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 Lnm-n(-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