Commutation relations between Ladder operators and Spherical Harmonics

In summary: Your starting point is the equality$$[L_z,A] = \hbar A$$which you apply to the spherical harmonic ##Y_{ll}##,$$[L_z,A] Y_{ll} = \hbar A Y_{ll}$$You expand the commutator, but the result should still be an equality.Thanks a lot everyone, I was able to solve it with all of yours help!
  • #1
PhysicsTruth
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Homework Statement
Consider an operator A such that it satisfies the following commutation relations-

##[L_+,A] = 0##
##[L_z,A] = \hbar A##
Using these, find ##L_z(AY_{ll})## and ##L^2(AY_{ll})## , where ##AY_{ll}## is an eigenfunction of ##L_z## and ##L^2##.

Also, deduce ##AY_{ll}##.
Relevant Equations
##L_+ = L_x +iL_y##
##L_z(Y_{ll}) = l\hbar (Y_{ll})##
I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?
 
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  • #2
Just figure out what you get when applying the given commutators to ##\mathrm{Y}_{ll}## and use the information that ##\hat{A} Y_{ll}## is also an eigenfunction of ##\hat{L}_z## and ##\hat{L}^2##.
 
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  • #3
vanhees71 said:
Just figure out what you get when applying the given commutators to ##\mathrm{Y}_{ll}## and use the information that ##\hat{A} Y_{ll}## is also an eigenfunction of ##\hat{L}_z## and ##\hat{L}^2##.
If I use the commutator between ##L_z, A##, I get

##(l\hbar +\hbar)A(Y_{ll})## for the 1st part. But I don't know how to figure out ##A(Y_{ll})## from the given information.
 
  • #4
PhysicsTruth said:
If I use the commutator between ##L_z, A##, I get

##(l\hbar +\hbar)A(Y_{ll})## for the 1st part. But I don't know how to figure out ##A(Y_{ll})## from the given information.
You should get an equality here. What is it?
 
  • #5
DrClaude said:
You should get an equality here. What is it?
I'm really sorry but I'm not being able to get you. Equality in which sense? Like it's not an equation, I'm just trying to find out ##L_z(AY_{ll})##. The eigenvalue is ##\hbar (l+1)##, but I also need to deduce what ##AY_{ll}## is.

Once again, I'm really sorry for not being able to follow you.
 
  • #6
PhysicsTruth said:
I'm really sorry but I'm not being able to get you. Equality in which sense? Like it's not an equation, I'm just trying to find out ##L_z(AY_{ll})##. The eigenvalue is ##\hbar (l+1)##, but I also need to deduce what ##AY_{ll}## is.

Once again, I'm really sorry for not being able to follow you.
Your starting point is the equality
$$
[L_z,A] = \hbar A
$$
which you apply to the spherical harmonic ##Y_{ll}##,
$$
[L_z,A] Y_{ll} = \hbar A Y_{ll}
$$
You expand the commutator, but the result should still be an equality.
 
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  • #7
Thanks a lot everyone, I was able to solve it with all of yours help!
 

What are commutation relations between ladder operators and spherical harmonics?

The commutation relations between ladder operators and spherical harmonics describe how these two mathematical objects interact with each other. Specifically, they show how the ladder operators, which are used in quantum mechanics to describe the creation and annihilation of particles, commute or do not commute with the spherical harmonics, which are used to describe the angular dependence of wave functions.

Why are commutation relations between ladder operators and spherical harmonics important?

These commutation relations are important because they allow us to understand the behavior of quantum mechanical systems. By knowing how the ladder operators and spherical harmonics interact, we can make predictions about the energy levels and angular momentum of particles in a given system.

How are commutation relations between ladder operators and spherical harmonics derived?

The commutation relations are derived using mathematical techniques from quantum mechanics and group theory. These techniques involve using the properties of the ladder operators and spherical harmonics, as well as the commutator operator, to determine the relationships between them.

What is the physical significance of the commutation relations between ladder operators and spherical harmonics?

The commutation relations have physical significance because they allow us to calculate the expectation values of physical observables, such as energy and angular momentum, in quantum mechanical systems. They also help us understand the symmetries and properties of these systems.

Are there any limitations to the commutation relations between ladder operators and spherical harmonics?

Like all mathematical models, the commutation relations have certain limitations and assumptions. They may not accurately describe all quantum mechanical systems, and they may not hold true in extreme conditions, such as at the quantum level. Additionally, they may not take into account all possible interactions between particles in a system.

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