A mathematical construct used to study the effect of applying two operators in succession.

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7
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2answers
288 views

How to replace $T$-product with retarded commutator in LSZ formula?

I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: ...
5
votes
1answer
170 views

Apparent spacetime dependence of creation and annihilation operators

I'm currently going through An Introduction to Quantum Field Theory by Hartmut Wittig I've stumbled upon. Having trouble with equation (2.29), I'm asking the question: Do creation and annihilation ...
-1
votes
0answers
23 views

Commutator relations for bosonic annihilation/creation operator [duplicate]

Does anyone know the commutator: $[a^n,(a^{\dagger})^n]=?$ where $[a,a^\dagger]=1$. I do not need a calculation, the solution is enough.
6
votes
4answers
542 views

Is uncertainty principle a technical difficulty in measurement?

Is the uncertainty principle a technical difficulty in measurement or is it an intrinsic concept in quantum mechanics irrelevant of any measurement? Everyone knows the thought experiment of measuring ...
0
votes
0answers
27 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_i,J_j]=i\hbar\epsilon_{ijk}J_k \end{equation*} has the structure of a Lie-algebra. It is ...
0
votes
2answers
57 views

Derivatives of Fields in E&M

In QED the field strength tensor $F_{\mu\nu}$ is given by the commutator of the covariant derivatives $$D_\mu=\partial_\mu-ieA_\mu$$ where $A_\mu$ is the gauge field. Explicitly we have ...
1
vote
0answers
45 views

When trying to see what symmetries an operator generates, how do you “decide” what coordinate to apply it to?

Suppose I have $\hat{O}_{1}=-i\hbar\partial_{x}$ then \begin{eqnarray} e^{-i\gamma\hat{O}_{1}/\hbar}x\,e^{i\gamma\hat{O}_{1}/\hbar}=x+\gamma \end{eqnarray} and \begin{eqnarray} ...
2
votes
2answers
68 views

Transition from 4-potential to E and B [closed]

In my lecture notes there is a step that i cannot follow: $$\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}] (\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})=: \sigma^{\mu\nu}F_{\mu\nu}=i\vec{\alpha} ...
0
votes
1answer
42 views

Derivation of Pauli Hamiltonian

In my lecture notes there is a step that i cannot follow: $$\frac{i}{2}\epsilon_{ijk}\sigma_k [\pi_i,\pi_j] = -e\epsilon_{ijk}\partial_iA_j\sigma_k$$ with $\vec{\pi}=\vec{p}-e\vec{A}(x)$ When I ...
2
votes
1answer
74 views

What is the meaning of commuting Hamiltonians?

I have two quantum mechanical Hamiltonians such that \begin{equation} [\hat{H}_1,\hat{H}_2] = 0, \end{equation} where $\hat{H}_1$ and $\hat{H}_2$ act on the same set of states. What is there to ...
2
votes
2answers
43 views

What does a left-right arrow in a tensor formula mean?

I need help with some some notation I've not seen before. Is using the left-right arrow in this formula $$[P^μ,M^{ρσ}]=i\hbar(g^{\mu\sigma}P^\rho-(\rho\leftrightarrow\sigma))$$ equivalent to writing ...
0
votes
1answer
92 views

Commutator and Hamiltonian [closed]

Assume that $[\hat{A},\hat{H}]_-=0$ and $[\hat{B},\hat{H}]_-=0$ but we know that $[\hat{A},\hat{B}]_-\neq 0$. Then there exists degenerate stationary states of $H$. How to prove it?
1
vote
1answer
50 views

Why is it “disconcerting” if the components of an operator do not commute?

A symmetrized operator is given by $$\hat{R}=\frac{1}{2\hat{H}}\hat{N}+\hat{N}\frac{1}{2\hat{H}}.$$ With $\hat{H}$ the Hamiltonian and $\hat{N}$ the first moment of energy. The defined $\hat{R}$ is ...
2
votes
1answer
70 views

Commutator of fermionic operators

The fermionic creation/annihilation operators are defined by the anti-commutation relations: $$ \{a_k^{\dagger},a_q^{\dagger}\} = 0 = \{a_k,a_q \} $$ $$ \{a_k^{\dagger},a_q\} = \delta_{kq} \, .$$ I ...
6
votes
3answers
656 views

Does it mean anything if the commutator of an operator with the Hamiltonian is equal to the Hamiltonian?

Question says it all, really. I have $[\hat{H},\hat{O}]=-2i\hbar\hat{H}$. Does this mean that the operator $\hat{O}$ (an observable) is special in some way?
5
votes
2answers
451 views

Heisenberg picture of QM as a result of Hamilton formalism

Consider the formula for the total time-derivative of a physical value in Poisson's formalism: $$\tag{1} \frac{dA}{dt} = -\{H, A\}_{P.B.} + \frac{\partial A}{\partial t}, $$ where $\{A, B\}_{P.B.}$ is ...
3
votes
1answer
117 views

Spin operator: tricky proof using gamma matrices

I have not dealt with the gamma matrices extensively so I am having a bit of trouble here. Basically I want to show that the spin operator defined by $$ \mathbf{\hat{S}} = \frac{1}{2}\gamma^5 ...
0
votes
3answers
99 views

How do I prove that the del squared operator commutes with the angular momentum operator? [closed]

I need to prove in Cartesian coordinates that $[\nabla^{2},\hat{L_{z}}]= 0$ I know that the angular momentum operator is defined as: $\hat{L_{z}}=x\hat{p_{y}}-y\hat{p_{x}}$ And the del squared is ...
1
vote
1answer
66 views

Translation Operators

Show that if the wave function $\langle x|\psi\rangle$ is modified by a position-dependent phase $\langle x|\psi\rangle \to e^{\frac{ip_ox}{\hbar}}\langle x|\psi\rangle$ then $\langle x\rangle ...
1
vote
1answer
66 views

Anticommutator difference [closed]

What is the value of this difference of anticommutators $$\{x^2,p^2\}-(\{x,p\}^2)/2$$ if the commutator $$[x,p]=i\hbar ~?$$ I have tried and obtained a value $$-3\hbar^2/2 - 2i\hbar px.$$ But ...
2
votes
0answers
80 views

Prove that $e^{\frac{i\lambda}{\hbar} S_x}S_ze^{-\frac{i\lambda}{\hbar}S_x}=S_z\cos(\lambda)+S_y\sin(\lambda)$ [closed]

Prove using Hadamard's lemma that $$e^{\frac{i\lambda}{\hbar} S_x}S_ze^{-\frac{i\lambda}{\hbar}S_x}=S_z\cos(\lambda)+S_y\sin(\lambda) $$ where $\lambda$ is a complex number. I get: ...
7
votes
2answers
246 views

A question about causality and Quantum Field Theory from improper Lorentz transformation

Related post Causality and Quantum Field Theory In Peskin and Schroeder's QFT p28, the authors tried to show causality is preserved in scalar field theory. Consider commutator $$ [ \phi(x), \phi(y) ...
1
vote
0answers
58 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...
2
votes
3answers
125 views

Help understanding proof in simultaneous diagonalization

The proof is from Principles of Quantum Mechanics by Shankar. The theorem is: If $\Omega$ and $\Lambda$ are two commuting Hermitian operators, there exists (at least) a basis of common eigenvectors ...
5
votes
2answers
3k views

Fundamental Commutation Relations in Quantum Mechanics

I am trying to compile a list of fundamental commutation relations involving position, linear momentum, total angular momentum, orbital angular momentum, and spin angular momentum. Here is what I have ...
0
votes
1answer
62 views

Commutator between square position and square momentum [duplicate]

I need (as a part of one exercise) to find commutator between $\hat{x}^2$ and $\hat{p}^2$ and my derivation goes as follows: $$[\hat{x}^2,\hat{p}^2]\psi = [\hat{x}^2\hat{p}^2 - ...
1
vote
0answers
38 views

Do the norms of the total and the orbital angular momentums commute? If yes, why is there a problem with 2p_{1/2}?

Question: For $\vec L$ the orbital angular momentum of an electron, $\bar S$ its spin, and $\vec J:=\vec L+\vec S$ the sum, do $\vec J^2$ and $\vec L^2$ commute? I assume it does: $[\vec J^2,\vec ...
0
votes
2answers
43 views

RH side of the Uncertainty principle: when is it a number and when an expectation value?

The uncertainty principle between the position $x$ and the momentum $p$ is given by: $$ \sigma_x \sigma_p \geq \hbar/2,$$ whereas for the $x$ and $y$ components of the angular momentum is given by: ...
0
votes
2answers
51 views

Why is the imaginary unit conventionally put on the right hand side of commutation relations? [closed]

Commutation relations in quantum mechanics are usually written in the form $$ [x_i,p_j] = i \hbar \delta_{ij} $$ with the imaginary unit $i$ put on the right hand side of the equation. But ...
0
votes
1answer
414 views

Quantum mechanics, operator commutes with Hamiltonian

My textbook said, if an operator $\hat{O}$ commutes with the Hamiltonian, then we can use the eigen vectors of the Hamiltonian as a basis of the Hilbert space, then express the operator $\hat{O}$ in ...
-1
votes
2answers
125 views

Proving that conservation of momentum doesn't apply to electron in H-atom

To prove that the conservation of linear momentum doesn't apply to electron in H-atom, is it sufficient to show that angular momentum operator ($\hat L$) and momentum operator ($\hat p$) do not ...
2
votes
0answers
40 views

Derivative of a function of an operator [closed]

I would like to calculate something like $$K\left(z\right)=\dfrac{dF\left(A\left(z\right)+B\left(z\right)\right)}{dz}$$ i.e. the derivative of a function $F$ of a sum of two operators $A$ and $B$ ...
2
votes
1answer
44 views

Connection between half and whole integer eigenvalues for orbital angular momentum [duplicate]

I have been trying to follow this derivation from Sakurai and Shankar, pulling from both. I would like to see how the following derivation can be extended to orbital angular momentum, and thus find ...
0
votes
1answer
35 views

Mutually Commutative

What is the definition of a Mutually Commutative set of operators? I've found articles describing a complete set of mutually commutative operators, but I can't actually find what mutually commutative ...
2
votes
1answer
1k views

Proof of weaker Baker-Campbell-Hausdorff Formula [duplicate]

Prove the weaker form of the BCH Formula: $$e^A e^B = e^{A + B + \frac{1}{2}[A,B]} $$ with the assumption $[A, [B, A]] = 0; [B, [B,A]] = 0$ Start with $f(\lambda) = e^{\lambda A} e^{\lambda B} ...
10
votes
1answer
514 views

Understanding Poisson brackets

In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
0
votes
0answers
31 views

Commutator for time?

I know that in quantum mechanics, we can define space as the operator $\hat{x}=i\hbar \frac{d}{dp}$ in momentum space,and that position does not commute with momentum. However, in general relativity, ...
2
votes
1answer
105 views

Given a QFT Hamiltonian, is there a unique Lagrangian?

Consider a QFT in one spatial dimension specified by the following Hamiltonian density: $\mathcal{H} = -i \phi^\dagger \frac{\partial}{\partial x} \phi + V(\phi^\dagger,\phi)$ where $\phi$ is a ...
1
vote
2answers
228 views

Derivation of (2.45) in Peskin and Schroeder

I'm having trouble understanding the step $$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int ...
0
votes
1answer
109 views

Commutation of Hamiltonian with momentum

In which case does the Hamiltonian $H$ commutes with the momentum $P$? Can anybody help me? With an example? (No particular or strange Hamiltonians and no particular momenta are involved). How can I ...
0
votes
1answer
85 views

Where does this commutator relation come from?

What is the origin of this relation: $$ [H,a_n^\dagger] = \epsilon_n a_n^\dagger $$ for Hamiltonian $H$, creation operator $ a_n^\dagger $, and eigenvalue $ \epsilon_n $. The square brackets denote ...
0
votes
3answers
73 views

Quantization conditions/ Real Scalar field

It is often written in books, the quantization conditions for classical field theory leading to Lagrangian of a real scalar field and thus to Klein Gordon equation. And these are introduced by ...
3
votes
2answers
670 views

Proving that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space

How can I prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?
1
vote
3answers
47 views

Commutation Relationship

For the Hamiltonian of the hydrogen atom, does the square of angular momentum, $$L^2 = L_x^2+L_y^2+L_z^2$$ commute with Hamiltonian operator, $$H = \frac{1}{2m}(p_x^2+p_y^2+p_z^2) + V(r)~?$$ Should ...
0
votes
1answer
145 views

Spin operators commutation

Why do the spin operators $ S_{x1}$ and $S_{x2}$ of two particles along the $x$-axis commute i.e $S_{1x}S_{x2}-S_{2x}S_{1x}=0 $ ?
3
votes
1answer
105 views

Proving $[a_k^\dagger, a_q^\dagger]=0$

I am trying to prove the commutation relations between the creation and annihilation operators in field theory. I was already able to show that $[a_k, a_q^\dagger]=i\delta(k-q)$. I want to show that ...
5
votes
3answers
814 views

Does the canonical commutation relation fix the form of the momentum operator?

For one dimensional quantum mechanics $$[\hat{x},\hat{p}]=i\hbar $$ Does this fix univocally the form of the $\hat{p}$ operator? My bet is no because $\hat{p}$ actually depends if we are on ...
1
vote
0answers
239 views

Commutation relation of a operator with Hamiltonian [duplicate]

Given that the eigenvalues of a Hamiltonian operator $H$ are bounded below, will a Hermitian operator $T$ exist such that $[T, H] = i\hbar{\bf 1}$ identity operator?
0
votes
1answer
110 views

Commutation relation of position and momentum using Dirac notation

This is likely a very trivial/silly question, but in following a derivation of the position and momentum commutation relation using the dirac notation, I am having trouble justifying a certain step. ...
2
votes
1answer
406 views

Generalization of canonical commutation relation

The canonical commutation relation $$[x,p] = i\hbar$$ can be generalized to $$[p_i,F(\vec{x})] = -i\hbar\frac{\partial F(\vec{x})}{\partial x_i}, \ [x_i, F(\vec{p})] = i\hbar\frac{\partial ...