Kramers theorem
Encyclopedia
The Kramers degeneracy theorem states that the energy level
s of systems with an odd number of electron
s remain at least doubly degenerate
in the presence of purely electric field
s (i.e. no magnetic field
s). It was first discovered in 1930 by H. A. Kramers
as a consequence of Breit equation
.
As shown by Eugene Wigner in 1932, it is a consequence of the time reversal invariance of electric field
s, and follows from an application of the antiunitary T-operator to the wavefunction of an odd number of electrons. The theorem is valid for any configuration of static or time-varying electric fields.
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...
s of systems with an odd number of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s remain at least doubly degenerate
Degenerate energy level
In physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, and in Quantum Mechanics it is represented mathematically by the Hamiltonian for the system having more...
in the presence of purely electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
s (i.e. no magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
s). It was first discovered in 1930 by H. A. Kramers
Hendrik Anthony Kramers
Hendrik Anthony "Hans" Kramers was a Dutch physicist.-Background and education:...
as a consequence of Breit equation
Breit equation
The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and...
.
As shown by Eugene Wigner in 1932, it is a consequence of the time reversal invariance of electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
s, and follows from an application of the antiunitary T-operator to the wavefunction of an odd number of electrons. The theorem is valid for any configuration of static or time-varying electric fields.