Let us suppose I initially have a rotationally invariant Hamiltonian: $$ H = H_{\text{hydrogen}} + H_2, [H_2,J_i]=0, i=\{1,2,3\}, J_i = L_i + S_i$$. Then I introduce a harmonic-like perturbation along the $z$-axis $V_z = 1/2 m \omega^2 z^2$. Thus, the rotational invariance reduces to axial invariance (along $\hat{z}$)

It is my misfortune I have zero group-theory knowledge yet, nevertheless, I'd like to know how this "symmetry reduction" affects the degeneracy of the energy eigenstates

The exercise tells me to find the energy corrections to a certain degenerate state, but I found out that the degeneracy is not removed eventually

$H_2 = A \mathbf{L}\cdot \mathbf{S}, A= const$ if anyone's wondering

Qmech is here, I humbly greet you :P

Hello,

Usually metals except gold and copper are silvery white and something like that

But some metals such as osmium and tin have bluish tinge

Why are they bluish

Especially tin

I think that's could be a good question for the Chemistry stack

Actually, that is a physics question. Not that chemistry cannot answer it, but the reason is purely quantum physics

I've just realized that my question is not well-written: first of all,the degeneracy I'm talking about involves only the subspace $\{|1,3/2,m_j,0,3/2\}$ but I don't know whether the breaking of the rotational symmetry into an axial one might actually remove the degeneracies of other energy eigenstates

If it is bluish only at certain angles, chances are that it is not even an actual usual colour of an object. It tends to be diffraction.

@naturallyInconsistent Yeah I just thought that, since I do not know how to answer (and I was the only one here), the chemistry stack could give him some good and interesting answers as well

I'm probably just ignoring other possible symmetries of my Hamiltonian :P

I think group theory could probably answer me this time, but no point in trying to uncover things that I cannot yet understand

Hi everyone! New to quantum mechanics (im a chem student) and Im wondering, in quantum physics all valid wavefunctions have to be eigenfunctions of the hamiltonian. Is this true of all operators representing classical quantities? Like will a wavefunction that is a solution to the Schrodinger equation necessarily be eigenfunctions of the momentum, position, etc. operators?

@Allie "in quantum physics all valid wavefunctions have to be eigenfunctions of the hamiltonian" no, they don't

aha, so why?

the eigenfunctions of the Hamiltonian are the stationary states, those that don't change with time

a general wavefunction is a sum of these states (since the eigenfunctions of self-adjoint operators form a basis of the Hilbert space)

ahhh, since Hpsi = Epsi is just the time independent equation

Hmmm, okay, I'll keep that in the back of my head and continue my reading. Thank you!

But also see physics.stackexchange.com/a/718563/123208

one more question, sorry, are the linear operators we work with in quantum mechanics associative?

just want to make sure im not making any assumptions im not allowed to make

@Allie yes, they are - but they're not commutative

yes, the commutative one im aware of, unless the commutator is 0. Thank you!!!!

socks and shoes type thing lol

Whats the diff between operations, relations, and functions again? A function is reducible to an n-ary operation/univalent relation I think?

math.stackexchange.com/a/1337361/234396 i dont get this answer

Idk if they're talking about totality, because if so then wiki also defines that for functions so idk the difference again

Functions are special kinds of relations, operations are special kinds of functions (though I don't think there's quite a formal definition of "operation" that works for all possible things we might call operation in any context). What exactly is unclear about that?

I guess I'm confused when we'd use operations over functions and relations over both. Munkres defines functions "abstractly" by defining a "rule" which is just a relational definition for a function

that is the usual definition of a function as a univalent total relation; what is your question?

I am mostly concerned about the differences between a function and an operation I suppose

why?

what's that distinction relevant for

To better understand the concepts of both especially in relation to sets. I'm just gonna see if there's a formal definition in a set theory lol