What Happens When Measuring Lz on a Non-Eigenstate Wavefunction?

[Kentaxel]

Homework Statement

I have a wavefunction ψ(r), I want to know what the probability of obtaining a specific value, say 0, Is upon measuring the z-component of the angular momentum L_z

What do i do?

The Attempt at a Solution

My first thought is to apply the L_z-operator to my function and see if this yields the value I am looking for, and if it does not the probability is simply zero. However when i actually do the calculations i don't even get my original function back, and I'm not sure how to interpret this result. i.e. I expected that i would get somethibg like L_zψ(r)=lψ(r) where l is the eigenvalue of L_z. But maybe this is a faulty reasoning?Edit: Maybe what I'm wondering is really what happens if you try to measure a variable of which my statevector is not an eigenfunction (as seems to be the case) or would such an action actually colapse the function to an eigenfunction of my L operator? In any case What sort of result is to be expected of the corresponding eigenvalue? nothing? 0? Or what?

 

[Morgoth]

you have several ways to work. Do you have the form of your ψ? if you do try writting it in the form of [SOMETHING that Lz cannot see]*[Eigenfunctions of L_z], which are the spherical harmonics Y[l,m].

try thinking in the same way you would, if you had ψ= Σ α_η φ_η(χ) and were asked for energy E_k...

φ_η(χ) eigenfunctions of Hamiltonian.

 

[Kentaxel]

Yes i see what you meen, i was thinking of it the wrong way. The function is in fact already given in a manner where it's ($\phi$,$\theta$)-dependence can easily be separated from its r-dependance. so all i have to do is calculate c_lm =<lm|Y> for whatever Y[l,m] corresponds to the desired eigenvalue of L_z and then square that c which would give the probability? Specificaly <l0|Y> for 0 and so on.

But there are allot of eigenfunctions that would correspond to m = 0, would these all give me the same coefficient c?

Furthermore as i understand it, the only way to obtain a value for L_z is if it's possible to separate the ($\phi$,$\theta$)-dependence so that i can be sure that this part is a linear combination of the spherical harmonics? Or can i deduce this in any other way? Actually how can i ever be sure that the function is at all a linear combination of the Y[l,m] ?

 

[Morgoth]

if you have for example- I will denote Y[l,m]-
Y[1,0], Y[0,0]
both will participate in giving you m=0.
if the one is 1/3 and the other is 1/3, then the total probability will be 2/3 :)

Generally you can find it by:
|<Your wanted state | ψ>|²
using dirac's notation since I see you used it too...

eg
|<l 0|ψ>|² will give you the possibility of your state ψ to fall (to be) on the state
|l 0> it's like seeing |ψ>,|l,0> as vectors, and taking the inner product of them gives you the projection of one over the other.
writing the above in integral form (going to the {r} representation):
|∫Y*[l,0] ψ(r) d³r|²

how did I do this?
I used that I= ∫|r><r|d³r
<l 0|I|ψ>= ∫<l 0|r><r|ψ>d³r= ∫<r|l 0>*<r|ψ>d³r = ∫Y*[l,0] ψ(r) d³r


Now since you can always write ψ(r)= R(r) Y(θ,φ) you will have that R(r) is not seen by the Y* but Y* sees the Y[l,m](θ,φ)s you have in your state, leaving only those that have the same quantum numbers as it alive (orthogonal and normalised)

how can you be sure that the function is a linear combination of the Y[lm]?
Because they form a complete set of basis of your space (which is the space of θφ), so you can form out of them all possible states.

 

[Kentaxel]

Very neat explanation, thank you so much it really helps allot!