How Do You Calculate the Expectation Value of L_z Using cos(φ)?

[hhhmortal]

Homework Statement

Hi, my problem is with part two of the question I've attached. I'm not exactly sure what they are expecting me to do, is it simply calculating the expectation value of L_z , from the wavefunction given (i.e. cos(φ))



Thanks.

 

[nickjer]

Not sure what part you have a problem with. I am assuming the 2nd paragraph. They basically want you to decompose the cosine into a superposition of eigenfunctions. The magnitude squared of the coefficients for each eigenfunction will give you the probability of that state.

 

[hhhmortal]

Not sure what part you have a problem with. I am assuming the 2nd paragraph. They basically want you to decompose the cosine into a superposition of eigenfunctions. The magnitude squared of the coefficients for each eigenfunction will give you the probability of that state.

Oh yes, forgot about decomposing cosine and sine.

I got another question, which is, if given a wave function like

u = Acosine(Pi/2a) + B sin(Pix/a)

How would I sketch the form of this squared (i.e. the probability distribution)?

 

[nickjer]

The first term looks like a constant, so you would just sketch the 2nd term and have it raised or lowered vertically by a constant. Unless you mistyped the first term.

 

[paranormal]

Hi there,

The question is asking you to calculate the expectation value of the angular momentum operator in the z-direction, known as L_z. This operator is defined as -iħ∂/∂φ, where φ is the azimuthal angle in spherical coordinates.

To calculate the expectation value, you will need to use the given wavefunction, cos(φ), and integrate it with the operator L_z. This will give you the average value of the angular momentum in the z-direction for this wavefunction.

I hope this helps. Best of luck with your homework!