-- Verify eigenfunctions for 2d hydrogen E(n) = -k^2 mu / (2 hbar^2 (n + 1/2)^2) -- range of m is -n...n psi(n,m) = 1 / a0 / sqrt(pi) * (n + 1/2)^(-3/2) * sqrt((n - abs(m))! / (n + abs(m))!) * (2 r / a0 / (n + 1/2))^abs(m) * Laguerre(2 r / a0 / (n + 1/2), n - abs(m), 2 abs(m)) * exp(-r / a0 / (n + 1/2)) * exp(i m phi) a0 = hbar^2 / (k mu) -- Laguerre polynomial (y is a local variable) Laguerre(x,n,m,y) = eval(y^(-m) exp(y) / n! d(exp(-y) y^(n + m), y, n), y, x) psi0 = psi(0,0) psi1 = psi(1,0) psi2 = psi(2,0) psi3 = psi(3,0) psi0 psi1 psi2 psi3 "Verify Schroedinger equation (1=ok)" Laplacian(psi) = d(psi,r,2) + d(psi,r) / r + d(psi,phi,2) / r^2 -hbar^2 / (2 mu) Laplacian(psi0) - k psi0 / r == E(0) psi0 -hbar^2 / (2 mu) Laplacian(psi1) - k psi1 / r == E(1) psi1 -hbar^2 / (2 mu) Laplacian(psi2) - k psi2 / r == E(2) psi2 -hbar^2 / (2 mu) Laplacian(psi3) - k psi3 / r == E(3) psi3 "Verify normalization (1=ok)" I(f) = do( f = f r, -- for integrating over polar coordinates f = defint(f, phi, 0, 2 pi), f = integral(f,r), 0 - eval(f,r,0) ) I(conj(psi0) psi0) I(conj(psi1) psi1) I(conj(psi2) psi2) I(conj(psi3) psi3)