- #1

MisterX

- 764

- 71

$$\vec{E}_{classical} = C\cos\left(\mathbf{k}\cdot \mathbf{r}-\omega t + \phi\right) $$

For the quantized fields we have been using

$$\mathbf{A}\left(\mathbf{r}, t\right) \propto \int \frac{d^3k}{\sqrt{\omega}} \sum_\alpha \boldsymbol{\epsilon}_\alpha a_\alpha\left(\mathbf{k}\right)e^{i\mathbf{k}\cdot \mathbf{r}}e^{-i\omega t} + \boldsymbol{\epsilon}^*_\alpha a^\dagger_\alpha\left(\mathbf{k}\right)e^{-i\mathbf{k}\cdot \mathbf{r}}e^{i\omega t}$$

$$\mathbf{E}\left(\mathbf{r}, t\right) = -\frac{\partial}{\partial t} \mathbf{A} \;\;\;\;\;\;\;\; \mathbf{B}\left(\mathbf{r}, t\right) = \boldsymbol{\nabla}\times \mathbf{A} $$

I notice that these operators all change the occupation numbers by [itex]\pm 1[/itex]. What is the significance of this? I would like to somehow see an oscillating, measurable quantity with frequency [itex]\omega [/itex] , perhaps akin to the classical plane wave. I suppose we might be interested in the above operators, and perhaps also the average number of photons with some particular [itex]\mathbf{k} [/itex] .

I have read of the "coherent states," which would be eigenvectors of [itex]a_\alpha\left(\mathbf{k}\right) [/itex] .

$$\mid \alpha \rangle = e^{\left|\alpha\right|^2/2}\sum_n \frac{\alpha^n}{\sqrt{n!}}\mid n \rangle \;\;\;\;\;\;\;\;a\mid \alpha \rangle = \alpha\mid \alpha \rangle$$

Which evolve in time as

[itex]U\left(t, 0\right)\mid \alpha \rangle = \mid \alpha e^{-i\omega t} \rangle[/itex]. However, [itex]a[/itex] is not an observable. I am not sure how to interpret the coherent states, or the action [itex]\mathbf{A} [/itex] upon them. I worked out that $$a^\dagger\mid \alpha \rangle = e^{\left|\alpha\right|^2/2}\sum_n \frac{n\alpha^{n-1}}{\sqrt{n!}} $$

I guess my questions are relating to how to interpret the operators [itex]\mathbf{A}, \mathbf{E},\, \text{and}\, \mathbf{B}[/itex] . How do we obtain something resembling the classical oscillating quantities? What states correspond most closely with a classical plane wave? What are the eigenvectors of [itex] \mathbf{A}[/itex]?