN)
..
...
.
... ..
.
ΨN(r 1 ) ΨN(r 2 ) ... ΨN(rN)∣∣
∣∣
∣∣
∣∣
∣∣(2)
Exchanging two columns changes the sign of the determinant. If two columns are the same, the
determinant is zero. There are stillN!, in the example79! (≈ 10100 = 1googol)terms.