User:Inconspicuum/Physics (A Level)/Quantum Behaviour

So far, we have identified the fact that light travels in quanta, called photons, and that these photons carry an amount of energy which is proportional to their frequency. We also know that photons aren't waves or particles in the traditional sense of either word. Instead, they are lumps of energy. They don't behave the way we would expect them to.

Many Paths
In fact, what photons do when they are travelling is to take every path possible. If a photon has to travel from point A to point B it will travel in a straight line and loop the loop and go via Alpha Centauri and take every other possible path. This is the photon's so-called 'quantum state'. It is spread out across all space.

However, just because a photon could end up anywhere in space does not mean that it has an equal probability of ending up in any given place. It is far more likely that a photon from a torch I am carrying will end up hitting the ground in front of me than it is that the same photon will hit me on the back of the head. But both are possible. Light can go round corners; just very rarely!

Calculating Probabilities
The probability of a photon ending up at any given point in space relative to another point in space can be calculated by considering a selection of the paths the photon takes to each point. The more paths considered, the greater the accuracy of the calculation. Use the following steps when doing this:

1. Define the light source.

2. Work out the frequency of the photon.

3. Define any objects which the light cannot pass through.

4. Define the first point you wish to consider.

5. Define a set of paths from the source to the point being considered, the more, the better.

6. Work out the time taken to traverse one of the paths.

7. Work out how many phasor rotations this corresponds to.

8. Draw an arrow representing the final phasor arrow.

9. Repeat steps 6-8 for each of the paths.

10. Add all the phasor arrows together, tip-to-tail.

11. Square the amplitude of this resultant phasor arrow to gain the intensity of the light at this point. It may help to imagine a square rotating around, instead of an arrow.

12. Repeat steps 4-11 for every point you wish to consider. The more points you consider, the more accurate your probability distribution will be.

13. Compare all the resultant intensities to gain a probability distribution which describes the probabilities of a photon arriving at one point to another. For example, if the intensity of light at one point is two times the intensity of light at another, then it is twice as likely that a photon will arrive at the first point than the second.

14. If all the points being considered were on a screen, the intensities show you the relative brightnesses of light at each of the points.

Examples
If we now take this method and apply it to several situations, we find that, in many cases, the results are similar to those obtained when treating light as a wave, with the exception that we can now reconcile this idea with the observable 'lumpiness' of light, and can acknowledge the fact that there is a certain probability that some light will not behave according to some wave laws.

Travelling from A to B
This is the simplest example to consider. If we consider a range of paths going from point A to point B, and calculate the phasor directions at the end of the paths, we get a resultant phasor arrow which gives us some amplitude at point B. Since there are no obstructions, at any point this far away from the source, we will get the same resultant amplitude.

It is important to note that different paths contribute to the resultant amplitude by different amounts. The paths closer to the straight line between the two points are more parallel to the resultant angle, whereas the paths further away vary in direction more, and so tend to cancel each other out. The conclusion: light travelling in straight lines contributes most to the resultant amplitude.

Young's Slits
Here, we just need to consider two paths: one through each slit. We can then calculate two phasor arrows, add them together to gain a resultant phasor arrow, and square its amplitude to gain the intensity of the light at the point the two paths went to. When calculated, these intensities give a pattern of light and dark fringes, just as predicted by the wave theory.



Reflection
This situation is very similar to what happens when light travels in a 'straight line'. The only difference is that we consider the paths which involve rebounding off an obstacle. The results are more or less the same, but the paths from which they were obtained are different. This means that we can assume the same conclusions about these different paths: that most of the resultant amplitude comes from the part of the mirror where the angle of incidence equals the angle of reflection. In other words, the likelihood is that a photon will behave as if mirrors work according to wave theory.



Refraction
Different paths have different lengths, and so photons take different amounts of time to traverse them (these are known as trip times). In the diagram on the right, the photons again traverse all possible paths. However, the paths with the smallest difference between trip times have phasor arrows with the smallest difference in direction, so the paths with the smallest trip times contribute most to the resultant amplitude. This shortest path is given by Snell's law. Yet again, quantum physics provides a more accurate picture of something which has already been explained to some degree.

Diffraction
Diffraction occurs when the photons are blocked from taking every other path. This occurs when light passes through a gap less than 1 wavelength wide. The result is that, where the amplitudes would have roughly cancelled each other out, they do not, and so light spreads out in directions it would not normally spread out in. This explains diffraction, instead of just being able to observe and calculate what happens.