P orbital orthonormality

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Homework Statement



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

Homework Equations





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??
 

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  • #2
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anyone? :(
 
  • #3
nrqed
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Homework Statement



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

Homework Equations





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 [itex] Y_1^0 [/itex] but the other two are
[tex] Y_1^{\pm 1} = \mp { \sqrt{\frac{3}{8 \pi}} \sin \theta e^{\pm i \phi} [/tex]
 

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