Kramers-Theorem

Named after Hendrik Anthony Kramers Kramers theorem, also named called Kramers degeneracy, is a theoretical quantum mechanical statement on the degree of degeneracy of the energy states of a system with half-integer total spin (for example, any number of bosons and an odd to fermions such as electrons). Accordingly, in the case that the system acts on an electric field maximum and the total of the spin system is a half-integer, each power state of at least two -fold degenerate, and also in any case an even number degenerate. Acts on the system under consideration, for example, explicitly a magnetic field, so the statement of the Kramers theorem does not apply.

It follows from the Kramers theorem, that the degeneracy of any energy state can never be completely abolished by merely applying an electric field.

Mathematical formulation

For the states of the system, the Bra- Ket notation is used. It is the semi -linear unitary operator which causes a time-reversal. For a system of particles with respective spin and thus the total spin.

A prerequisite is the Hamiltonian of the many-body system describes time-reversal invariant. It follows for an arbitrary total spin that if an eigenstate of the energy eigenvalue, then even such a eigenstate of the eigenvalue is:

The fact that for half-integer total spin is linearly independent of, and follows from the semi -linearity of, especially the property for:

The Kramers theorem applies in the presence of electric fields, since these do not affect the time-reversal invariance of the Hamiltonian, while the presence of magnetic fields cancels the time-reversal invariance of the Hamiltonian. (For the form of the Hamiltonian see charged, spinless particle in an electromagnetic field, other additive terms for the consideration of the spins can not recover the time-reversal invariance. )