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I want to grasp the idea of a photon. While researching, I have come upon many different ways of describing a photon, but have found "quantum of the electromagnetic field" to be most satisfying. However, I still have a few questions about this description.

I. What does 'quantum' mean in this context? Quantum of what physical quantity?

II. What features do photons exhibit as a wave? (wavelength, speed, et cetera)

III. What features do photons exhibit as a particle? (mass, spin, et cetera)

I would especially thank if anyone could explain the momentum of a photon as a wave and as a particle. For anybody wondering, I am a high school student interested, but not fluent in physics.

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I. What does 'quantum' mean in this context? Quantum of what physical quantity?

Photons of frequency $\nu$ have energy $E = h\nu$. This means: if a photon of this frequency is absorbed or emitted, exactly this amount (quantum) of energy is transferred.

Photons of frequency $\nu$ have momentum $p = \frac{h\nu}{c}$. This means: if a photon of this frequency is absorbed or emitted, exactly this amount (quantum) of momentum is transferred. Photons move in the direction of their momentum vector.

Circularly polarized photons have angular momentum $\ell_z=\pm \hbar$. This means, if they are absorbed or emitted, exactly this amount (quantum) of angular momentum is transferred.

II. What features do photons exhibit as a wave? (wavelength, speed, et cetera)

As a wave they have frequency $\nu$ and wavelength $\lambda$, related by $$ \nu\lambda = c,$$ where $c$ is the speed of light. The speed of light is the same in all inertial frames of reference. As a wave they also have an amplitude.

As waves they can interfere (e.g. double slit experiment), be reflected from mirrors, be transmitted through matter.

III. What features do photons exhibit as a particle? (mass, spin, et cetera)

The particle properties of photons are energy, momentum, and angular momentum. They have zero rest mass. Their spin (angular momentum) depends on their polarization, but it is integer, so they are bosons. As particles, they can, for example, hit objects and exert pressure. They can be emitted and absorbed transferring discrete portions of energy, momentum, and angular momentum.

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Classical solutions of Maxwell's equations describe light, electromagnetic radiation.

enter image description here

Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. Note that the electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together

The photon belongs to the quantum mechanical frame, together with the electrons and neutrinos, see the table here of the elementary particles in the standard model of particle physics..

It can be derived mathematically that the classical electromagnetic radiation with the varying Electric and Magnetic fields emerges from a superposition of zillions of photons, where the frequency of the light gives the energy of the individual photon.

I.What does 'quantum' mean in this context? Quantum of what physical quantity?

Classical light has energy, the photon with its energy = to h*nu is a quantum of this energy. When light (EM radiation) is emitted, by atoms or charges,it is emitted as a number of photons, each photon a packet of energy building up the classical light.

II. What features do photons exhibit as a wave? (wavelength, speed, et cetera

The photon is a particle, i.e. it leaves a footprint when interacting with matter at an (x,y,z) like a classical particle would, and it is also a probability wave, i.e. many photons of the same energy passing through a double slit show an interference pattern, which is a wave property. Its speed is the velocity of light in vacuum, by construction of the standard model, and there has been no falsification of this "axiom".

enter image description here

on the left it is the individual footprints of photons, but the accumulations shows the classical light interference.

III. What features do photons exhibit as a particle? (mass, spin, et cetera)

Check the table. A photons mass is zero, it obeys special relativity rules as all elementary particles.

The momentum of the photon is given by the four vector that describes it , substituting in the special relativity equations.

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I. A photon is the quantum, or the basic building block, of the electromagnetic field. For example, visible light, which is an electromagnetic field, is a large collection of photons.

Photons exhibit wave-particle duality. This means that they have some properties that exhibit their wave-like properties manifestly, while some other properties exhibit the particle-like properties.

II. Wave-like properties - Some frequency/wavelength and processes like reflection, refraction, diffraction, interference, polarization and dispersion

III. Particle-like properties - Blackbody radiation, Photoelectric effect, Compton scattering, pair production, non-zero momentum (and energy) causing radiation pressure and bending of light under the action of gravity, spin angular momentum

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