Theoretical Nanophysics 7/2023

In the vector space of 4x4 real symmetric matrices, how many parameters do we need to tune to find a ground-state degeneracy?

• 1

• 2

• 3

• 4

2x2 Hermitian matrices form a 4-dimensional real vector space. 2x2 Hermitian matrices with a degenerate spectrum form a subspace of that vector space. What is the dimension of that subspace?

• 0

• 1

• 2

• 3

What are the two axes used when we plot the Wigner semicircle law?

• energy and bandwidth

• level spacing and probability

• energy and density of states

• bandwidth and level spacing

Which setup is NOT described by random matrix theory?

• disordered systems showing localization

• chaotic quantum biliards without spin-orbit interaction

• ballistic but chaotic scattering regions

• chaotic quantum billiards with spin-orbit interaction

Which level-spacing distribution applies in the absence of level repulsion?

Order our three matrix ensembles according to the strength of level repulsion!

• GOE < GSE < GUE

• GUE < GOE < GSE

• GSE < GOE < GUE

• GOE < GUE < GSE

In the lecture, we have called the parameter J the `bandwidth'. What is the actual energy bandwidth in the GUE? (Assume that the matrix size N tends to infinity.)

• J

• 2J

• 3J

• 4J

How many real numbers parametrize a 4x4 matrix of the Gaussian Symplectic Ensemble?

• 6

• 13

• 16

• 28

What is the effective interaction potential between the levels in the Gaussian matrix ensembles?

• 1/distance

• Yukawa

• van der Waals

• logarithmic

Fine tuning of how many parameters of a GSE matrix is required to reach a level degeneracy?

• 2

• 3

• 4

• 5