load("orthopoly")$

be(n):= 2/n;

psi(n,l,m,r,th,ph):= sqrt((2/n)^3*factorial(n-l-1)/(2*n*factorial(n+l)))*exp(-be(n)*r/2)*(be(n)*r)^l*gen_laguerre(n
 -l-1,2*l+1,be(n)*r)*spherical_harmonic(l,m,th,ph);

psi(1,0,0,r,th,ph);

int3(f):= integrate(ingegrate(integrate(f,ph,0,2*%pi),th,0,%pi),r,0,inf);

int3(conjugate(psi(1,0,0,r,th,ph))*psi(1,0,0,r,th,ph));

int3(f):= integrate(integrate(integrate(f,ph,0,2*%pi),th,0,%pi),r,0,inf);

int3(conjugate(psi(1,0,0,r,th,ph))*psi(1,0,0,r,th,ph));

int3(conjugate(psi(1,0,0,r,th,ph))*psi(2,0,0,r,th,ph));

shy(l,m,th,ph):= sqrt(factorial(l-m)/factorial(l+m))*assoc_legendre(l,m,cos(th))*exp(
 -m*phi);

shy(1,0,th,ph);

shy(l,m,th,ph):= sqrt(factorial(l-m)/factorial(l+m))*assoc_legendre_p(l,m,cos(th))*exp(
 -m*phi);

shy(l,m,th,ph):= sqrt(factorial(l-m)/factorial(l+m))*assoc_legendre_p(l,m,cos(th))*exp(
 -%i*m*phi);

shy(l,m,th,ph):= sqrt(factorial(l-m)/factorial(l+m))*assoc_legendre_p(l,m,cos(th))*exp(%i*m*phi);

shy(1,0,th,ph);

spherical_harmonic(1,0,th,ph);

psi(n,l,m,r,th,ph):= sqrt((2/n)^3*factorial(n-l-1)/(2*n*factorial(n+l)))*exp(-be(n)*r/2)*(be(n)*r)^l*gen_laguerre(n
 -l-1,2*l+1,be(n)*r)*shy(l,m,th,ph);

int3(conjugate(psi(1,0,0,r,th,ph))*psi(2,0,0,r,th,ph));

remvalue(all);

spherical_harmonic(0,0,th,ph);

spherical_harmonic(0,1,th,ph);

spherical_harmonic(1,0,th,ph);

spherical_harmonic(1,1,th,ph);

psi(n,l,m,r,th,ph):= sqrt((2/n)^3*factorial(n-l-1)/(2*n*factorial(n+l)))*exp(-be(n)*r/2)*(be(n)*r)^l*gen_laguerre(n
 -l-1,2*l+1,be(n)*r)*sherical_harmonic(l,m,th,ph);

int3(conjugate(psi(1,0,0,r,th,ph))*psi(2,0,0,r,th,ph)*sin(th)*r^2);

int3(conjugate(psi(1,0,0,r,th,ph))*psi(1,0,0,r,th,ph)*sin(th)*r^2);

psi(n,l,m,r,th,ph):= sqrt((2/n)^3*factorial(n-l-1)/(2*n*factorial(n+l)))*exp(-be(n)*r/2)*(be(n)*r)^l*gen_laguerre(n
 -l-1,2*l+1,be(n)*r)*spherical_harmonic(l,m,th,ph);

int3(conjugate(psi(1,0,0,r,th,ph))*psi(1,0,0,r,th,ph)*sin(th)*r^2);