Theoretical Nanophysics 6/2023

The phrase `Kramers pair' referes to ...

• an energy eigenvalue, or two energy eigenstates

• two different energy eigenvalues

• an energy eigenstate

• a matrix and its adjoint

Kramers pairs are characteristic of which symmetry class?

What is the difference between time reversal for integer-spin / half-integer-spin particles?

• T is unitary / antiunitary

• T squares to +1 / -1

• T is linear / antilinear

• T involves / does not involve complex conjugation

For spin-1/2 particles, e.g., electrons, time reversal ...

• flips both momentum and spin

• flips momentum but does not flip spin

• flips spin but does not flip momentum

• does not flip either momentum or spin

If the Hamiltonian of an integer-spin particle is time-reversal symmetric, then it can be written as a matrix that is ...

• orthogonal

• unitary

• real

• antisymmetric

Why do we call the `orthogonal ensemble' of Hamiltonians `orthogonal'?

• any two such Hamiltonians are orthogonal to each other

• they are orthogonal matrices

• any two rows of such a Hamiltonian are orthogonal to each other

• they are diagonalized by orthogonal matrices

How many real numbers parametrize the Hamiltonian matrices of the 3x3 orthogonal ensemble?

• 3

• 6

• 9

• 12

What is the domain of the Wigner-Dyson distribution for the 2x2 Gaussian Orthogonal Ensemble?

• nonnegative real numbers

• triplets of real numbers

• complex numbers

• the [0,1] interval

Which quantum-mechanical model has Hamiltonians from the orthogonal class?

• spinless electrons, no magnetic field

• spinless electrons with magnetic field

• spinful electrons with spin-orbit interaction and no magnetic field

• spinful electrons with spin-orbit interaction and magnetic field

Which function resembles the Wigner-Dyson level statistics of the 2x2 Gaussian Orthogonal Ensemble?