One electron approximation

5.1 One electron approximation

we will have a closer look on a many particle system in which the one electron approximation is allowed, i.e.

each electron will occupy a one electron state independently of all the other electrons
electrons in different one electron states do not ”recognize” each other
within one state the Pauli principle has to be obeyed, i.e. each state can only be occupied once; taking into account the spin two electrons can occupy one state.
Since the one electron states have been calculated by diagonalization of the Hamiltonian, there exists no overlap between different one electron states.
Consequently there exists no scattering between those states
the system does not change in time

All perturbations from outside

will change the Hamiltonian for a short time
will introduce particles which are not Eigenstates of the Hamiltonian
will induce particles which couple to the electrons and lead to scattering of electrons into different states

Electrons will be scattered into mixed states $ψ = \sum_{n} c_{n} f_{n}$

this mixed state will relaxate into a pure state $f_{n}$ with a probability $| c_{n} |^{2}$
Since a macroscopic state is filled with many electrons, this is equivalent to the statement that a mixed state will relaxed into pure states according to the $| c_{n} |^{2}$ .

The dominant and most fundamental perturbation of all systems is the temperature; e.g. the Brownian motion of the particles will excite electrons leading to a constant interaction between electrons. There exists a continuous flux of electrons into neighboring states. This flux is defined by $| c_{n} |^{2}$ , i.e. the mixture of pure states. The coefficients $| c_{n} |^{2}$ are calculated from the matrix elements

⟨ f_{m} | P e r t u r b a t i o n | f_{n} ⟩

(5.1)