# All Questions

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46 views

### What is the implication of overlap between eigenstates of two operators in Quantum Mechanics?

For instance, what does it mean that a certain position eigenstate is also an energy eigenstate? I understand that measurable (Observables) in Quantum mechanics are the operators. Their eigenvalues ...
38 views

### Commutation relations in Gupta-Bleuler quantization

Quantization of the free electro-magnetic field has essential differences in comparison to quantization of say scalar or massive vector fields. In fact there are different approches to it. One of ...
63 views

### Proof for $\langle i[A,B]\rangle$ [closed]

I have to prove the following equation: $$\langle i[A,B]\rangle = 2\mathfrak{Im}\left[\int dV(\overline{B\psi)}(A\psi)\right]\,,$$ where A,B are hermitian operators. Here is my calculation, but I don'...
21 views

### Source for mathematical methods [duplicate]

I am just curious about any question and example sources for linear vector spaces, bra-ket notation, operators, commutators and hilbert spaces.
66 views

### Does $[L_z,H] = 0$ imply the state is also an Eigenstate of $H$ is also an eigenstate of $L_z$?

Given that the Hamiltonian $\mathcal{H}$ is rotationally invariant then we know $[L_z,\mathcal{H}] = 0$. Does that imply that an eigenstate of H is also an eigenstate of $\mathcal{H}$? More ...
53 views

### Does total $\hat{S}^2$ always commute with total $\hat{S}_z$ even for interacting spins?

I was given the following operator $\hat{f}$ describing the interaction of two spin-$\frac12$ particles: $$\hat{f}=a+b{\hat{\bf S}_1}\cdot{\hat{\bf S}_2}.$$ I was told that I can prove that $\hat{f}$...
48 views

### The charge given by a commutator

I saw in the text that $[Q,X]=cX$ and says the operator $X$ has charge $c$ under the generator $Q$. I tried to understand why the coefficient $c$ means the charge. So I used this relation to get the ...
24 views

### Symmetry in Fock-space 2-body interaction

The simplest two body interaction term for fermions is $$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$ and I'm trying to determine the symmetries on $U$. Unfortunately I keep getting ...
83 views

### Angular momentum coupling

I read about angular momentum coupling on wikipedia and there are a few things i dont understand. What does this mean "spin and orbital angular momentum of a single object belong to different Hilbert ...
66 views

### What is the QFT state with two distinguishable fermions present?

I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$. The state describing a single electron is given ...
164 views

### “Commuting observables share common eigenstates”

I am struggling to find a precise definition of this line from my quantum mechanics textbook: If $[A,B] = 0$, then the operators commute, and "commuting operators share common eigenstates". This ...
133 views

### Operators commutation and relation between eigenvalues

If $H$ and $L_i$ are commuting ( $[H, L_i] = 0$ ) could we deduce that the eigenvalues of $H$ depend/ do not depend on $m$ and $\ell$ ( eigenvalue of $L_z, L^2$ )? I don't think so since it does not ...
72 views

### Joint Spectral Measure theorem

I want to gain an intuition to understand the joint spectral measure theorem. In the case that operators involved in this theorem have purely discrete spectrum, the theorem should be reduced to the ...
144 views

### Ladder operators vs creation/annihilation operators

I am trying to figure out the difference between the ladder operators (for harmonic oscillator) $a^\dagger$, $a$ and the creating/annihilation operators $c^\dagger$, $c$. Are they the same? I have ...
186 views

### What does it mean for 2 observables to be compatible?

If I have 2 observable operators $A$ and $B$, if $A$ and $B$ commute: $[A, B] = 0$, then they must necessarily form a complete set of commuting observables (CSCO). Essentially, if 2 observables are ...