Chapter 2 Old Quantum Theory Let us begin with the story of how the quantum revolution began. It was Max Planck who first realised that in quantum physics energy has to be quantised. In other words, it comes in small discrete chunks. Before that, physicists be- lieved that energy was a continuous quantity and could have any value, just like most other physical quantities, such as position, momentum and so on. Then Einstein realised a very profound implication of this: light must be composed of fundamental particles called photons. So, light, according to Einstein, is not just a continuous wave, but also has a particle component to it. Finally, de Broglie postulated that matter is not just made up of particles either, but is also wavelike. Electrons are, therefore, not just particles but can and do behave like waves1. But, let us not get ahead of ourselves. The whole story starts in 1859, the same year in which Darwin published his Origin of Species. Kirchhoff, a famous German experimental physicist, declared that determining the energy spectrum of a black body was the holy grail of physics. He made some prelimi- nary measurements of it, but they were nowhere near precise enough to answer this question. But why did he think that this was so important in the first place, and what do we mean by a black body? 2.1. Black Body Radiation There are two basic ways in which heat propagates in a given medium: conduc- tion and radiation. Conduction is a relatively simple process. It is governed by a diffusion equation and the rate of change of temperature is proportional to the temperature gradient dT d2 = α T (2.1) dt − dx2 where α is a constant that depends only on the properties of the medium, i.e. it does not depend on the temperature. Once the temperature is the same 1We will explain exactly what a wave is later on. 7 8 CHAPTER 2. OLD QUANTUM THEORY everywhere, the right hand side of (2.1) vanishes and there is then no conduction, which is why we said that it was basically a relatively simple process2. But there is still radiation, which is independent of any temperature gradient. Radiation processes are much more complicated to study precisely because they can and do take place at constant temperature. Even in equilibrium the behaviour of light is not that simple. So much so that the intense study of its properties led to the advent of quantum mechanics. Physicists like simple models and extreme situations. Limits of various types abound in physics and as far as radiation is concerned there are two useful situations. One is when we have a body that reflects all radiation that falls on it. This is called a white body. Realistic bodies will of course only strongly reflect in some range of radiation and will absorb for other wavelengths. A mirror is a good example of a white body for visible radiation. The other extreme is a body that absorbs all the radiation that falls on it – a black body. Surprisingly there are a lot of examples of good approximations to black bodies – the Sun and the Earth to name a couple. Suppose that we look at the radiation that leaves the black body. What kind of properties would it have? What would we see if we were inside a black body? Imagine that you are sitting in your room at night, reading by the night lamp. Your room is not in thermal equilibrium. Why? First of all there are lights on and they emit energy keeping you out of equilibrium. Switch the lights off. Even then, there are objects in your room generating heat and emitting radiation that is not necessarily visible to you. Yourself, for example. You are not in a thermal equilibrium, you generate heat, you are alive. But, suppose that you get rid of all sources and sinks of heat. What properties would the remaining radiation have that is in thermal equilibrium with the matter inside your room? The remaining radiation would be extremely uniform (the same at every point), isotropic (the same in all directions) and highly mixed, i.e. it would contain many frequencies. Kirchhoff’s aim was to describe the spectrum of such a body by finding the energy density as a function of frequency or wavelength. Towards the end of 19th century it became clear that classical physics could not explain the experimental results of black body radiation. The simple reason was this: classical physics predicts that every atom in the black body should emit radiation at all possible frequencies and that, at temperature T , the energy of each frequency is equal to kT where k is Boltzmann’s constant3. This energy was classically predicted to be independent of the frequency. Since there are infinitely many possible frequencies, classical physics predicts that the total amount of radiation emitted by a black body is infinite. This prediction is clearly wrong. It is known as the ultraviolet catastrophe since it was a catastrophe for classical physics. The word ultraviolet is there because the density of radiation is quadratically higher at higher frequencies so that the classical prediction becomes worse and worse at higher frequencies, i.e. towards the ultraviolet end 2This is oversimplifying this a bit, but nothing too drastic. 3Boltzmann’s constant can be thought of as the conversion unit between the microscopic 23 energy of particles and their (macroscopic) temperature. It has the value k 1.38 10− Joules per Kelvin. ≈ × 2.1. BLACK BODY RADIATION 9 of the visible spectrum. Anyway, by 1900 it was completely clear that classical physics could not explain black body radiation. This was around the time that Planck entered the scene. Now, Planck was already a well established professor who made his name in thermodynamics. He wrote some beautiful papers on it and a few books4. However, soon after his work on thermodynamics, he received a book by Gibbs, a famous American physicist, and realised that Gibbs had done all this stuff at least ten years before him. Now if this was not depressing enough, Planck was also trying to prove the second law of thermodynamics when radiation is in equilibrium with matter, but without any success at all5. So, disappointed by Gibbs and by his failure to prove the second law6, Planck decided to attack the problem of explaining the black body spectrum. He spent five years doing this and got nowhere. Then in 1900 he became desperate. He took the best available experimental results of the energy spectrum and extrapolated from it a formula for the energy density. He guessed completely correctly that the energy at a particular frequency is given by 1 E = hf . (2.2) ehf/kT 1 − Here h is a constant (now known as Planck’s constant in his honour) with a 34 value h 6.63 10− Js, f is the frequency of the radiation, k is Boltzmann’s constant≈ and T×is the temperature of the black body. Note that the overall units correspond to energy since Planck’s constant has units of Joules times seconds (Js) and frequency has the units of inverse seconds (1/s), which means that the product has units of Joules, i.e. energy. The rest of the expression is dimensionless and we will understand its exact meaning a bit later. We can easily show that for high temperatures this goes into the classical limit of kT . At high T , we have hf/kT 1 and so the exponential can be approximated as ehf/kT 1 + hf/kT . Substituting this into the expression for E above and cancelling≈ the hf factors, we get E kT , which proves the classical limit. ≈ The meaning of this quantity is the average energy emitted by black body at the particular frequency. The word average is important simply because the amount of energy fluctuates. This means that at one time you would mea- sure one amount and another time another, but when you average over many measurements you would get the above formula. We would now like to calculate the total energy output of the black body. For this we need to know how many frequencies there are around the frequency f. Since every frequency corresponds to a different state of black body radiation, this leads to the notion of the density of states. The density of states (per unit 4These are classics... 5We now know that Planck was probably being a bit hard on himself. No one can prove the second law of thermodynamics from underlying mechanics. This is still one of the greatest open problems of theoretical physics. 6and probably reaching the mid-life crisis... 10 CHAPTER 2. OLD QUANTUM THEORY −13 x 10 2 ) 1 − Hz 3 − 1.5 T=30000K 1 Energy density (Jm 0.5 T=20000K 0 0 1 2 3 4 5 6 Frequency (1015 Hz) Fig. 2.1. The black body energy density given by Eq. (2.4) for T = 20000K and T = 30000K. volume) is given by 8π g(f) = f 2. (2.3) c3 We are stating this result without deriving it, but the logic is very simple. We can think of different states as lying on a sphere of radius f. The density of states is then proportional to the area of this sphere, which gives us the f 2 factor. Taking into account the fact that there are two polarisations corresponding to each frequency of light gives us the formula above. For now, you don’t need to worry about where the other factors come from.