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P orbital orthonormality

  1. Nov 20, 2007 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    3. 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??
  2. jcsd
  3. Nov 20, 2007 #2
    anyone? :(
  4. Nov 20, 2007 #3


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    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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