I Explicit non-local form for the vector potential?

Amentia
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Hello everyone,

I was looking at the light matter interaction Hamiltonian and I worked out a simple calculation where I was surprised to see that I had to introduce an explicitly non-local vector potential if I want to go further:

$$\langle\psi| \boldsymbol{\hat{A}}(t)\cdot\boldsymbol{\hat{p}}|\phi\rangle = \int\int d^{3}rd^{3}r' \langle\psi|\boldsymbol{r'}\rangle\langle\boldsymbol{r'}|\boldsymbol{\hat{A}}(t)|\boldsymbol{r}\rangle\cdot\langle\boldsymbol{r}|\boldsymbol{\hat{p}}|\phi\rangle$$

Giving:

$$\langle\psi|\boldsymbol{\hat{A}}(t)\cdot\boldsymbol{\hat{p}}|\phi\rangle = \int\int d^{3}rd^{3}r' \psi^{*}(\boldsymbol{r'})\langle \boldsymbol{r'}|\boldsymbol{\hat{A}}(t)|\boldsymbol{r}\rangle\cdot\left(\frac{\hbar}{i}\right)\boldsymbol{\nabla}_{\boldsymbol{r}}\phi(\boldsymbol{r})$$

I would rewrite ##\langle \boldsymbol{r'}|\boldsymbol{\hat{A}}(t)|\boldsymbol{r}\rangle## as ##\boldsymbol{\hat{A}}(\boldsymbol{r'},\boldsymbol{r},t)##. But only ##\boldsymbol{\hat{A}}(\boldsymbol{r},t)## has been considered as a correct form for the vector potential in the literature (usually the dependence with r is a plane-wave). Perhaps I have done something wrong in my calculation although it looks simple? Or is there something in the physics related to vector potentials that I have been missing until now?

Thank you for any thoughts about that!
 
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Amentia said:
I would rewrite ##\langle r'|\boldsymbol{\hat{A}}(t)|r\rangle## as ##\boldsymbol{\hat{A}}(\boldsymbol{r'},\boldsymbol{r},t)##.
It's actually
$$\langle r'|\boldsymbol{\hat{A}}(t)|r\rangle=\boldsymbol{A}(\boldsymbol{r'},\boldsymbol{r},t)$$
i.e. there is no hat on the right-hand side because the matrix element of an operator is a c-number, not an operator. Furthermore, this c-number function is of the form
$$\boldsymbol{A}(\boldsymbol{r'},\boldsymbol{r},t)=\delta(\boldsymbol{r'}-\boldsymbol{r}) \boldsymbol{A}(\boldsymbol{r},t)$$
which removes non-locality. All this is nothing but a simple generalization of
$$\langle r'|\boldsymbol{\hat{r}}|r\rangle=\delta(\boldsymbol{r'}-\boldsymbol{r}) \boldsymbol{r}$$
 
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Thank you, this is what I was looking for because I was trying to see how a delta function could be introduced here. However, is it always correct in general? If the dependence of A was not a plane-wave or a simple analytical function of r, could we still use this trick? It could happen that the exact form of A is not known in some complex medium, a nanosystem with position-varying dielectric constant, etc.

Edit: Also A is still an operator in quantum mechanics here... I do not like hybrid notations but we usually find it written both with the dependence on r and the creation and annihilation operators. I assume the initial A is supposed to be some kind of tensor product of operators acting on two different vector spaces.
 
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Amentia said:
Thank you, this is what I was looking for because I was trying to see how a delta function could be introduced here. However, is it always correct in general? If the dependence of A was not a plane-wave or a simple analytical function of r, could we still use this trick? It could happen that the exact form of A is not known in some complex medium, a nanosystem with position-varying dielectric constant, etc.
Think of A as
$$\hat{\bf A}(t)={\bf A}(\hat{\bf r},t)$$
where ##{\bf A}({\bf r},t)## is a known function. It doesn't need to be simple or analytic, it only needs to be known. If ##{\bf A}({\bf r},t)## is unknown, then you cannot analyze the system. It's like studying Schrodinger equation with unknown potential ##V(x)##; when it's unknown then you cannot say much about the system.
 
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Thank you, I think it is a reasonable assumption to make in physics and that completely solves my question!
 
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Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
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