# 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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### Canonical transformations in Quantum Mechanics

Heisenberg's famous commutator of a pair of conjugated canonical variables is formulated for position and conjugated momentum $$[q,p] = i\hbar.$$ Intuitively I would guess that it would also work ...
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### How does Dirac prove that the canonical coordinates $q_r$ form a complete commuting set of observables?

I'm reading "The principles of quantum mechanics" by Paul Dirac, and I've reached the point where he introduces the momentum operator. After showing that the $p_r$'s commute with each other ...
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### Set of Compatible Observables and Simultaneous Eigenkets Basis

I have managed to prove that a finite set of compatible observables [i.e. pairwise commuting (possibly unbounded) selfadjoint operators] on an Hilbert space $\{H\}$ admits an orthonormal basis of ...
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### Maximality and Completeness for Set of Compatible Observables

Let $\mathcal{A} = \{A_j\}_{j=1}^{N}$ be a finite family of compatible observables (they commute pairwise) on an Hilbert space $H$. Consider the following definitions: $\mathcal{A}$ is Maximal if and ...
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### If something commutes with $x$, it commutes with any $f(x)$

I often read that if some operator commutes with (say) the position operator $x$, it commutes with any function $f(x)$ of it. For analytical functions, this is obvious, but I was wondering if there ...
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### How can difference of two operators produce a constant?

In the canonical commutation relations, $$[\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i \hbar$$ Both $\hat{x}$ and $\hat{p}$ are operators and so are $\hat{x}\hat{p}$ and $\hat{p}\hat{x}$. ...
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