The Stack Overflow podcast is back! Listen to an interview with our new CEO.

# 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.

646 questions
Filter by
Sorted by
Tagged with
471 views

### Commutation of two vector operator [closed]

Consider vectors $\overrightarrow { A }$ and $\overrightarrow { B }$ as operators or vector of operators. If this commutation holds$$[\overrightarrow { A },\overrightarrow { B }]=0$$ Then, is that ...
119 views

887 views

### Why do the ladder operators in harmonic oscillators work?

The Hamiltonian can be diagonalized by transforming $x$ and $p$ to $a$ and $a^\dagger$. I understand how one proceeds from there to find the spectrum of $a^\dagger a$, the ground state $|0\rangle$ and ...
992 views

### What is the commutator of the exponential derivative operator and the exponential position operator?

What is the commutator of the exponential derivative operator and the exponential position operator? \begin{align} \left[\exp(\partial_x),\exp(x)\right] =\exp(\partial_x)\exp(x) -\exp(x)\exp(\...
490 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 ...
936 views

### Schrödinger equation for time dependent Hamiltonian and conjugation

The Schrödinger equation for the evolution operator reads: $$\frac{\partial U}{\partial t} = -\frac{i}{\hbar}HU$$ where for a time dependent Hamiltonian which need not commute with itself at ...
2k views

### How to recognize a Complete Set of Commuting Operators (CSCO)

A question about 'completeness'. These two operators are commuting, but I want to know more about their completeness. How do you know if {H}, {B}, {H,B} and/or {$H^2$,B} are forming (a) Complete Set(...
10k views

### Commutator $[\hat{p},F(\hat{x})]$ of Momentum $\hat{p}$ with a Position dependent function $F(\hat{x})$?

I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\hbar$ times the derivative of the position dependent quantity. Can someone point me towards a ...
1k views

421 views

### Is obtaining the coordinate representation of momentum operator from commutator more fundamental than generator of translation

Related post: What is the most general expression for the coordinate representation of momentum operator? There are two methods of obtaining the coordinate representation of momentum in quantum ...
639 views

### The Physical Meaning behind a Commutator [duplicate]

I've just been introduced to the idea of commutators and I'm aware that it's not a trivial thing if two operators $A$ and $B$ commute, i.e. if two Hermitian operators commute then the eigenvalues of ...
2k views

### A few simple questions about Grassmann numbers: commutation relations and derivatives

I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions ...
86 views

### Is is possible to have a pair commuting observables only in a single direction?

In quantum mechanics, for two observables to be compatible, successive measurements of the observables, say $A$ and $B$, should yield the same result as earlier, i.e if we do the measurements with the ...
1k views

### Commutation relations of the generators of the Lorentz group

$$J^{\mu\nu} = i(x^\mu\partial^\nu-x^\nu\partial^\mu). \tag{3.16}$$ We will soon see that these six operators generate the three boosts and three rotations of the Lorentz group. To determine ...
230 views

### Differential operators inside commutators

I started learning QED and I'm making my way through some introductory literature. I encountered a problem in a section that derives the commutator between the fields, $\left[A_i(x);E^j(y) \right]$. ...