Quantum Mechanics, Angular Momentum Measurement Problem

[Dav0s]

The problem involves measurement in a quantum system with angular momentum. There are two parts I am struggling with. Apologies for the images, I tried to put it all in latex but failed!

Part 1:

[PLAIN]http://i2.sqi.sh/s_1/1xo/part1.jpg

I have found a long derivation in a book involving a function Y_lm, but I am sure I must be missing a simple trick to derive the eigenvalues simply from this data given? I have found that the eigenvalue should be mh, but I am stumped as to how I can find this from L². Any hints are welcome.

Part 2:

[PLAIN]http://i2.sqi.sh/s_1/1xp/part2.jpg

I have no problem with the first part of this question, finding the eigenvalues with respect to L² and L_z, but the second part has me stumped.

My eigenvalues are as follows, found using a form of L² and L_z given previously in the question.

Firstly for L²


2\,{h}^{2}, 2\,{h}^{2}+{\frac {{h}^{2}}{{\sin}^{2}\theta}}, 2\,{h}^{2}-{\frac {{h}^{2}}{{\sin}^{2}\theta}

And for L_z

0, -h, h

My problem is, I'm not sure what a simultaneous measurement is? I can see that the answers given are simply the eigenvalues for L_z, and one of the eigenvalues for L², but I'm not quite sure how they are connected?

I am also not sure about the final part of this question. Should I try to write \Psi as a linear combination of u1, u2 and u3?


Thanks in advance for any help, I think my main problem is interpreting what I should be finding, I don't think the maths will be that difficult.

 

[gabbagabbahey]

For part one, assume that \psi(r,\theta,\phi)=R(r)\Theta(\theta)\Phi(\phi) is an eigenfunction of \hat{L}_z[/itex] (i.e. Assume the eigenfunctions are separable). The eigenvalue equation for \hat{L}_z[/itex] will then give you a separable 1st order ODE to solve for \psi.

 

[vela]

My eigenvalues are as follows, found using a form of L² and L_z given previously in the question.

Firstly for L²

2\,{h}^{2}, 2\,{h}^{2}+{\frac {{h}^{2}}{{\sin}^{2}\theta}}, 2\,{h}^{2}-{\frac {{h}^{2}}{{\sin}^{2}\theta}

The second and third values are wrong. The eigenvalues are constants. They won't depend on θ.

My problem is, I'm not sure what a simultaneous measurement is? I can see that the answers given are simply the eigenvalues for L_z, and one of the eigenvalues for L², but I'm not quite sure how they are connected?

I am also not sure about the final part of this question. Should I try to write \Psi as a linear combination of u1, u2 and u3?

Yes, that's exactly what you want to do.

When you have a state, how do you determine what the possible outcomes of a measurement are and their probabilities? Your textbook should cover this topic if you don't know the answer. It's one of the basic things you need to know in quantum mechanics.

 

[flower321]

First part is so simple first you write the l^2 in the form of x,y and z-coordinates and then add it

you see the book of neuredine zettlie of quantum mechanics

 

[Dav0s]

Thanks for the responses.

For part one, assume that \psi(r,\theta,\phi)=R(r)\Theta(\theta)\Phi(\phi) is an eigenfunction of \hat{L}_z[/itex] (i.e. Assume the eigenfunctions are separable). The eigenvalue equation for \hat{L}_z[/itex] will then give you a separable 1st order ODE to solve for \psi.

<br /> <br /> Thanks, I have had a go but not sure I am on the right lines.<br /> <br /> I start with;<br /> <br /> -i\hbar \frac{\partial}{\partial \theta} \psi = X \psi<br /> <br /> where X is the eigenvalue? Is this right? If I proceed this way, I get the eigenfunction as an exponential.<br /> <br /> <blockquote data-attributes="" data-quote="vela" data-source="post: 2700409" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> vela said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> The second and third values are wrong. The eigenvalues are constants. They won't depend on θ.<br /> <br /> Yes, that's exactly what you want to do.<br /> <br /> When you have a state, how do you determine what the possible outcomes of a measurement are and their probabilities? Your textbook should cover this topic if you don't know the answer. It's one of the basic things you need to know in quantum mechanics. </div> </div> </blockquote><br /> The form of \hat{L}^2 I am given gives me the sin^2(\theta) term. I have redone my calculations and I still get this term. What should I get? I'm sorry I cannot see where I have gone wrong here.<br /> <br /> Am i correct in assuming the possible outcomes of a measurement are the eigenvalues? And the probability of these eigenvalues being obtained is the modulus squared of the constant in the linear combination? <br /> <br /> I still don't quite understand what a "simultaneous measurement" is?<br /> <blockquote data-attributes="" data-quote="flower321" data-source="post: 2700572" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> flower321 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> First part is so simple first you write the l^2 in the form of x,y and z-coordinates and then add it <br /> <br /> you see the book of neuredine zettlie of quantum mechanics </div> </div> </blockquote><br /> Sorry I don't quite understand what you mean. Can you be a bit more specific?

 

[vela]

I start with;

-i\hbar \frac{\partial}{\partial \theta} \psi = X \psi

where X is the eigenvalue? Is this right? If I proceed this way, I get the eigenfunction as an exponential.

You should be differentiating with respect to ϕ, not θ, and to be a bit more precise, you found that the eigenfunction is proportional to an exponential. Did you assume the form of ψ that gabbagabbahey suggested?

The form of \hat{L}^2 I am given gives me the sin^2(\theta) term. I have redone my calculations and I still get this term. What should I get? I'm sorry I cannot see where I have gone wrong here.

You should just get 2\hbar^2 for all three. You might want to post your work here so we can see where you went astray.

Am i correct in assuming the possible outcomes of a measurement are the eigenvalues? And the probability of these eigenvalues being obtained is the modulus squared of the constant in the linear combination?

Yes and yes.

I still don't quite understand what a "simultaneous measurement" is?

When they say you do the measurements simultaneously, they mean you're doing both measurements on the same initial state. If you did one measurement followed by the other, the first measurement would cause the wavefunction to collapse to a new state, and the second measurement would then be performed on this new state.

 

[Dav0s]

You should be differentiating with respect to ϕ, not θ, and to be a bit more precise, you found that the eigenfunction is proportional to an exponential. Did you assume the form of ψ that gabbagabbahey suggested?

Sorry, just a mistype, should have been ϕ.

So I get -i\hbar R(r)\Theta(\theta)\frac{\partial \Phi}{\partial \phi}=X R(r)\Theta(\theta)\Phi(\phi)

Cancelling the R and θ functions, I get

-i\hbar \frac{\partial \Phi}{\partial \phi}=X \Phi(\phi)

Then solving for \Phi I get

\Phi = exp{\frac{-X\phi}{i\hbar}}

Is this correct? If so, where do I go from here?

You should just get 2\hbar^2 for all three. You might want to post your work here so we can see where you went astray.

Finally sorted it and got the same answer for all 3, sorry, was just some bad calculus.

When they say you do the measurements simultaneously, they mean you're doing both measurements on the same initial state. If you did one measurement followed by the other, the first measurement would cause the wavefunction to collapse to a new state, and the second measurement would then be performed on this new state.

Thanks for clearing this part up, I think I was just reading too much into it.

Thanks for the continued support, I'm very grateful.