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I'd like to proceed in a linear fashion, taking each part on one by one. For the first part, we can write the Hamiltonian as ##H = \sum_{n}^{N} w(c_{An}^{\dagger}c_{Bn}+c_{Bn}^{\dagger}c_{An})+v(c_{Bn}^{\dagger}c_{A(n+1)}+c_{A(n+1)}^{\dagger}c_{Bn})##. We can convert the creation and...

Yeah, I've done that thankfully. Thanks for the heads up!

##\alpha## is considered to be the absorption coefficient for a beam of light of maximum intensity ##I_0##. It's related to the complex part of the refractive index as we have shown above. Now, I have a doubt. Should I solve for ##k## from the quadratic equation in terms of the linear optical...

Thanks. I have some problem to understand this intuitively. Can you just explain a bit as to why we end up getting this sole matrix when we change our basis and fix it along a direction?

I did get this after a wholesome unitary transformation to the new eigenbasis, which is the same as the ##\sigma _z## matrix.

So, would the representation just be ##\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}## in that case?

In that case, I can just perform an Unitary transformation to make it diagonal. So I need to know this.

Can someone just tell me how to write a Spin Matrix along any vector rotated by an angle ##\theta## from ##z## axis in terms of the eigenbasis of ##\sigma _z## Pauli spin matrix? Is it ##cos(\theta /2) |+> + sin(\theta /2) |->##, where ##|+>, |->## are eigen vectors of ##\sigma _z## ?

Okay, so the rotation around any unit axis ##\hat{n}## of a spinor is given by- ##Icos(\theta /2) + i(\hat{n} \cdot \sigma)sin(\theta /2)## So here, it's getting rotated about the ##y## axis, so should I get the representation as required if I use this rotation form?

To be honest, the matrix doesn't have non-trivial solutions explicitly. It's only when one of the values is decided can the other value be determined. So, it might be diagonal only for a particular value of ##\theta## or ##\phi##. Also, ##\phi## hasn't been mentioned explicitly, so can't really...

I've tried to use the 1st equation as a matrix to determine, but it clearly isn't a diagonal matrix. My guess is that I need to find the spin matrix along the direction ##\hat{n}##, but do I need to find the eigenstates of ##\sigma \cdot \hat{n}## first and check if they form a diagonal matrix...

Thanks a lot everyone, I was able to solve it with all of yours help!

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.

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.

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?