- #1

Amentia

- 110

- 5

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!

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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