Transcription of Quantum Physics III Chapter 2: Hydrogen Fine Structure
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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. a0 53 pm. ( ). me2.
When we form ℓ⊗swe are tensoring a a full ℓmultiplet to an smultiplet (here, of course, s= 1/2). All states in ℓ⊗sare eigenstates of Lˆ2 and eigenstates of Sˆ2, so ℓand sare good (constant) quantum numbers for all jmultiplets that arise in the tensor product. Each j multiplet has states with quantum numbers (j,mj).
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