# 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.

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### Is it true that we can't measure position and momentum together?

The uncertainty principle states that there always will be mean variance if we measure position or momentum. It does not state that the measurement is wrong. It only states that there always will be a ...
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### Commuting the time evolution operator [closed]

Given the time evolution operator $U(t, t_0)$, I don't understand why it is true that for a time-independent operator Q, $$[Q, U(t, t_{0})] = 0 \Leftrightarrow [Q, H(t)] = 0$$ where H is the ...
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### Expanding commutator in terms of an eigenbasis

I'm trying to get some extra intuition into commutators by considering an eigenbasis expansion (where this is possible). However, when I try to expand the operators into their own eigenbasis, and do a ...
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### How to solve differential equation involving commutator and anti-commutator?

In one of my exercise, I got following differential equation for density matrix $\rho$, $$\frac{d\rho}{dt}=-i[H_1,\rho]+\{H_2,\rho\}$$ where $H_1$ and $H_2$ are the Hermitian Hamiltonian, and $[.,.]$...
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### Unexpected divergence in expectation value

I'm currently trying to calculate the expectation value $$\langle\psi(p,s)|\bar{\psi}(x)\Gamma_\rho \psi(x)|\psi(p,s)\rangle,$$ where $\Gamma_\rho$ is understood to be ...
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### Is the commutator relation $[\hat{x}, \hat{p}_x]=i\hbar 1\!\!1$ an *assumption* in the quantum theory?

This question is somewhat related to (but not by any means the same as) the question I asked recently. In his Lectures on Quantum Theory, Isham essentially says (reference given below) that if an ...
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### How do you know if a operator commutes with the hamiltonian?

In the question there is a central potential within a Hamiltonian, and I have to find the appropriate quantum numbers. They say that $j, m, s$, $\ell$ are the appropriate quantum numbers to describe ...
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### Why does Peskin and Schroeder move normal ordering move outside a commutator?

The equation trying to prove that Wick's theorem by induction in P&S on page 90 implies that normal ordering can be moved outside a commutator (at least with a positive frequency field), which I ...
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### Three operators commuting with each other

It is well known that if two operators commute, the it is possible to find common eigenfucntions for them. What if we have 3 operators that commute with each other? Will we find common eigenfunctions ...
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### Is every pair of conjugate variables associated with a Fourier transform?

For example, in quantum mechanics, the commutator of the position and momentum is $$[\hat{P_i} ;\hat{Q_j} ] =i\hbar\delta_{ij}\neq 0, i\neq j$$ I know that the position space representation of the ...
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### What can be said about the commutator of an operator with itself at different times?

In general for some smooth and bounded $\hat{V}$ $$\left[\hat{V}(t_1), \hat{V}(t_2) \right] \neq 0 \text{ if } t_1 \neq t_2$$ But what more can be said about commutators of this type? I am ...
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### Can current and voltage be linked by an uncertainty relation when electrons tunnel through a barrier?

Quantum tunneling has been shown to be linked to uncertainty relations for some observables involved in the system. For instance, if we consider electrons tunneling through a potential barrier it can ...
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### Degeneracy and Complete Sets of Commuting Observables

I want to understand how the degeneracy of an operator is related to the existence of a complete set of commuting operators that includes it. I know that if a set of operators commute, they possess a ...
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### Does the canonical commutation relation give a unique solution for the momentum operator? [duplicate]

So lets say we are in a 1d system and in the position basis just for simplicity. The CCR is: $$[x,p]=i$$ and the momentum operator is $-i\partial_x$. Is this solution unique or are there other ...
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### The commutation relation between the square components of angular momentum

So my question is as follows. I was reading about Angular Momentum from Griffiths, Introduction to Quantum Mechanics and it is a well known fact that the components of angular momentum do not commute- ...
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### Commutation relations/sturcture constants for Lorentz algebra

I am trying to compute the curvature for a gauge theory based on the pure (local) Lorentz group. The final hurdle is working with monstrous structure constants. My objective is to show that \begin{...
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### Commutation relation for deviation of two hermitian operators

On page 35, right after equation 1.4.60, Sakurai says that the commutator $$[\bigtriangleup A, \bigtriangleup B] = [A,B]$$ where $\bigtriangleup A = A - \langle A \rangle$, and $A$ is a hermitian ...
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### Eigenfunctions of compatible observables that are not shared

I'm using D.J. Griffiths's Introduction to Quantum Mechanics (3rd. ed), reading about the angular momentum operators $\mathbf L=(L_x,L_y,L_z)$ and $L^2$ in chapter 4. The author discusses ...
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### How does preservation of the Lorentz algebra demonstrate Lorentz invariance of a QFT?

In his book "Quantum Field Theory of Point Particles and Strings", Brian Hatfield makes the following claim (on p. 46) after canonically quantizing the free scalar field theory: We started with a ...
### For $[A,B]=0$, if an eigenfunction of $A$ not an eigenfunction of $B$, does that imply degeneracy of one operator?
When two operators $A$ and $B$ commute, there can be functions which are eigenfunctions of $A$ but not that of $B$. For example, in case of the one-dimensional harmonic oscillator, any linear ...