Wave Function Spherical Coordinates Probabilities

[brooke1525]

Homework Statement

A system's wave function has the form

$$\psi(r, \theta, \phi) = f(t, \theta)cos\phi$$

With what probability will measurement of $$L_z$$ yield the value m = 1?

Homework Equations

$$L_z|\ell, m> = m|\ell, m>$$

The Attempt at a Solution

I feel like there may be a typo, in that that "t" should be "r" in the wave function. Is there a general expression for $$\psi_{n,\ell,m}$$ that I should know/use?

 

[tim_lou]

use the postulate of quantum mechanics. it says when you measure something, you'll get one of its eigenvalues with probabilities equal to the |coefficient|^2 of the corresponding eigenstates.

 

[brooke1525]

What is the proper eigenvalue equation to use?

 

[borgwal]

The one you displayed under "2. Homework Equations "!

 

[brooke1525]

So the probability L_z will give m=1 comes from:

<Psi | L_z | Psi> = m

Then, the probability of getting 1 is 1^2=1? Likewise, the probability of getting 0 is 0? I'm pretty sure I'm wrong here, can someone please correct me?

 

[borgwal]

$$<\Psi | L_z | \Psi>$$ is the expectation (i.e., average) value of L_z.

The probabilities to find specific values m for L_z follow from writing your wavefunction in the form

$$|\Psi>=\sum_m c_m |l,m>$$

as in your special case you only have 1 value of l in the superposition.
In this case the probability to find the value "m" is given by

$$P_m=|c_m|^2$$

More generally, you'd write

$$|\Psi>=\sum_m \sum_l a_{lm}|l,m>$$

and the probability to find the result "m" would be given by a sum over l:

$$P_m=\sum_l |a_{lm}|^2$$(And use that the wavefunction should be normalized properly!)

 

[brooke1525]

Why is there only one value of l in my case? I'm assuming I need to use some separation of variables in order to put the wave function in the appropriate form? I think I'm having a hard time seeing what I'm supposed to do with that wave function that was given to me. And that "t" is throwing me off...

 

[brooke1525]

Can anybody give me a clue here? I'm preparing for a test tomorrow, so I'd really, really appreciate the help!

 

[malawi_glenn]

brooke, use the spherical harmonics, you should have a table of them. Then you rewrite f(t,theta*cos(phi) as a function of them. Then use the fact that L_Z is an eigenoperator on those functions.

"First few spherical harmonics"
http://en.wikipedia.org/wiki/Spherical_harmonics

Now you study where the phi depandance is on these functions, and you might want to rewrite your cos(phi) to exponentials using eulers formula..http://en.wikipedia.org/wiki/Euler's_formula

 

[malawi_glenn]

So for clarification.

i) the Spherical Harmonics is a complete basis for angular functions, you can express any function dep. on anges as a sum of those sphreical harmonics, compare with Fourier series.

ii) The spherical harmonics is eigenfunctions to L_z operator with eigenvalue m.

iii) You don't need to know the values of L in this problem, just look at the phi dependence of the function you are given and have a look at the spherical harmonics on the page i gave you or in your book. You will find that only two possible spherical harmonics exists.

iv) Rewrite your phi-dependence in terms of exponentials