Quantum Physics III Chapter 2: Hydrogen Fine Structure

Chapter 2. Fine Structure c B. Zwiebach Review of Hydrogen atom The Hydrogen atom Hamiltonian is by now familiar to you. You have found the bound state spectrum in more than one way and learned about the large degeneracy that exists for all states except the ground state. We will call the Hydrogen atom Hamiltonian H (0) and it is given by p2 e2. H (0) = . ( ). 2m r We take this to be the known Hamiltonian, meaning that we know its spectrum. This Hamiltonian is supplemented with corrections that can be studied in perturbation theory. That study is the subject of this Chapter . We begin, however, with some review and comments. The mass m in H (0) is the reduced mass of the electron and proton, which we can accurately set equal to the mass of the electron. If one wishes to consider the case of an an electron bound to a nucleus with Z protons one lets e2 Ze2 in H (0) . The Bohr radius is the length scale build from ~, m, and e2.

, (2.1.4) and the rest energy of the electron is mc2. Then, e2 a 0 = me4 ~2 = mα2~2c2 ~2 = α2 mc2. (2.1.5) This states that the energy scale of hydrogen bound states is a factor of α2 smaller than the rest energy of the electron, that is, about 19000 times smaller. We can thus rewrite the possible energies as: En = −1 2α 2 mc2 1 n. (2.1.6)

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