[SOLVED] Quantum Physics Ex 5-Eigenproblem

Exercise 1: Eigenproblem

Consider a random Hermitian matrix A of size N.

• Diagonalize A and store the N eigenvalues λ_i in crescent order.
• Compute the normalized spacings between eigenvaluess_i = ∆λ_i/∆^¯λ where

∆λi = λi+1− λi,

and ∆^¯λ is the average ∆λ_i.

• Optional: Compute the average spacing ∆^¯λ locally, i.e., over a di erent number of levels around λ_i (i.e. N/100,N/50,N/10…N) and compare the results of next exercise for the di erent choices.

Exercise 2: Random Matrix Theory

Study P(s), the distribution of the s_i de ned in the previous exercise, accumulating values of s_i from di erent random matrices of size at least N = 1000.

• Compute P(s) for a random HERMITIAN matrix.
• Compute P(s) for a DIAGONAL matrix with random real entries.
• Fit the corresponding distributions with the function:

P(s) = as^α exp(−bs^β)

and report α,β,a,b.

• Optional: Compute and report the average hri of the following quantity

for the cases considered above. Compare the average hri that you obtain in the di erent cases.

Hint: if necessary neglect the       rst matrix eigenvalue.