# Photons in non-relativistic quantum mechanics

I have just finished my very first quantum theory of matter course and everything we did was strictly non relativistic. No QFTs whatsoever, no creation and annihilation operators, no mention to the Dirac equation. The most relativistic thing we did was the spin orbit interaction, and even then it wasn't fully justified (the factor $$\frac{1}{2}$$ popped out of nowhere and we treated everything with standard non relativistic perturbation theory). Despite that the word photon kept coming up. I know it is a strictly relativistic thing but I just care about getting an intuitive idea of what it is and not a rigorous one (I'll leave that for the future).

1. If I have an em field is it correct to say that the photon is spread out inside this field?

2. When a photon is emitted from an atom is it an electromagnetic wavepacket with a wavelength kinda constant but not really (or it wouldn't be a wavepacket but rather a monochromatic wave)? Does it make sense to calculate the amplitude of this wavepacket using $$h \nu = \text{Volume in space of the wave packet} \times \epsilon_{0} E^{2}$$ (i. e equating the energy of the photon to the classical em energy)?

3. Is an electromagnetic wave made of a single photon what the wavefunction of a single electron is to the electron?

4. Suppose I have a em wave made of a single photon in a large cavity. Suppose the em wave is constant in the entire cavity. Can I localize the photon with a measurement and say "It's there, with a precision (say) + or minus .1 mm". What happens to the em field then? Does it contract to that single point (+/- .1 mm) to expand again in future times to cover the whole cavity again? Is there a way to predict this time evolution with a (relativistic) Schroedinger-like equation?

• – John Rennie May 29 at 16:23
• – John Rennie May 29 at 16:24
• – John Rennie May 29 at 16:24

In the standard model of particle physics which is a quantum field theoretical model, the photon is a point particle, of mass zero, spin 1, and $$energy = hν$$, in the table of particles. All the fundamental elementary particles are point particles.

1) If I have an em field is it correct to say that the photon is spread out inside this field?

No each photon is a point particle, its (x,y,z,t)given by the probability of the complex conjugate square of its wavefunction

2) When a photon is emitted from an atom is it an electromagnetic wavepacket

It is a point particle for each individual emission. There is a width to the lines emitting, so the spectral lines have a frequency width.

3) Is an electromagnetic wave made of a single photon what the wavefunction of a single electron is to the electron?

No. The classical electromagnetic wave is an emergent phenomenon from the quantum field theoretical base. It can be shown mathematically in quantum field theory that the classical electromagnetic field is built up by zillions of photons of the appropriate frequency.

4) Suppose I have a em wave made of a single photon in a large cavity. Suppose the em wave is constant in the entire cavity.

An electromagnetic wave is composed out of zillions of photons, you can get an estimate of the huge numbers involed by taking the energy of such a wave and dividing it by $$hν$$

Can I localize the photon with a measurement and say "It's there, with a precision (say) + or minus .1 mm".

After the fact, footprints of photons can be seen in single photon experiments, see here. . The classical interference is built up as the single photons (point footsteps) accumulate.

• The wavefunction of the photon is a controversial topic. I don’t find any evidence that Raymer and Smith ever managed to get their 14-year-old preprint published. Do you know if they did? – G. Smith May 29 at 16:57
• @G.Smith It is controversial because there are various ways of defining one. I tend to trust that CERN would not keep an inaccurate document on its document server. – anna v May 29 at 18:00
• In my opinion, if it has never been published in a reputable journal after 14 years, it has zero credibility, regardless of whose server it happens to be sitting on. – G. Smith May 29 at 18:46
• @G.Smith see Smith and Raymer:"Photon wave functions, wave-packet quantization of light, and coherence theory" New Journal of Physics 9 (2007) 414 PII: S1367-2630(07)56746-7 – hyportnex May 29 at 20:10
• @hyportnex Thank you for this reference, which I consider more much credible. – G. Smith May 29 at 20:25