- #1
Isaac Pepper
- 31
- 1
Homework Statement
Consider an electron in a state described by angular wavefunction $$\psi(\theta,\phi)=\sqrt{\frac{3}{4 \pi}}\sin \theta \cos \phi$$ Here θ and φ are the polar and azimuthal angles, respectively, in the spherical coordinate system.
i. Calculate the probability that a simultaneous measurement of the electron orbital angular momentum squared (L2) and the Z component of the electron orbital angular momentum (Lz) will give 2ħ2 and ħ respectively.
ii. What is the probability of measuring L2 = 2ħ2 ?
iii. What is the probability of measuring Lz = 0 ?
Homework Equations
$$\psi(\theta,\phi) = \sum_{l=0}^{l=\infty} \sum_{m=-l}^{m=+l}c_{l,m} Y_{l,m} (\theta, \phi)$$
$$c_{l,m} = \int Y^{*}_{lm}(\theta, \phi) \psi (\theta, \phi) d\Omega$$
The Attempt at a Solution
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In my notes I am told that $$\vert c_{l,m} \vert ^2$$ is the probability that a simultaneous measurement of L2 and Lz on a particle described by the wavefunction ψ gives l(l+1)ħ2 AND mħ.
The probability that a measurement of L2 will give l(l+1)2 is simply the sum of the probabilities for each possible m state: $$P(l) = \sum_{m=-l}^l \vert c_{l,m} \vert ^2$$
Looking up the wavefunction in a table, it seems to be a spherical harmonic with l=1 and m=0
which means m can range between -1 and 1.
Now for question i. the answer would simply be $$\vert c_{1,1} \vert ^2$$ ...but how can I calculate what c is?
I guess my problem is similar for part ii. as the answer would be $$P(l) = \sum_{m=-l}^l \vert c_{l,m} \vert ^2$$ where m can be -1, 0 and 1...but I'm stumped on how to calculate cl,m
Any help would be greatly appreciated !
Thank you
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