Quantum Physics Ex 5-Eigenproblem Solved

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Exercise 1: Eigenproblem

Consider a random Hermitian matrix A of size N.

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

λi = λi+1− λi,

and ∆¯λ is the average ∆λi.

  • Optional: Compute the average spacing ∆¯λ locally, i.e., over a di erent number of levels around λi (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 si de ned in the previous exercise, accumulating values of si 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) = asα exp(−bsβ)

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.

  • EX5-lkcb3o.zip