# P orbital orthonormality

## Homework Statement

I have to show that the p orbital wavefunctions are orthonormal to eath other in l=1 subspace.

## The Attempt at a Solution

looking at my notes, I thought the expressions for p orb wavefunctions were:
Psi_px=sqrt(3/4pi) cos(phi) sin(theta)
Psi_py=sqrt(3/4pi) cos(theta) sin(theta)
Psi_pz=sqrt(3/4pi) cos(theta)

I can show that they are all orthogonal to each other (for instance, integral of psi_px times psi_py gives 0) over the range of 0<theta<pi, 0<phi<2pi.

However, I cannot show that they are all normal (square of each psi is equal to 1).... do I have the wrong psi expressions or am I integrating wrong??

anyone? :(

nrqed
Homework Helper
Gold Member

## Homework Statement

I have to show that the p orbital wavefunctions are orthonormal to eath other in l=1 subspace.

## The Attempt at a Solution

looking at my notes, I thought the expressions for p orb wavefunctions were:
Psi_px=sqrt(3/4pi) cos(phi) sin(theta)
Psi_py=sqrt(3/4pi) cos(theta) sin(theta)
Psi_pz=sqrt(3/4pi) cos(theta)

I can show that they are all orthogonal to each other (for instance, integral of psi_px times psi_py gives 0) over the range of 0<theta<pi, 0<phi<2pi.

However, I cannot show that they are all normal (square of each psi is equal to 1).... do I have the wrong psi expressions or am I integrating wrong??

Your third expression is indeed the spherical harmonic $Y_1^0$ but the other two are
$$Y_1^{\pm 1} = \mp { \sqrt{\frac{3}{8 \pi}} \sin \theta e^{\pm i \phi}$$