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Random Matrices in Nuclear Physics

The document discusses random matrices in the context of nuclear physics, specifically focusing on the Hamiltonian matrix and its eigenvalues. It outlines a method to model complex nuclear states using matrices with random entries, leading to the exploration of the probability distribution of eigenvalues. The document also introduces Wigner's semicircle rule, which describes the distribution of eigenvalues for certain random matrices, highlighting its relevance in various fields of mathematics and physics.

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haidaica9981
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12 views1 page

Random Matrices in Nuclear Physics

The document discusses random matrices in the context of nuclear physics, specifically focusing on the Hamiltonian matrix and its eigenvalues. It outlines a method to model complex nuclear states using matrices with random entries, leading to the exploration of the probability distribution of eigenvalues. The document also introduces Wigner's semicircle rule, which describes the distribution of eigenvalues for certain random matrices, highlighting its relevance in various fields of mathematics and physics.

Uploaded by

haidaica9981
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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probability theory → random variables FFFFF

Random Matrices
Let |αi represent the state of a nucleus, and H be the Hamiltonian matrix for the nucleus. We do not
necessarily know that these are eigenstates of H, although since H is a real-valued symmetric matrix
we know it has real eigenvalues. In reality there are a huge number of states for a complex nucleus like
238 U, far more than we can ever hope to measure exactly. In the 1950s, Wigner and Dyson realized that

we could model a system like this using a matrix with a large number of random entries. That way,
even if we don’t get any particular transition energy right, we may expect on average to see the overall
behavior of the system; e.g., within a keV or so, how much resonance do we expect to see? Thus, the
basic question we want to answer is the following: given a random H, what is the probability distribution
of the eigenvalues?
Let A be a real-valued symmetric n × n matrix with i.i.d. N (0, 1) entries; not surprisingly, A is called
a random matrix. Assume that n is very large, but finite.

(a) Show that (for a generic symmetric matrix B) and k ∈ N,


n
X
k
tr(B ) = λi k .
i=1

(b) Let λ be an eigenvalue of the random matrix A. Show that


  
 k/2 1 k
n · k even
hλk i = 1 + k/2 k/2 .
0 k odd

This is a challenging combinatorics problem, so be careful!

Suppose we could find a probability distribution on λ that had the properties found in (b); then we
might say that that is indeed the same as the distribution of the eigenvalues of A themselves. Mathemat-
ically, that is not technically correct and further proof would be required, but for this problem that is all
that is necessary. Consider the following probability distribution on λ:
r
1 λ2 √
p(λ) = √ 1− Θ(2 n − |λ|).
π n 4n

(c) Show that this probability distribution has the same moments as you found in part (b).

The fact that this is the proper distribution to describe the eigenvalues of A is known as Wigner’s
semicircle rule (as the graph of p(λ) is a semicircle). While we didn’t prove it, this is also true for many
different types of random variables Aij .
Random matrix theory is a fairly modern field of mathematics with applications to number theory
and computational linear algebra, as well as nuclear physics.

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