Hi; I am curious about the correspondence between the E&M fields and photons in Quantum Electrodynamics. I want to calculate the electric and magnetic fields produced by a single photon. This may look like a rather long post containing an elaborate question designed to stump everyone, but it isn't. It is long simply because I am defining my notation here.(adsbygoogle = window.adsbygoogle || []).push({});

The way I approached this is by realizing that the best I can do is find theexpectedvalues for each of the four components of the electromagnetic vector potential, [itex]A_\mu[/itex]. From there, I differentiate appropriately to arrive at theexpectedvalue of the electric and magnetic fields.

If my interpretation is right, the vector field operator, when sandwiched between two kets in Dirac's notation,shouldyield the expected value of the vector field for the system described by the ket, just as how one would expect ordinary operators in quantum mechanics to behave. So, in my case, I would like to calculate

[tex]\langle A(x)\rangle_\mu=\langle\Psi|\hat{A}_\mu(x)|\Psi\rangle[/tex],

where [itex]|\Psi\rangle[/itex] represents the state of the system, which is the one-photon wavepacket,

[tex]|\Psi\rangle=\int \frac{d^3k}{(2\pi)^3}f(k)\hat{a}^{r\dagger}_k|0\rangle[/tex].

Here, [itex]f(k)[/itex] is the fourier decomposition of the modes, which allows me to construct my wavepacket, and [itex]\hat{a}^{r\dagger}_k[/itex] is the creation operator which creates a photon with momentum, [itex]k[/itex] and with a polarization vector [itex]r[/itex], when it acts on the vacuum state, [itex]|0\rangle[/itex]. The planewave solution to differential equation that describes free photon propagation is

[tex]\hat{A}_\mu=\int\frac{d^3p}{(2\pi)^3}\frac{1}{2\omega_p}\sum_r\big[\hat{a}^r_p\epsilon_\mu^r(p)e^{-ip\cdot x}+\hat{a}^{r\dagger}_p\epsilon^{r*}_\mu(p)e^{ip\cdot x}\big][/itex].

The integration, here, runs over all possible momentum, and [itex]\omega_p[/itex] is the frequency of mode [itex]p[/itex] to be interpreted as the energy of the particle. The summation runs over all polarization states, and [itex]\epsilon^r_\mu[/itex] is the polarization vector.

So, all I have to do is substitute these relations into the first centered equation, right? Unfortunately, the bra and the ket states each contribute a single ladder operator and the field operator contributes one for each term. Thus, the expression contains many terms with an odd number of ladder operators sandwiched between the vacuum states, which we all know vanishes. So I find myself predicting a 0 expectation value for the electromagnetic field, which is clearly wrong. If I try to the extend this argument for many photon states, I arrive at the same dilema since the bra and the ket together will always contribute an even number of ladder operators.

PLEASE, does anyone have any ideas? My intuition tells me this should be a straightforward calculation, so please help!!! :grumpy:

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# Electric and Magnetic Fields due to Photon

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