Particular Preferred Paths

- Photons tend to take the shortest paths, because those are the quickest to complete, meaning the phasors line up more at the end
- (And a greater resultant phasor amplitude means a greater probability of photon arrival – see the other post)
- The shortest/quickest paths are the ones that are closest to a straight line, which is why light as we know it always goes in straight lines
- Fermat’s Least Time Principle states that the path taken by light is always the quickest path

Photon Persuasion (altering probability to work in your favour…)

- Even though photon paths are completely random and only obey probability, there are ways of getting them to go where you want…
- You can improve the probability of photons arriving in certain places by altering the time it’ll take photons to travel there, in order to get them to arrive in phase
- If you’re creating a lens and you want photons to converge in somewhere on the other side, you can slow the photons on the shortest paths down, using thick glass, to get them to arrive in phase with the photons on the shorter paths
- Sometimes, reducing the possible paths helps
- You can get photons to pick particular paths by… eliminating (blocking) all the paths which would cancel it out
- Using this method, and controlling the angle of incidence, slit width/separation and distance to screen, diffraction gratings can be designed to produce maxima in certain places (I bet that’s difficult, though…)

Electrons Also Quantum

- Photons are the quanta of electromagnetic waves
- Electrons are also quanta – they’re particles, but they do wave stuff
- You can prove this by diffracting electrons
- The because of the wavelength of electrons the spaces between atoms make quite nice diffraction gratings for them
- The gaps between atoms aren’t two nice, parallel, vertical slits, though, so the interference pattern will be circular – a load of rings, of varying brightnesses
- You can use an electron gun to provide electrons, a bit of graphite as your grating, and a phosphor screen as the screen (phosphor so that electrons show up on it as light)

How Things Are Different

- There are many differences between photons and electrons, so they do quantum a bit differently (“doing quantum” is not a thing)
- Photons don’t have mass, but electrons have a tiny (by our standards) mass
- Photons travel at C, the speed of light (3*10^8 ms^-1, but you knew that) whereas electrons travel at any old speed below the speed of light… let’s call it v
- (C is a constant, but v definitely isn’t!)
- Photons have no charge, but electrons have a small (by our standards) negative charge
- Photons don’t orbit atoms, but electrons do (so we talk about free electrons, when we’re quantuming them [“quantuming” something is not a thing. I honestly thought I had a better vocabulary than this. I must be tired. Yeah, I guess I am. I’ve been blogging this stuff practically all day])
- They both explore all possible paths, though, so at least they have something in common…

The Stuff About An Electron (*the properties of quantum electrons*, perhaps)

- For a free electron, the speed depends on the energy
- Kinetic energy is given by 0.5 * m * v^2 (where m is mass and v is speed/velocity)
- The mass of an electron is 9.11 *10^(-31) kg (but you’ll find that in the formula booklet)
- The ‘frequency’ of an electron can be worked out using Planck’s constant:
- f = E / h (where f is the frequency, E is the kinetic energy, and h is Planck’s constant)
- The way of imagining a wavelength of an electron must have been invented by someone called de Broglie, I suppose, because the wavelength of an electron is called the de Broglie wavelength, and can be worked out as follows:
- λ=h/mv (where lambda is the de Broglie wavelength, h is Planck’s constant, m is the mass – of an electron – and v is the speed. mv happens to be the momentum, too)

Effects on Electron Diffraction

- nλ = d sin(θ), still, where n is an integer, lambda is the wavelength (de Broglie, in this case), d is the slit separation and theta is the angle), but the wavelength, and, thus, the angle, change depending on the kinetic energy of the electron
- The slit separation must be roughly equal to the de Broglie wavelength if the diffraction is to work properly
- As you increase the voltage supply to the electron gun in the setup I described earlier, the rings get closer together/smaller/closer to the middle
- This is because the wavelength has been changed, and the wavelength relative to slit width dictates how spread out the interference pattern will be
- Giving the electron gun more volts presumably makes it fire electrons at higher (kinetic) energies, which then have more momentum…
- …using the e=h/mv equation from above, you can see that electrons with higher momentums have shorter wavelengths…
- …and electrons with shorter wavelengths diffract less when going through the same slit
- What this explanation was ignoring, though, is the potential difference / charge on the electrons
- Also, as the momentum of electrons gets very large, their quantum behaviour apparently gets less noticeable

###### Related articles

- Photoelectric Effect, Photomultipliers and Resolution – AS Physics Revision (mattg99.wordpress.com)
- Physicists Unite! I Need Your Help… (underwaterraven.wordpress.com)
- Planck’s Constant and Quantum Theory (mattg99.wordpress.com)
- Photon Engineering and Quantum Electrons (mattg99.wordpress.com)

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