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with the experimental data. As far as the last chapter was concerned, the ‘particle in a box’ simplification was good enough because it contains all the key points that we wanted to highlight. However, there is a feature of the full calculation that we shall need, which comes about because the real hydrogen atom is extended in three dimensions. For our particle in a box example, we only considered one dimension and obtained a series of energy levels labelled by a single number that we called n. The lowest energy level was labelled n = 1, the next n = 2 and so on. When the calculation is extended to the full three-dimensional case it turns out, perhaps unsurprisingly, that three numbers are needed to characterize all of the allowed energy levels. These are traditionally labelled n, l and m, and they are referred to as quantum numbers (in this chapter, m is not to be confused with the mass of the particle). The quantum number n is the counterpart of the number n for a particle in a box. It takes on integer values (n = 1, 2, 3, etc.) and the particle energies tend to increase as n increases. The possible values of l and m turn out to be linked to n; l must be smaller than n and it can be zero, e.g. if n = 3 then l can be 0, 1 or 2. m can take on any value ranging from minus l to plus l in integer steps. So if l = 2 then m can be equal to −2, −1, 0, 1 or 2. We are not going to explain where those numbers come from, because it won’t add anything to our understanding. Suffice to say that the four waves in Figure 6.9 have (n,l) = (1,0), (2,0), (2,1) and (3,0) respectively (all have m = 0).1

As we have said, the quantum number n is the main number controlling the values of the allowed energies of the electrons. There is also a small dependence of the allowed energies upon the value of l but it only shows up in very precise measurements of the emitted light. Bohr didn’t consider it when he first calculated the energies of the spectral lines of hydrogen, and his original formula was expressed entirely in terms of n. There is absolutely no dependence of the electron energy upon m unless we put the hydrogen atom inside a magnetic field (in fact m is known as the ‘magnetic quantum number’), but this certainly doesn’t mean that it isn’t important. To see why, let’s get on with our bit of numerology.

If n = 1 then how many different energy levels are there? Applying the rules we stated above, l and m can both only be 0 if n = 1, and so there is just the one energy level.

Now let’s do it for n = 2: l can take on two values, 0 and 1. If l = 1, then m can be equal to −1, 0 or +1, which is 3 more energy levels, making 4 in total.

For n = 3, l can be 0, 1 or 2. For l = 2, m can be equal to −2, −1, 0, +1, or +2, giving 5 levels. So in total, there are 1 + 3 + 5 = 9 levels for n = 3. And so on.

Remember those numbers for the first three values of n: 1, 4 and 9. Now take a look at Figure 7.1, which shows the first four rows of the periodic table of the chemical elements, and count how many elements there are in each row. Divide that number by 2, and you’ll get 1, 4, 4 and 9. The significance of all this will soon be revealed.

Figure 7.1. The first four rows of the periodic table.

Credit for arranging the chemical elements in this way is usually given to the Russian chemist Dmitri Mendeleev, who presented it to the Russian Chemical Society on 6 March 1869, which was a good few years before anyone had worked out how to count the allowed energy levels in a hydrogen atom. Mendeleev arranged the elements in order of their atomic weights, which in modern language corresponds to the number of protons and neutrons inside the atomic nucleus, although of course he didn’t know that at the time either. The ordering of the elements actually corresponds to the number of protons inside the nucleus (the number of neutrons is irrelevant) but for the lighter elements this makes no difference, which is why Mendeleev got it right. He chose to arrange the elements in rows and columns because he noticed that certain elements had very similar chemical properties,