# Questions tagged [commutator]

A mathematical construct quantifying the difference in effect of applying two operators in two alternate successions. It is the defining product of a Lie algebra, the efficient underlying description of Lie groups, of use in several areas of physics, most notably quantum field theory.

629 questions
Filter by
Sorted by
Tagged with
0answers
83 views

### Definitions of operators and commutativity in quantum mechanics

If $[\hat A,\hat B] = 0$, where $\hat A$ and $\hat B$ are operators, then the operators commute. This also means that, when applied to a wavefunction, that one can measure observables $A$ and $B$ in ...
1answer
125 views

### Do all well-measured observables effectively commute?

Do all well-measured observables effectively commute with each other? The rest of this long post clarifies what I mean by that simple-looking question. Consider quantum field theory in Minkowski ...
1answer
102 views

1answer
72 views

### If $\psi$ acting on $a_+$ and $a_-$ operator just moves up and down the ladder, why is $[a_-, a_+] = 1$ and not 0? [duplicate]

If $\psi$ is acted upon by both the operators one by one, it should return the same wave function. Thus order in which you increase or decrease the energy shouldn't matter. Then why is it so that the ...
3answers
375 views

### Is the Heisenberg-picture commutator $[x(t),p(t)]$ between position and momentum always equal to $i\hbar$?

Some misconceptions over here, For $x=$position and $p=$ momentum, I know $[x,p]=i\hbar$ but does $[x(t),p(t)]$ still have the same relation where $t$ here represents time.
1answer
172 views

### Propagator Causality with commutators all the way

We know that two fields commute - by locality and causality - iff there is spacelike separation $\left[\phi_l^k(x) , \phi_m^{k'}(y)\right] = 0$ for $(x-y)^2<0$ In the canonical quantization of ...
1answer
82 views

### A problem in three sequential selective measurements in case of incompatible observables

In case (a) the B filter takes care of B's eigenvalues and we sum all of their probabilities to calculate the probability of obtaining $|c'\rangle$. In case (b) B filter is not used and this creates ...
1answer
119 views

### The creation and annihilation operators in quantum mechanics

What is the result of the commutation relation between the creation operator and a power of the annihilation operators in simple harmonic oscillator problem?
1answer
270 views

1answer
72 views

1answer
66 views

### How to prove commutation relation between charge and current in current algebra?

I am reading Gauge Theory of Elementary Particle Physics by Tapei Cheng and Lingfong Li. Proceeding equation 5.54, there is a statement which says Then from Lorentz covariance, we can include the ...
0answers
34 views

### Commutation relation, field and conserved charges

One more question on basic commutation relation for fields. Let $\phi(x)$ be a scalar field and $$P^\alpha = \int d^3x T^{0\alpha},$$ where $T^{\alpha\beta}$ is the energy momentum tensor. Now, ...
2answers
195 views

### Physical meaning of commutation

I was reading the solution to quantum harmonic oscillator by J.J. Sakurai. He uses the annihilation and creation operators and there's a key step (I think) which is $$[a,a^{\dagger}]=1$$ I know we can ...
1answer
81 views

### Can non-central hamiltonians commute with $\vec{L}^2$?

Central potentials $V(r)$ trivially commute with the operator $\vec{L}^2$ in quantum mechanics because the latter is a function of the angular coordinates $(\theta,\phi)$ only. Non-central potentials, ...
1answer
102 views

### Find the commutator $[AX+BY,Z]$

The problem I'm asked to solve is on quantum mechanics: Find the commutator $[xH+pH, p^2]$, where $H$, $x$ and $p$ are the Hamiltonian, space and momentum operator respectively A the moment, ...
1answer
145 views

### Show that $\vec L$ and $\vec S$ commute with each other

It is stated in Griffiths in a hint to a question that $\vec L$ and $\vec S$ commute with each other but no proof is given. $\vec L$ is given in the differential form and $\vec S$ is given in matrix ...
1answer
93 views

### Momentum operator ambiguous?

In nonrelativistic quantum mechanics, are different operators possible as a candidate for the momentum operator, given that one has fixed one position operator and a hilbert space that this position ...
2answers
137 views

### How to prove the translation generator commutes with the spinors in SUSY algebra?

I was reading Modern Supersymmetry by John Terning, the book starts with SUSY algebra and says $$\left[ P_{\mu} , Q_{\alpha} \right] = \left[ P_{\mu} , Q_{\alpha}^{\dagger} \right] = 0$$ I am ...
1answer
157 views

### Solving the von-Neumann equation explicitly [closed]

The von-Neumann equation reads: $$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho]$$ The solution: $$\rho(t)=U\rho U^{\dagger}$$ with $U=e^{-\frac{iHt}{\hbar}}$ is easily obtained when starting from the ...
1answer
83 views

### Strange question involving finding a relation between a commutator and the time derivative of an operator

In order to get to the parts I am stuck at, I will add the examiners' solutions to each subquestion, which is needed to get to the subquestion that I am querying. The following is a bizarre question ...
1answer
36 views

### Energy $E_n$ for eigenstates $n$ and position probabilistic distribution

Energy and position operator does not commute. Uncertainty relation and Energy-Position interference So how come "For a particle in a box with given states $n$," and we obtained the $E_n$ for exact ...
0answers
63 views

1answer
125 views

### Misner, Thorne and Wheeler, Box 9.2 Commutator … doesn't make sense to me

I apologize for the goofy commutator $\left[\left[\_,\_\right]\right]$ notation. MathJax doesn't like my \llbracket \rrbracket notation. And I religiously use $\left[\dots\right]$ for function ...
2answers
577 views

### What is the mistake in calculating such a commutator? [duplicate]

$B$ is an Hermitian operator in Hilbert space, and $|b\rangle$ is the eigenstate of $B$. We can have $[A, B] = 1$ where A is arbitary operator. Then we can calculate as below: \begin{align} &\...
0answers
68 views

### In which cases is the Schrodinger representation more useful?

Dirac in his brilliant book derived quantum mechanics using non-commuting operators $\hat{q}$ and $\hat{p}$. He related these to the Schrodinger representation using a wavefunction $\psi(x)$ and the ...
1answer
80 views

### Is $:A: \; \;= A - \left<0\right|A\left|0\right>$ a correct definition of normal ordering?

My course notes say that normal ordering is defined as $$:A: \;\; = A - \left< 0\right| A \left| 0\right>.\tag{1}$$ This works for $A = aa^\dagger$ and all already normal ordered expressions. ...
3answers
116 views

### How to see that $[\textbf{p}^2,\textbf{L}^2]=[\textbf{p}^4,\textbf{L}^2]=0$ without doing any messy algebra?

The Hamiltonian of a particle moving under the influence of a central potential $V(r)$ given by $$H=\frac{\textbf{p}^2}{2m}+V(r)$$ commutes with $\textbf{L}^2\equiv L_x^2+L_y^2+L_z^2$. Without doing a ...
1answer
84 views

### How to prove a set of matrices form a representation of Lie algebra?

When reading Paul Langacker's The Standard Model and Beyond, I am quite confused on equation 3.29, which says with a set of fields $\Phi _a$, where $a$ goes from 1 to $n$, are chosen to be transformed ...
1answer
189 views

1answer
48 views

0answers
58 views

### Ladder functions and operators - commutation and product

I have been studying An introduction to QFT by Peskin and Schroesder, in my free time, to learn about the fascinating subject of QFT. I have a couple of basic ...