- #1
Demon117
- 165
- 1
Homework Statement
This really is not a homework problem but I am studying for the qualifying exam upcoming. I came across an objective that I am not familiar with. I'm given a wave function made of a linear combination of spherical harmonics with complex coefficients. I'm asked to calculate the possible values of measurement for [itex]L^{2}[/itex] and [itex]L_{z}[/itex] which is of course straight forward. What I am unsure of though is it asks me to calculate the probability of obtaining those values. Perhaps I missed this in my undergraduate coursework somehow, but I'll list the wavefunction.
Homework Equations
[itex]\psi=A[(1+2i)Y_{3}^{-3}+(2-i)Y_{3}^{2}+\sqrt{10}Y_{2}^{2}][/itex]
The Attempt at a Solution
I've already calculated the normalization constant A, and found it via integration and the orthogonality condition to be
[itex]A=\frac{1}{4}\sqrt{\frac{7}{6\pi}}[/itex]
I know the values of angular momentum [itex]L^{2}[/itex] are [itex]12\hbar^{2}[/itex] corresponding to [itex]\left|3,-3\right\rangle[/itex], and [itex]\left|3,2\right\rangle[/itex]. Also, [itex]6\hbar^{2}[/itex] corresponding to [itex]\left|2,2\right\rangle[/itex]. I am utterly lost on how to calculate this probability. I have tried this:
[itex]|\left\langle 3, m\right|A[(1+2i)\left|3,-3\right\rangle+(2-i)\left|3, 2\right\rangle+\sqrt{10}\left|2,2\right\rangle]|^{2}[/itex] for m = -3, 2
&
[itex]|\left\langle 2, 2\right|A[(1+2i)\left|3,-3\right\rangle+(2-i)\left|3, 2\right\rangle+\sqrt{10}\left|2,2\right\rangle]|^{2}[/itex]
This gives me values that do not add up to 1 as expected. Is this right, and if so perhaps I am doing it incorrectly? Any pointers would be appreciated.