A-level Physics (Advancing Physics)/Light as a Quantum Phenomenon

We have already seen how light behaves like both a wave and a particle, yet can be proven not to be either. This idea is not limited to light, but we will start our brief look at quantum physics with light, since it is easiest to understand.

Quantum physics is the study of quanta. A quantum is, to quote Wiktionary, "The smallest possible, and therefore indivisible, unit of a given quantity or quantifiable phenomenon". The quantum of light is the photon. We are not describing it as a particle or a wave, as such, but as a lump of energy which behaves like a particle and a wave in some cases. We are saying that the photon is the smallest part of light which could be measured, given perfect equipment. A photon is, technically, an elementary particle. It is also the carrier of all electromagnetic radiation. However, its behaviour - quantum behaviour - is completely weird, so we call it a quantum.

Dim Photos
The easiest evidence to understand is dim photographs. When you take a photo with very little light, it appears 'grainy', such as the image on the right. This means that the light is arriving at the camera in lumps. If light were a wave, we would expect the photograph to appear dimmer, but uniformly so. In reality, we get clumps of light distributed randomly across the image, although the density of the random lumps is higher on the more reflective materials (the nuts). This idea of randomness, according to rules, is essential to quantum physics.



Photoelectric Effect
The second piece of evidence is more complex, but more useful since a rule can be derived from it. It can be shown experimentally that when electromagnetic radiation (e.g. light) is shone onto the surface of a metal, it can cause electrons to be emitted. Shining the right type of light with a positively charged terminal nearby can generate a current if done in a vacuum.

Scientists found that the amount of light increased the amount of current, which was to be expected. They also found that lower frequency light, which has less energy also gave lower current. Again, this was not particularly surprising.

What was puzzling, however, was that when light went below a specific minimum frequency (dependent on the metal), no current was observed whatsoever. This confused scientists. No matter how much they increased the intensity with low frequency light, no electrons were emitted. This was surprising because even with sufficient energy the photoelectric effect stopped.

This does not fit in with a wave model of light. In a wave, the energy is evenly distributed along the wave front. Higher frequency waves deliver more energy, but higher intensity also means more waves arriving, which can compensate. The energy transferred depends on both. Despite light showing all of the properties of waves (diffraction, refraction, spreading out progressively) the fact metals need light with a minimum frequency to be able to free electrons is evidence that light is not a wave. The energy along the wave front should just add up and release the electron.

The minimum frequency threshold suggests there is something about the "waves" themselves that allows electrons to be released. If energy were evenly distributed around the wave front then they would. The energy would be evenly shared and build up until electrons could be released. Rethinking light as being particles, i.e. photons, better explains this. Photons have to have a one-to-one, quantised, particle interaction with electrons. A low energy photon never releases an electron.

This is analogous to how people pay. A wave model of light freeing electrons would be like people pooling their cash in order to buy something. If they don't have enough money, they can find more people to agree to contribute to the cost until there is enough money. Higher intensity light could provide more waves to contribute enough energy to share electrons. However, this is not what happens. The photon model is analogous to a group of people trying to pay for something by credit card. No matter how many people with insufficient credit try to pay, no card is accepted, and no one gets to buy the item.

 Key points: 
 * To release electrons, light must be above a threshold frequency.
 * Higher intensity light below the threshold frequency will never release photons despite having the same, or even more, energy.
 * When light is above the minimum frequency, higher intensity light will give more electrons.

The Relationship between Energy and Frequency
The photoelectric effect allows us to derive an equation linking the frequency of electromagnetic radiation to the energy of each quantum (in this case, photons). This can be achieved experimentally, by exposing the metallic surface to light of different colours, and hence different frequencies. We already know the frequencies of the different colours of light, and we can calculate the energy each photon carries into the surface, as this is the same as the energy required to supply enough potential difference to cause the electron to move. The equation for the energy of the electron is derived as follows:

First, equate two formulae for energy:

$$P = \frac{E}{t} = IV\,$$

Rearrange to get:

$$E = ItV\,$$

We also know that:

$$Q = It\,$$

So, by substituting the previous equation into the equation for energy:

$$E = QV = e\Delta V\,$$,

where P = power, E = energy, t = time, I = current, V = potential difference, Q = charge, e = charge of 1 electron = -1.602 x 10−19 C, ΔV = potential difference produced between anode and cathode at a given frequency of radiation. This means that, given this potential difference, we can calculate the energy released, and hence the energy of the quanta which caused this energy to be released.

Plotting frequency (on the x-axis) against energy (on the y-axis) gives us an approximate straight line, with a gradient of 6.626 x 10−34. This number is known as Planck's constant, is measured in Js, and is usually denoted h. Therefore:

$$E = hf\,$$

In other words, the energy carried by each quantum is proportional to the frequency of the quantum. The constant of proportionality is Planck's constant.

Questions
1. How much energy does a photon with a frequency of 50 kHz carry?

2. A photon carries 10−30J of energy. What is its frequency?

3. How many photons of frequency 545 THz does a 20W bulb give out each second?

4. In one minute, a bulb gives out a million photons of frequency 600 THz. What is the power of the bulb?

5. The photons in a beam of electromagnetic radiation carry 2.5μJ of energy each. How long should the phasors representing this radiation take to rotate?

/Worked Solutions/