Symmetry of the Hydrogen Atom

We began studying physics together, and Sandro was surprised when I tried to explain to him some of the ideas that at the time I was confusedly cultivating. 

That the nobility of Man, acquired in a hundred centuries of trial and error, lay in making himself the conqueror of matter, and that I had enrolled in chemistry because I wanted to remain faithful to this nobility. That conquering matter is to understand it, and understanding matter is necessary to understanding the universe and ourselves and that therefore Mendeleev s Periodic Table, which just during those weeks we were laboriously learning to unravel, was poetry, loftier and more solemn than all the poetry we had swallowed down in liceo and come to think of it, it even rhymed  

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The construction of spherical harmonics can be extended to other dimensions. For example, V. Fock uses four-dimensional spherical harmonics in his article on the S O (4) symmetry of the hydrogen atom — see Chapter 9. Spherical harmonic functions of various dimensions are used in many spherically symmetric problems in physics. 

This essential sameness is at play when people speak of the S O (4) symmetry of the hydrogen atom, which we will discuss in Chapter 8. The hydrogen atom is not a four-dimensional system, much less a system rotating in four dimensions. Yet the largest known symmetry group of the hound states of hydrogen is isomorphic to the four-dimensional rotation group SO(4). 

Note that none of the results of this section mention energy, so that we cannot even predict a certain number of shells at or below a certain energy level. In contrast, the 5o(4) symmetry of the hydrogen atom, presented in Section 8.6), make predictions about energy levels. 

In this section we have presented the celebrated so (4) symmetry of the hydrogen atom. Thus the hydrogen atom has more symmetry than is evident from its spatial symmetry. Is this a happy accident or a sign of a deeper symmetry in our world The author does not know. The fourth dimension here is an abstract theoretical construct, not a physical reality. However— and this is one of the main points of this text—the abstract symmetry has real physical consequences. 

Congratulations By working through this section you have verified the celebrated so 4) symmetry of the hydrogen atom. A pause for celebration would be quite appropriate. 

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