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I am struggling a bit to understand the concept of gauge invariance/dependence with canonical momentum. For instance, if we consider a Hamiltonian of a particle in an electromagnetic field described with potentials ($\textbf{A}, \varphi$): $$\mathcal{H}=-\frac{\hbar^2}{2m}\left(\nabla-i\frac{q}{\hbar}\textbf{A}\right)^2+q\varphi$$ it can be shown that kinetic momentum, $\nabla-i\frac{q}{\hbar}$, is gauge invariant, while the canonical momentum is not. On the other hand, consider a Hamiltonian of a particle in some scalar potential, $V$: $$\mathcal{H}'=-\frac{\hbar^2}{2m}\nabla^2+V(\textbf{r})$$ In this second case, what can be said about the canonical momentum, $-i\hbar\nabla$, in terms of the gauge status? Is it gauge dependent, independent, or the gauge status makes no sense here?

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The gauge choice of a scalar potential corresponds to adding a constant. This choice does not affect the momentum, i.e. the canonical momentum is unchanged by switching gauge, and thus always equals the kinetic momentum.

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