Castles and quantum mechanics

How are castles and quantum mechanics related? One connection is rook polynomials.

The rook is the chess piece that looks like a castle, and used to be called a castle. It can move vertically or horizontally, any number of spaces.

A rook polynomial is a polynomial whose coefficients give the number of ways rooks can be arranged on a chess board without attacking each other. The coefficient of xk in the polynomial Rm,n(x) is the number of ways you can arrange k rooks on an m by n chessboard such that no two rooks are in the same row or column.

The rook polynomials are related to the Laguerre polynomials by

Rm,n(x) = n! xn Lnmn(-1/x)

where Lnk(x) is an “associated Laguerre polynomial.” These polynomials satisfy Laguerre’s differential equation

x y” + (n+1-x) y‘ + k y = 0,

an equation that comes up in numerous contexts in physics. In quantum mechanics, these polynomials arise in the solution of the Schrödinger equation for the hydrogen atom.

Related: Relations between special functions

One thought on “Castles and quantum mechanics

  1. Nice, I did not know about rook polynomials! A different connection between castles and quantum mechanics: during the last Ghent Light Festival, a result of a quantum mechanical computation (of nanowire structures) was projected on the Medieval castle “Gravensteen”. Picture here.

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