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Energy levels of complex systems: background

Freeman Dyson
Scientist

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This is another chapter in which again, I took somebody else's problem and converted it into a mathematical theory. It was Wigner's idea that one would describe a complex system, such as a heavy nucleus involving a lot of strongly interacting particles, by a random matrix. That is to say, a matrix is supposed to be just the Hamiltonian of the system, which describes the physical nature of the system. Since we really know nothing about the system except that it's a very complicated black box, or in this case a black sphere with a whole lot of things going on inside which we can't observe, let's just pretend we know nothing and say that we have absolutely a state of total ignorance about the Hamiltonian - then what can we say should be true on the average? So that's the question which Wigner asked. If you take the average of all possible Hamiltonians in a certain class, then how should they behave? And that turns out to be a very interesting question. First of all, you have to define very carefully the ensemble of matrices that you're talking about so that you have a precise definition of probabilities, and then you have to figure out how to calculate averages. And so Wigner was able to do that. He defined an ensemble, which is just what we now call the Gaussian orthogonal ensemble, which essentially says that the probability of a certain Hamiltonian which is represented by a matrix, is just the independent Gaussian distribution of each of the matrix elements, multiplied by a normalisation factor to make the probability come to one. And that's it. So, simply, as all the matrix elements are statistically independent and Gaussian, that's it. That gives you a well-defined ensemble, and he imposed the condition that they should be real and symmetric, which means the Hamiltonian is time reversal invariant. And then you can get well-defined questions and well-defined answers, and Wigner was able to show that in this case - Gaussian orthogonal ensemble - that the Eigen values of the matrix which correspond to the energy levels of the system have a semicircular distribution. That means the probability distribution of the Eigen values goes like the square root of one over one minus x 2 - no, the square root of one minus x 2, where x is the difference between the energy and the average energy. And so if you plot the curve of probability of density of Eigen values against energy, it's a semicircle. So that was a non-trivial result, which Wigner proved around 1946, I think. Then it was taken up by two Frenchmen, Mehta and Godard, and they went much further, and they were able to calculate many other things, and they found the spacing distribution, which is a much harder thing to calculate. To calculate the statistical distribution of the spacings between nearest neighbour levels, which is a very non-trivial problem, they were able to calculate that by very clever tricks.

Freeman Dyson (1923-2020), who was born in England, moved to Cornell University after graduating from Cambridge University with a BA in Mathematics. He subsequently became a professor and worked on nuclear reactors, solid state physics, ferromagnetism, astrophysics and biology. He published several books and, among other honours, was awarded the Heineman Prize and the Royal Society's Hughes Medal.

**Title: **Energy levels of complex systems: background

**Listeners:**
Sam Schweber

Silvan Sam Schweber is the Koret Professor of the History of Ideas and Professor of Physics at Brandeis University, and a Faculty Associate in the Department of the History of Science at Harvard University. He is the author of a history of the development of quantum electro mechanics, "QED and the men who made it", and has recently completed a biography of Hans Bethe and the history of nuclear weapons development, "In the Shadow of the Bomb: Oppenheimer, Bethe, and the Moral Responsibility of the Scientist" (Princeton University Press, 2000).

**Tags:**
Hamiltonian, Gaussian orthogonal ensemble, Eugene Wigner, Peter Goddard, Madan Lal Mehta

**Duration:**
3 minutes, 51 seconds

**Date story recorded:**
June 1998

**Date story went live:**
24 January 2008