3.5 Why does a quantum mechanical state change?

Since all Eigenstates of the Hamiltonian are stable, one needs at least a short perturbation of the system to get an excited state; e.g.:

• A beam of

  • electrons

  • electromagnetic radiation

  • positrons

  • neutrons

  • ...

• The scattering may be elastic or inelastic, i.e. momentum and/or energy may be exchanged.

• Electrons can be scattered by core electrons

• There exists an interaction between electrons and phonons

• The thermal energy of the system itself causes an excited state since only at \(T = 0\) the Fermi energy describes the highest occupied energy state. We find a dynamic equilibrium between states steadily increasing or decreasing the energy.

• Surfaces and defects in a crystal change the Hamiltonian of the perfect solid leading to additional electronic transitions.

For all these processes according to Eq. (3.18) the transfer matrix element can be calculated; they quantize the probability for these transitions.
The excited state may loose energy by the same processes as described above:

• by electromagnetic radiation

• the same particles, which moved into the solid, loose energy and momentum and will be detected in the scattered beam.

• thermal processes (e.g. phonons)

• at defects

• on surfaces

REMARK: Principally you do not need superconductivity to find zero ohmic resistance; once a current flow started (described by the \(k\)-distribution of electrons, which are Eigenstates of the Hamiltonian) it would be stable for infinite time. You need defects within the crystal or phonons to reduce the current by scattering.
For every well defined measurement and if knowing the underlying processes, quantum mechanics provides the adequate calculation rule.

1. Which is the starting state?

2. Which is the end state?

3. Which expectation value do I have to calculate?

Examples:
Scattering experiment:

1. incoming free particle

2. outgoing free particle

3. Scattering cross section

Recombination processes:

1. State in the first band

2. State in the second band

3. Cross section of a trap (a surface)/Probability for a radiative transition

• This allows to calculate which processes have the highest transition probability

• Often approximation methods are used to calculate matrix elements

• Essential for all calculations is the Hamiltonian of the perfect system, i.e. the Eigenvalues and Eigenvectors. Quantum mechanical Hamiltonians almost always look like:

  \begin{equation*} H = \sum_i E_i |f_i\rangle \langle f_i| \end{equation*}

  (3.21)

© J. Carstensen (Quantum Mech.)