- #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 (L

^{2}) and the Z component of the electron orbital angular momentum (L

_{z}) will give 2ħ

^{2}and ħ respectively.

ii. What is the probability of measuring L

^{2}= 2ħ

^{2}?

iii. What is the probability of measuring L

_{z}= 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 L

^{2}and L

_{z}on a particle described by the wavefunction ψ gives l(l+1)ħ

^{2}AND mħ.

The probability that a measurement of L

^{2}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 c

_{l,m}

Any help would be greatly appreciated !

Thank you

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