Coefficients of wave function of a hybrid orbital

[Samuelriesterer]

Assuming the 2s and 2p wavefunctions are normalized, determine the coefficients in the hybrid orbital:

Ψ(sp3) = aΨ(2s) + aΨ(2px) + aΨ(2py) + aΨ(2pz) (the other 3 hybrids have – signs for some of the coefficients.

I have no clue where to start. I know this is a tetrahedral hybrid orbital but can't seem to grasp this question.

 

[fzero]

Are the other 3 hybrids completely specified for you? If so, you should impose orthonormality to determine the coefficients for this wavefunction.

 

[Samuelriesterer]

No this is the whole problem. I keep thinking it must be entirely simple but I guess it's not. (My teacher has a way of omission to the point of ridiculous)

 

[fzero]

Well, you've written down a fairly specific expression with the same coefficient $a$ in front of each term. Assuming that's given, then $a$ should be fixed by normalizing the wavefunction. You should verify that the 4 hybrid orbitals are orthonormal to complete the exercise.

 

[Samuelriesterer]

If I assume orthonormality and the other 3 hybrids have some coefficients as negative, then am I right to conclude that the coefficients for this one are all +1/2?

 

[fzero]

Yes, normalizing this requires $a=1/2$.