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  1. de Broglie's motivation to introduce matter waves was an analogy with wave-corpuscular dualism of light: he conjectured that also electrons may behave sometimes like particles and sometimes like waves.

  2. The hypothetical connection between these two points of view on electron came again from an analogy with optics. Geometric optics with its notion of rays is a limit case of the wave optics when the wave length is very small.

Do I understand correctly that classical trajectories of an electron are analogous to rays in optics? Is it true that in optics rays may be considered as trajectories of photons in some sense?

A refrerence would be very helpful.

MKO
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    I think this is basically a duplicate of What is the relation between electromagnetic wave and photon? – John Rennie Dec 08 '20 at 08:30
  • @JohnRennie : I think the answers there do not answer my questions. – MKO Dec 08 '20 at 08:55
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    In a sense, I think that the analogy between classical trajectories of electrons and rays in optics could be valid (think about electron optics, electrostatic lenses and so on to steer electron beams). To me, the breakdown of this viewpoint, both for electrons and photons is represented by interference. For optics, think about a Mach-Zender interferometer: since photons can't split, if rays were their trajectories, which path would they follow? Even worst in the single-photon case. If I remember correctly, you can find something about this in Gerry and Knight's "Introductory quantum optics" – JackI Dec 08 '20 at 10:50
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    I think yes they can, but taking into account the answer by Carl Witthoft. Like drawing electrons around a nucleus, once we know that is not the correct picture it can turn out useful nevertheless. – Alchimista Dec 08 '20 at 14:19
  • Classical trajectories don't work very well at the quantum level. It's bad enough for particles with mass like protons or electrons, but even worse for photons, since you can't even get a probability of the location of a photon in flight. – PM 2Ring Dec 08 '20 at 14:28
  • I am back to point out that the dualism for a photon should be in its associated material wave rather than the electromagnetical one which is light. So my qualitative comment above might be about an even more pictorial use than what I have originally thought. – Alchimista Dec 09 '20 at 11:45
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    @JohnRennie I think this question is not a duplicate. It is complementary to the indicated question, and no answer in the supposed duplicate addresses the question here. The key point is not the relation between quanta and waves but between quanta, classical trajectories, and geometric optics. – GiorgioP-DoomsdayClockIsAt-90 Dec 10 '20 at 07:24

2 Answers2

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Do I understand correctly that classical trajectories of an electron are analogous to rays in optics?

Basically, yes. Schrödinger's guiding idea, when he arrived at its celebrated equation, was the relationship between geometric and wave optics. In particular, in a 1926 paper ( Quantization as a Problem of Proper Values-II ) he clearly explained his idea of looking at Hamilton-Jacobi's equation of Classical Mechanics as the analogous of the Eikonal approximation of undulatory theory of light. However, notice that the concept of a ray of geometric optics does not completely coincide with a particle trajectory. It should be better thought of as the evolution of a small section of a wave-front. It turns out that in many cases the evolution of such a small section looks like a classical trajectory but not always. A typical geometric optics phenomenon like the simultaneous reflection and refraction at a surface is far from the behavior of a classical particle.

Is it true that in optics rays may be considered as trajectories of photons in some sense?

Definitely not. At least, not if by photon one is meaning the entity described by Quantum Electrodynamics (QED). The reason is twofold.

  1. Light waves emerge as a superposition of a huge number of photons. In a way, the motion of a wave-front cannot give information about the trajectories of the individual constituents more than the water wave motion provides information about the trajectories of the individual water molecules.
  2. in the case of QED photons there is an additional problem connected with the problem of defining its position.
GiorgioP-DoomsdayClockIsAt-90
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No.
The classical ray diagram is completely deterministic, in that the 'ray' of light goes exactly where the line points.

Photons, and high-velocity particles with mass, show up at a non-deterministic location based on wave & diffraction theory.

Carl Witthoft
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    I agree that strictly speaking photons have no trajectories. The question was whether it is possible in some approximation under special circumstances. Like geometric optics is a limit case of wave optics. – MKO Dec 08 '20 at 18:07