How Do You Calculate Angular Momentum Components?

[Kjjm]

Homework Statement

Prove that 〈Lx〉 = 〈Ly〉 = 0 for the spherical harmonics
Y1 = [ (3/8𝜋)^1/2 ]* sin 𝜃 * 𝑒^𝑖𝜙 and
Y2 = [(15/32𝜋)^1/2]*[( sin𝜃 )^2 ]* 𝑒^2𝑖𝜙
This result generally holds for any other rotational states.

Relevant Equations

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[jbriggs444]

You will need to show effort in order for help to be given.

It is not clear to me how a spherical harmonic (per Wikipedia, functions over a spherical shell which are orthogonal in some sense) is relevant to angular momentum.

 

[PeroK]

You will need to show effort in order for help to be given.

It is not clear to me how a spherical harmonic (per Wipedia, functions over a spherical shell which are orthogonal in some sense) is relevant to angular momentum.

They are eigenfunctions of the QM angular momentum operators. Conventionally, $L^2$ and $L_z$.

 

[anorlunda]

Our rules require you to show us your work before we offer help.

 

[Kjjm]

Sorry. I am not good at english. so i worked it so hard, but i can’t write much about it. Its my fault. Sorry

 

[Orodruin]

It is not clear to me how a spherical harmonic (per Wipedia, functions over a spherical shell which are orthogonal in some sense) is relevant to angular momentum.

Let me add to what @PeroK said:

The angular momentum operators generally form the Lie algebra of SO(3) and you can construct a reprsentation of SO(3) acting on functions on a spherical shell by the obvious action on that function space.

Now, this representation can be split into irreps of SO(3) and it turns out that it actually contains a single copy of each SO(3) irrep, which can be labelled by the total angular momentum, or $\ell$. You can go further on and restrict your symmetry to a single SO(2) rotation (typically chosen to be about the z axis), which splits irrep $\ell$ of SO(3) into $2\ell+1$ one-dimensional irreps (they have to be one-dimensional as SO(2) is Abelian) of SO(2). These irreps are still functions on the sphere with particular properties, namely the spherical harmonics. The connection to angular momentum is through the role of angular momentum as the generator of rotations (just as linear momentum generates translations), i.e., the symmetry group involved.

 

[BvU]

Sorry. I am not good at english. so i worked it so hard, but i can’t write much about it. Its my fault. Sorry

First thing to do is tell us what you have learned to use as operators L_x and L_y .
Then what you should do to find <L_x> and idem y.
All this under the heading: Homework equations:
It's there for a good reason !