Angular momentum operator Puns

I'm working through a textbook I found on quantum mechanics and I'm having some difficulty with the problem sets. I am not a student, I'm just interested in physics, and I figure I can't really say I understand it to any degree if I can't do the math. I've reached out to the publisher for a solutions manual but they won't provide it because I'm not an instructor, which I understand but is still annoying.

Here is the problem (please excuse any formatting issues):

The operator for the z-component of angular momentum is given by:

Lz = -iℏ (∂/∂φ)

You're asked to determine whether or not these functions are eigenfunctions of the operator:

1. ψ = sin(φ) e^(iφ)
2. ψ = sin^k(φ) e^(ikφ), k=constant
3. ψ = sin(φ) e^(-iφ)

When I work the problems, it doesn't look like ANY of them are eigenfunctions of Lz, but I'm sure that isn't right, because you're supposed to use the results for later problems. Here is my work:

1. Lz ψ = -iℏ (∂/∂φ) sin(φ) e^(iφ)

= -iℏ [cos(φ) e^(iφ) + i sin(φ) e^(iφ)]

= -iℏ [cos(φ) + i sin(φ)] e^(iφ)

= -iℏ e^(i2φ)

(or expressing it in terms of ψ...)

= [-iℏ e^(iφ) / sin(φ)] ψ

2. Lz ψ = -iℏ (∂/∂φ) sin^k(φ) e^(ikφ)

= -iℏ [k sin^(k-1)(φ) cos(φ) e^(ikφ) + ik sin^k(φ) e^(ikφ)]

= -iℏ [k sin^(k-1)(φ) e^(ikφ)] [cos(φ) + i sin(φ)]

= -iℏ [k sin^(k-1)(φ) e^(i2kφ)]

= [-iℏ k e^(ikφ) / sin(φ)] ψ

3. Lz ψ = -iℏ (∂/∂φ) sin(φ) e^(-iφ)

= -iℏ [cos(φ) e^(-iφ) - i sin(φ) e^(-iφ)]

= -iℏ [cos(φ) - i sin(φ)] e^(-iφ)

= -iℏ e^(-i2φ)

= [-iℏ e^(-iφ) / sin(φ)] ψ

None of the last quantities in brackets look like constants to me, so they can't be solutions to Lz ψ = ℓ ψ, for some constant ℓ, right? Meaning they aren't eigenfunctions of Lz, right? But maybe my math is bad, maybe I'm overlooking some identity, maybe I'm failing to understand something fundamental...This is an early chapter problem, so I'm pretty discouraged about my ability to ever grasp this material. I've lost momentum and am just bogged down in a frustrating math puzzle, one that probably any undergraduate could solve easily...In any case, I would appreciate some assistance.

https://cdn.discordapp.com/attachments/743934084560060517/883881586939006996/unknown.png

I was presented with this question with zero context or clarification and no way to get any.

I'm not so much looking for a solution, moreso if anyone can parse what the question is actually asking/what tools does someone usually use to solve this?

From googling I've found that J_(-1)/J_(0) appears to be a ratio of angular momentum operators (https://en.wikipedia.org/wiki/Ladder_operator#Applications_in_atomic_and_molecular_physics), and it's being equated to some generic complex number. Am I trying to do a Taylor expansion or something? I'm not sure why the equation has i - sqrt(-1) either since that's just zero. It seems more like a statement than a question so I'm generally confused on where to go from here.

Is there something similar to the glaube state as left and right hand eigenstate of the destruction and creaton operator of the harmonic qiantum oscillator?

They don't commute

http://en.wikipedia.org/wiki/Angular_momentum_operator

At the bottom of the Wikipedia page is the formula for the angular momentum L squared operator in spherical coordinates.

Isn't this operator undefined when theta is 0 (and thus sin theta is 0)?

I'm having trouble with this simple concept I thought I understood.

the total angular momentum is supposed to be just that. If I have two particles, each spin 1/2, then the total angular momentum (ignoring orbital angular momentum) is 1, right? Yet my book wants to tell me that 0 is also an option.

If j1=1/2, and j2=1/2,

My book uses F for the total a.m.. How can F ever be anything but 1? The book says it can go as low as j1-j2, but if F is the total angular momentum, and the j's never change, then how can the total j1+j2 change?

It would seem that it has something to do with linear combinations of uncoupled states, such as 1/sqrt2(|j1j2m1m2> - |j1j2m1m2>), but it still makes no sense to me because the other superposition state 1/sqrt2(|j1j2m1m2> + |j1j2m1m2>) does not yield F=2/sqrt2.

Can the total angular momentum ladder operator be expressed as [; \hat{J}\pm = \hat{L}\pm + \hat{S}_\pm ;]?

The problem in question is here. I believe I can do this by expressing the total angular momentum state as a superposition of [; |l=1, m_l = -1, s = 1/2, m_s = 1/2> ;] and [; |l=1, m_l = 0, s = 1/2, m_s = -1/2> ;]. Then using the [;\hat{J}_\pm ;] operators I should be able to get enough information to normalise the state. I'm not sure how I can apply these operators to the linear superposition though.

Hey, everyone. This is my first post on reddit. I'm trained as an engineer, but I've been slowly teaching myself QM through Griffith's book in conjunction with Atkin's Physical Chemistry. QM introduced me to the ladder operator techniques in order to get different eigenstates for the quantum harmonic oscillator and the angular momentum problems.

I understand that in the quantum harmonic oscillator problem, the raising and lowering operators come from an attempt to factor the Hamiltonian (Griffith, pg. 42):

H * psi = E * psi 1/2m * [p^2 + (mwx)^2] = E * psi

Where p is the momentum operator and x is the position operator. An attempt to factor [p^2 + (mwx)^2] yields:

a+ = constant * (i * p + mwx)

a- = constant * (i * p - mwx)

These are the raising and lowering ladder operators for the quantum harmonic oscillator and can be used to represent the Hamiltonian as:

H = h_bar * w * (a+a- + 1/2)

H = h_bar * w * (a-a+ - 1/2)

However, when we get to angular momentum, every single source that I have come across simply defines the raising and lowering operators as:

L+ = Lx + iLy

L- = Lx - iLy

without any particular reason as to how they got to these ladder operators. If I were taking a course at a university, I would simply ask the professor, but unfortunately I do not have that resource available right now. So, I want to ask what is the rationale for defining these two ladder operators this way?

My attempt at a solution to this would be to take into account the angular momentum operators L^2 and Lz. Both operators are hermitian and therefore give real values. L^2 can also be defined as:

L^2 = Lx^2 + Ly^2 + Lz^2

Since it is established that Lz is hermitian, Lz^2 must be nonzero and real. Therefore:

[L^2 - Lz^2] * psi = some real value = [Lx^2 + Ly^2] * psi

Factoring [Lx^2 + Ly^2] leads to:

L+ = Lx + iLy

L- = Lx - iLy

In an attempt to find an eigenstate that will satisfy both L^2 and Lz simultaneously. Am I on the right track here?

edit: Edited for weird symbol stuff.

Asking because upon review of my notes for the derivation of the hydrogen atom’s spectrum, it looks as though the considerations taken along the way to derive the spectrum (of hydrogen), namely the reasonings stemming from expressing the radial functions in terms of power series, are ultimately what fixed this upper limit of L to what I see it generally referenced as, n-1.

If this result is general, can someone explain why, atleast qualitatively? Thanks in advance.

thank you!

Seeing the varying kinetic elements of buildings (with attention to things like pendulums) I got curious if reaction wheels have ever been used on land based structures.

Does anyone know of any applications?

Are rotating body type physics ever applied to building design?

Angular momentum is conserved, so there must be some quantity of angular momentum for the entire universe that never changes. What is the significance of this, if any?