Time Reversal and Xxxxxxx Degeneracy Sample Clauses

Time Reversal and Xxxxxxx Degeneracy. In 1930 H. A. Xxxxxxx in his study of the Schrödinger equation of an electron with spin in the absence of a magnetic field, found a mapping T that given a solution |ψ⟩ with energy E gives another solution T |ψ⟩ with the same energy [11]. For systems with odd number of spin half electrons these solutions are orthogonal and therefore lead to a degeneracy in the spectrum, the Kramers degeneracy. A couple of years later Xxxxxx pointed out that the mapping Xxxxxxx found is simply time reversal and that the degeneracy is a manifestation of the presence of time reversal symmetry [12]. Symmetries in quantum mechanics can be represented either by unitary and linear operators or antiunitary and antilinear operators, according to a theorem also due to Xxxxxx [13]. We will see that time reversal is nec- essarily in the latter, somewhat less familiar category. There is a crucial difference between the two groups in the fact that while unitary symmetries lead to a conserved quantity (e.g. translation symmetry to conservation of momentum and rotation symmetry to conservation of angular momentum) antiunitary symmetries in general do not. The effect of antiunitary sym- metries (time reversal) is thus more subtle, as reflected in the Kramers degeneracy, but just as important. In addition to the Kramers degeneracy of energy eigenvalues, the pres- ence of time reversal imposes a symmetry on Hamiltonians and scattering matrices. Furthermore, in scattering, transmission eigenvalues are twofold degenerate. The exact symmetries of the Hamiltonian are usually given in terms of quaternions (or Pauli sigma matrices) in which they take a simple form. All the above mentioned properties are of importance in any quan- tum theory of transport. In the literature, these have become a common knowledge and are used as such. For a newcomer, it can take some time to dig up definitions and proofs of these important properties, in particular since topics such as antiunitary operators and quaternions are often not included in textbooks. In the case of the Xxxxxxx degeneracy of trans- mission eigenvalues, the proofs that exist in the literature are somewhat convoluted and not given directly in terms of the scattering matrix. In this section we therefore represent definitions and proofs in a unified man- ner, and an alternative proof of the Kramers degeneracy of transmission eigenvalues. We start by a review of the mathematical concepts of antiunitary opera- tors and quaternions. Time reversal is t...