Kramer's theorem

The by Hans Kramers called Kramers theorem , also called Kramers degeneracy called, is a theoretical , quantum mechanical statement on the degeneracy of energy ; states of a system with half-integer total spin (z. B. any number of bosons and odd at Fermions like electrons ). Accordingly, in the event that at most one electric field acts on the system and the total spin of the system is half-integer, every energy state is at least twice degenerate and, moreover, in every case even-numbered degenerate. Affects the system under consideration z. If, for example, a magnetic field is explicitly stated , the statement of Kramer's theorem does not apply.

From the Kramers theorem it follows that the degeneration of any energy state can never be completely eliminated by simply applying an electric field.

Mathematical formulation

The Bra-Ket notation is used for the states of the system . Let it be the semilinear , unitary operator that causes a time reversal . For a system of -particles with respective spin and thus total spin applies . ${\displaystyle \left|\psi \right\rangle }$ ${\displaystyle T}$ ${\displaystyle N_{i}}$ ${\displaystyle s_{i}}$ ${\displaystyle S=\sum \nolimits _{i}N_{i}s_{i}}$ ${\displaystyle T^{2}=\exp \left(2\pi iS\right)}$

As a prerequisite, the Hamilton operator , which describes the many-body system, is time-reversed invariant . From this it follows for any total spin that if an eigenstate of is an energy eigenvalue , then such an eigenstate of is an eigenvalue : ${\displaystyle \left(THT^{\dagger }=H\right)}$ ${\displaystyle S}$ ${\displaystyle \left|\psi \right\rangle }$ ${\displaystyle H}$ ${\displaystyle E}$ ${\displaystyle T\left|\psi \right\rangle }$ ${\displaystyle H}$ ${\displaystyle E}$

${\displaystyle THT^{\dagger }=H\quad \wedge \quad H\left|\psi \right\rangle =E\cdot \left|\psi \right\rangle \qquad \Rightarrow \qquad HT\left|\psi \right\rangle =E\cdot T\left|\psi \right\rangle }$

The fact that for a half- integer total spin is linearly independent of follows from and the semilinearity of , especially the property for : ${\displaystyle \left|\psi \right\rangle }$ ${\displaystyle S}$ ${\displaystyle T\left|\psi \right\rangle }$ ${\displaystyle T^{2}=\exp \left(2\pi iS\right)=-1}$ ${\displaystyle T}$ ${\displaystyle T\lambda ={\overline {\lambda }}T}$ ${\displaystyle \lambda \in \mathbb {C} }$

${\displaystyle \exists n\in \mathbb {N} :N=2n+1\qquad \Rightarrow \qquad \nexists \alpha \in \mathbb {C} :T\left|\psi \right\rangle =\alpha \cdot \left|\psi \right\rangle }$

The Kramers theorem is valid in the presence of electric fields, since these do not influence the time reversal invariance of the Hamilton operator, while the presence of magnetic fields cancels the time reversal invariance of the Hamilton operator. (For the form of the Hamilton operator, see charged, spinless particle in the electromagnetic field ; further additive terms for taking the spins into account cannot restore the time-reversal invariance.) ${\displaystyle H}$ ${\displaystyle H}$

Individual evidence

1. ^ Albert Messiah : Quantum Mechanics. Volume 2, 2nd edition. Walter de Gruyter, Berlin 1985, ISBN 3-11-010265-X , p. 165.
2. ↑ ^{a b} Franz Schwabl : Quantum Mechanics for Advanced Students. 5th edition. Springer-Verlag, Berlin / Heidelberg 2008, ISBN 978-3-540-85075-5 , p. 232.

literature

• LD Landau , JM Lifschitz : Textbook of theoretical physics. Volume 3: Quantum Mechanics. 9th edition. Akademie-Verlag, Berlin 1990, ISBN 3-05-500067-6 .