# 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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### How to show that if two operators $A$ and $B$ commute, then simultaneous accurate measurement is possible?

I have proved that if two operators commute then their simultaneous accurate measurement is possible using the uncertainty equation but I am unable to do so without using it. I have tried and reached ...
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### Uniqueness of Fock space

Given a single-particle Hilbert space, it's not hard to construct a Fock space using tensor products and symmetrization/anti-symmetrization projection operators, from which we can define creation/...
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### The Different Propagators for the KG Field

I am currently working through the first few chapters of Peskin and Schroeder and have arrived at the sections where the different propagators are discussed. In regular Quantum Mechanics, the ...
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### Complete set of common eigenfunctions of two commuting operators (reverse theorem) [closed]

Problem 3.16 of introduction to quantum mechanics by David J. Griffiths states that [\hat{P},\hat{Q}] \neq 0 \implies \hat{P} \text{ and } \hat{Q} \text{ do not have a complete set of ...
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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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### Doubt about commutator of operators

I have a doubt regarding commutator algebra. I've seen this expression $$[A,B^n] = nB^{n-1} [A,B]\tag{1}$$ and have used this often for position and momentum operators. However, I want to know when ...
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### A Mathematical Formulae for Peskin and Schroeder's Exercise 2.2 (a)

I am self studying Quantum Field Theory and I am using the book An Introduction to Quantum Field Theory by Peskin and Schroeder. Currently I am working on problem 2.2 (a). In the textbook problem, the ...
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### Biconditionality of the compatibility theorem for commuting operators

The compatibility theorem states that, if the operators $\hat A$ and $\hat B$ representing the observables $A$ and $B$ do commute, then there exists a common eigenbasis for their eigenstates, and ...
Consider an unitary transformation $$\hat{D}(f) = e^{-\frac{i}{2\hbar}f(t)\hat{q}^2}$$ from the book I find that $\hat{D}\hat{p}\hat{D}^{\dagger} = \hat{p} + f(t)\hat{q}$, ...
Given the commutation relation $$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$ and define the Fourier transform as \tilde{\phi}(t,\...