All Questions
208 questions
2votes
1answer
46views
Explicit form of Fock state acting on Weyl operators
In Bratelli and Robinson's $\textit{Operator Algebras and Quantum Statistical Mechanics Vol. 2}$, on page 24, they state that the following is an easy calculation $$\omega_F(W(f)) = \langle \Omega, W(...
0votes
2answers
88views
Commutativity of the expectation value of the product of two hermitian operators
I have two Hermitian operators $\hat{A}$ and $\hat{B}$. Is $$\langle \psi | \hat{A} \hat{B} | \psi \rangle \stackrel{?}{=} \langle \psi | \hat{B} \hat{A} | \psi \rangle\quad ?$$ If not in general, are ...
- 57
0votes
2answers
71views
Proving quantum operator relationship used in derivation of Radial Schrodinger Equation
In the derivation of the Radial Schrodinger Equation for central potentials, I have seen the following relationship used: $$ r^2p^2 = L^2+(\bf{r}\cdot\bf{p})^2-i\hbar(\bf{r}\cdot\bf{p}) $$ and I have ...
0votes
1answer
41views
Exploiting Creation Operator Commutation Relation in HOM Interference Calculation
In this paper where the authors derive the formula for coincidence probability in a Hong-Ou Mandel (HOM) interference effect as a function of time delay $\tau$, they arrive at an equation (15) with ...
- 345
2votes
0answers
43views
Taylor condition on the general formula for momentum commutator [closed]
My quantum homework asked me the following question: Prove that for any $f(x)$ such that $f$ admits a Taylor expansion, the following is true: $$[f(x), \hat{p}] = i\hbar\frac{\mathrm{d}f}{\mathrm{d}x}...
- 21
0votes
1answer
83views
Conmutators and Jacobi's Identity
I've come across an exercise asking me to calculate: $$[[A,B],[C,D]]$$ knowing $[A,C]=[B,D]=0$ and $[A,D]=[B,C]=1$ I've already solved it by "brute force", separating the commutator as ...
- 2,164
0votes
1answer
439views
Proof of Spin commutation relation for Holstein-Primakoff-Transformation
I have run into an issue while trying to prove the Holstein-Primakoff commutations \begin{align*} [S^+_i,S^-_j]=2 \delta_{ij} S^z_i, \ [S^z_i, S^-_j]=-\delta_{ij} S^-_i \end{align*} where \begin{...
- 111
0votes
1answer
193views
Help with commutator algebra with fermionic operators
I am struggling to understand how the following is true for the fermionic creation/annihilation operators $a^\dagger, a$: $$[a^\dagger a, a]=-a$$ If someone could walk me through the math derivation ...
- 119
2votes
2answers
416views
Uncertainty on the sum of two non-commuting operators
Suppose that I have an observable $$ \hat{E} = \sin(\alpha) \hat{Q} + \cos({\alpha}) \hat{P} $$ with $\hat{Q}, \hat{P}$ being non-commuting operators satisfying $$ [\hat{Q}, \hat{P}] = i \hbar $$ It ...
- 1,150
2votes
1answer
102views
Why does $[L_z,L_{\pm}]\neq 0$ imply $[J^2,L_z]\neq 0$?
In a lecture about the angular momentum operator, it is stated that the operator $L_z$ commutes with itself, with $L^2$, with all of spin angular momentum operators, but not with $L_{\pm}$, so; $$[J^2,...
0votes
1answer
743views
Understanding exception to: Two non-commuting Hermitian operators commute with the hamiltonian implies degenerate energy eigenvalues
For context, I am working through the exercises in Modern Quantum Mechanics by Sakurai and Napolitano Second Ed. I have previously completed (years ago in undergrad) the Griffiths 3rd ed. Introduction ...
- 1,793
1vote
1answer
54views
Three Mutually Non-commuting Dynamic Variables
On the usual quantum-mechanical Hilbert space, the operators q and p commute to a constant: [q,p]=i. I'm looking for an elementary example of some Hilbert space for which 3 operators, r, q, and p, ...
- 21
1vote
1answer
377views
Exponential of the sum of two non-commuting operators where their commutator is proportional to one of them
I was shown the following property: Given two operators $A$ and $B$, and $$[A,B]=-\frac{k}{2}B,$$ being $k$ an arbitrary constant, then: $$ \exp(A+B)=\exp(A)\exp\left(\frac{-2}{k}B \left(1-e^{\frac{-...
- 73
3votes
2answers
175views
Quantum fidelity commutativity proof
I am looking for a proof that the quantum fidelity $$F(\rho, \sigma) = \left(\text{tr} \sqrt{\sqrt{\rho}\sigma\sqrt\rho}\right)^2$$ is commutative, i.e. $F(\rho, \sigma) = F(\sigma, \rho)$. I have ...
- 237
2votes
1answer
308views
Is this operator Hermitian? Commutator of non-Hermitian operators [closed]
In the derivation of a Master Equation, I am left with two additional terms: $$ \sigma_j [\sigma^{\dagger}_k,\rho] - [\rho, \sigma_k]\sigma_j^{\dagger} \quad ,$$ where $\sigma_j = |g\rangle \langle e|...
- 147
