Atomic Physics - Orbital Angular Momentum Probability

[Isaac Pepper]

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 (L_z) will give 2ħ² and ħ respectively.

ii. What is the probability of measuring L² = 2ħ² ?
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

[/B]
In my notes I am told that $$\vert c_{l,m} \vert ^2$$ is the probability that a simultaneous measurement of L² and L_z 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 c_l,m

Any help would be greatly appreciated !
Thank you

 

[DrClaude]

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 (L_z) will give ħ² and ħ respectively.

ii. What is the probability of measuring L² = ħ² ?
iii. What is the probability of measuring L_z = 0 ?

Are you sure about the parts I indicated in red?

Looking up the wavefunction in a table, it seems to be a spherical harmonic with l=1 and m=0

Why m = 0?

Now for question i. the answer would simply be $$\vert c_{1,1} \vert ^2$$ ...but how can I calculate what c is?

$$c_{l,m} = \int Y^{*}_{lm}(\theta, \phi) \psi (\theta, \phi) d\Omega$$

 

[Isaac Pepper]

Sorry! Edited my post, it was meant to be 2ħ².

I am having trouble understanding the integral though... What are Y_l,m... and dΩ :(
I have a table with values of Y_0,0, Y_1,0, Y_1,1 etc...

So for example $$Y_{0,0} = \sqrt{\frac{1}{4 \pi}}$$

$$c_{0,0} = \int \sqrt{\frac{1}{4 \pi}} \sqrt{\frac{3}{4 \pi}} \sin \theta \cos \phi d\Omega$$
where $$\int d\Omega = \int_0^{2 \pi} d\phi \int_0^{\pi} \sin \theta d\theta$$

$$c_{0,0} = \frac{\sqrt{3}}{4\pi} \int_0^{2 \pi} \cos \phi d\phi \int_0^{\pi} \sin^2 \theta d\theta$$

Leading to a sin φ evaluated between 2π and 0... Which is 0, so c_0,0=0 ? Am I on the right track?

If I then apply this with the other values, I could work out each value for c, and put them into the equations to get the probabilities?

So i. would be c_1,1²
ii. would be c_1,-1²+c_1,0²+c_1,1²

Correct?

 

[Isaac Pepper]

Working through the numbers, I have got an answer of $$c_{1,0}=0$$ $$c_{1,1}=-\frac{1}{\sqrt{2}}$$ and $$c_{1,-1}=\frac{1}{\sqrt{2}}$$

Which means the answer to i. is 1/2, ii. is 1/2+1/2 = 1
But how do I calculate the probability of measuring L_z=0?

The probability of m=1 AND l =1 was 1/2 (from part i.)
and the probability of l=1 is 1 from part ii.
This suggests that the probability of m=1 is 1/2...?
Which I'm assuming is the same for the probability of m=-1
As such the probability of m=0 is 0, correct?

 

[DrClaude]

The probability of m=1 AND l =1 was 1/2 (from part i.)
and the probability of l=1 is 1 from part ii.
This suggests that the probability of m=1 is 1/2...?
Which I'm assuming is the same for the probability of m=-1
As such the probability of m=0 is 0, correct?

That's one way to do it. You can also get it directly from the fact that all coefficients $c_{l,0}$ are zero, therefore the probability of measuring $m=0$ is zero.