Find angular momentum outcomes and their probabilities

[James_The_Ern]

Homework Statement

Basically, I'm dealing with part d) in this document: https://s3.amazonaws.com/iedu-attachments-message/b663095a5021cb6aee55657de728a8d7_bfbe0ba9d2f10f8ac9ef9d049934c1da.jpg. I have found that the angular momentum only depends on spatial coordinate and it doesn't on time. Is explanation that the operator itself has a derivative of angle but not time sufficient? Or should I state that if I take the squared wavefunction, e^(-iEt/h_bar) term becomes 1?

Back to the question. I've been doing a lot of research online and I struggle to find how I should approach this problem in terms of finding possible outcomes and their probabilities. Don't know where to start, actually. Looking at part e), I assume I have to take particular energy values, since energy is quantized, but which ones? And how many? And how do I find the probabilities?

Homework Equations

L = ih d/dx, I took this operator for the whole superposition state.

The Attempt at a Solution

I only applied the operator to the wave function. I also noticed that this wave function is probably not an eigenfunction of L, since the derivative doesn't get me a Lf = mf relationship, which is also confusing. Really need some clear help. Would be appreciated a lot!

 

[PeroK]

That link won't open for me. Can you post your question explicitly?

 

[James_The_Ern]

That link won't open for me. Can you post your question explicitly?

Not sure how to write the functions via this forum properly, so I just uploaded the image to another host. http://i64.tinypic.com/dg12wz.jpg
Does that work? Sorry for the inconvenience.

 

[PeroK]

I assume you can calculate the expected value of angular momentum? But that isn't generally enough to give you the probability of each value

What about eigenfunctions of the AM operator. Can you calculate those?

 

[James_The_Ern]

I assume you can calculate the expected value of angular momentum? But that isn't generally enough to give you the probability of each value

What about eigenfunctions of the AM operator. Can you calculate those?

I also thought about the expectation value, but you're right, this has nothing to do with the probabilities, just the average angular momentum value.

Right, so what I did was applying L operator to the wave function which gave me (with the dependence on time term):

-ħ/√π (-½ e^i(φ-Et/ħ) - ½ e^i(5φ-Et/ħ) + 2ie^-iEt/ħsin(2φ)). Now, it doesn't seem like an eigenfunction, because there's sin term now instead of initial cos. So I can't find an eigenvalue. Or am I doing a mistake somewhere?

 

[PeroK]

I also thought about the expectation value, but you're right, this has nothing to do with the probabilities, just the average angular momentum value.

Right, so what I did was applying L operator to the wave function which gave me (with the dependence on time term):

-ħ/√π (-½ e^i(φ-Et/ħ) - ½ e^i(5φ-Et/ħ) + 2ie^-iEt/ħsin(2φ)). Now, it doesn't seem like an eigenfunction, because there's sin term now instead of initial cos. So I can't find an eigenvalue. Or am I doing a mistake somewhere?

Finding eigenfunctions of an operator doesn't relate to the wave function in question. It's a general eigenvalue problem. Given the number of marks you may be expected to know these? Otherwise, they are not too hard to calculate. They satisfy

$Lf = \lambda f$

 

[James_The_Ern]

Finding eigenfunctions of an operator doesn't relate to the wave function in question. It's a general eigenvalue problem. Given the number of marks you may be expected to know these? Otherwise, they are not too hard to calculate. They satisfy

$Lf = \lambda f$

I don't quite grasp what you mean. Why don't they relate to the wave function? If we use this relationship of Lf = λf, I basically plugged in f as a wave function given and I can see that no λ fits here. Sorry for such dumb questions, it's just that I don't get a useful clue where to start and what to do, where my mistakes are.

 

[PeroK]

I don't quite grasp what you mean. Why don't they relate to the wave function? If we use this relationship of Lf = λf, I basically plugged in f as a wave function given and I can see that no λ fits here. Sorry for such dumb questions, it's just that I don't get a useful clue where to start and what to do, where my mistakes are.

An operator is an operator and has its own eigenfunctions, independent of the state of the system.

The AM operator has a set of eigenfunctions in this system which you can calculate as a general eigenvalue problem.

Then, it's often a good idea to express your wavefuction in terms of the eigenfunctions of the operator you are interested in.

Hint: In this case you have been given the wave function almost in terms of AM eigenfunctions. So, there is not much to do.

 

[PeroK]

PS what do you have for the answer to part c)?

 

[James_The_Ern]

I got 1/√π (½ e^{i(φ - Et/ħ)} - ½ e^-iEt/ħcos(2φ) + ½ e^{i(5φ - Et/ħ)})

 

[PeroK]

I got 1/√π (½ e^{i(φ - Et/ħ)} - ½ e^-iEt/ħcos(2φ) + ½ e^{i(5φ - Et/ħ)})

You seem to have assumed that those are all eigenfunctions of the Hamiltonian. How did you know that? In any case, you should have different values for $E$ in each case.

 

[James_The_Ern]

So should I take each of the three terms separately and see if they are eigenfunctions? Then only two of them are. So, let's say if one term isn't an eigenfunction of an angular operator, does that mean that it doesn't have an angular momentum and I'm only considering two of the other functions in terms of their values and probabilities?

 

[PeroK]

So should I take each of the three terms separately and see if they are eigenfunctions? Then only two of them are. So, let's say if one term isn't an eigenfunction of an angular operator, does that mean that it doesn't have an angular momentum and I'm only considering two of the other functions in terms of their values and probabilities?

You need to take a step back and learn more about eigenfunctions. The eigenfunctions of the operator of an observable form a complete orthogonal set. That means every function can be expressed as a linear combination of the eigenfunctions.

In this case, you were given the wave function as a linear combination of energy eigenfunctions. If you hadn't been given this you would have needed to do some work to express your wavefuction in this form.

Now, the eigenfunctions of the AM operator do not quite coincide with energy eigenfunctions. There's a bit of theory here about degeneracy and commuting operators that I won't go into. But you should learn this at some point.

The important point is that every function can be expressed as a linear combination of AM eigenfunctions.

So, you do have a bit of work to do to express your wave function in eigenfunctions of the AM operator.

To help you out note that:

$2\cos(2\phi) = e^{2i\phi} + e^{-2i\phi}$

 

[James_The_Ern]

Thanks a lot, I also thought there should be some relationship for it! That makes sense. So, let's say I'm able to find an eigenvalue, how do I move from here to probabilities? Do you have any clue? It's easy to find the expectation value, but not separate probability values.

 

[PeroK]

Thanks a lot, I also thought there should be some relationship for it! That makes sense. So, let's say I'm able to find an eigenvalue, how do I move from here to probabilities? Do you have any clue? It's easy to find the expectation value, but not separate probability values.

The coefficient of each energy (or AM) eigenfunction determines the probability of getting the relevant eigenvalue as a measurement. That's why you need a normalised wave function.

The probability is the modulus squared of the relevant coefficient.