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• Users: Cryxic
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1. Normalising wave functions.

Your result is close but awkward on the first part. Go back and redo the integral. You should get the square root of 2 divided by a, not 1 divided by a. The bounds of integration are 0 to a. Given what you've presented here, the width of the well is a. You have the right idea for the second...
2. Dealing with addition of cosntant to wave equation? Spherical Harmonics

Shankar's Principle of Quantum Mechanics has a similar question, and that question asks for the possible L(z) angular momenta and their probabilities. That's what I assumed this person was trying to find, although that's not stated anywhere. Either way, it's all simple. The r^2 part will just...
3. Dealing with addition of cosntant to wave equation? Spherical Harmonics

But even if it wants the azimuthal quantum numbers (little L)...that's easy too: just 0 and 2 (given it's a 2nd degree equation)...
4. Dealing with addition of cosntant to wave equation? Spherical Harmonics

No the question is about the z part of the angular momentum (implicitly), otherwise you'd be right. But in this case, no phi dependence, and so measured L(z) will always be 0.
5. Dealing with addition of cosntant to wave equation? Spherical Harmonics

Everyone is making this too complicated. The allowed angular momentum is just 0. There is no phi dependence in that wavefunction! You just have r and z. You're always going to get 0 with 100% chance.