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 *x ^{k}* in the polynomial

*R*

_{m,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

*R*_{m,n}(*x*) = *n*! *x ^{n}*

*L*

_{n}

^{m–n}(-1/

*x*)

where *L _{n}^{k}*(

*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 SchrÃ¶dinger equation for the hydrogen atom.

**Related**: Relations between special functions

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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.