Energy Levels for the Hydrogen Atom (from Ph234)



Lecture 13 (October 27, 2008): (version 1.0; 24-Oct-2008::00:00)

Fine Structure of Hydrogen (summary)

The last lecture was mainly spent on deriving the fine-structure splitting of hydrogen. We found that the total FS shift in the energy levels is the sum of two separate pieces, one due to the relativistic correction to the electrons energy, and one due to the coupling between the electron magnetic moment and the B-field that the electron sees in its own rest frame. The sum of these is

[pic] (13.1)

This is a rather striking result. Due to the partial cancellation between these two terms, the final result is rather simple; in fact it is the same as the relativistic correction alone, but with j replacing l.

It is worth noting that we found earlier that we could use either of two bases for hydrogen, the [pic] basis or the [pic] . In the first, j is not a good quantum number, and the second [pic] and [pic] are not good quantum numbers. With the spin-orbit effect included, the energy eigenstates are proportional only to basis vectors in the second of these. Although [pic] is still a good quantum number, [pic] and [pic] are not (because both [pic] and [pic] precess around [pic]).

The Simple Harmonic Oscillator and other examples

(Shankar section 17.2)

Shankar uses the 1D simple harmonic oscillator as the first example in discussing perturbation theory. I skipped this in lecture, but it is a very good and instructive example, since it can be solved exactly as well as by using perturbation theory. Also, it gives a good review of the SHO raising and lowering operators. I urge you to work through this in detail, and make sure you understand everything. (And it might make a really good problem on the midterm or the final).

Deriving the Thomas-Reiche-Kuhn sum rule (Shankar Exercise 17.2.3) is also a good exercise, and I have put the solution to this under “Examples” for week 5.

The next exercise in Shankar (17.2.5) asks you to find the first-order correction to the helium ground state, treating the ee interaction term as a perturbation. Shankar says this is “hard”. In fact, we have already calculated all the integrals in Lecture 9 as an example of the “Variational Method” (and it was hard).. We assumed a wavefunction of the form [pic], and derived the expectation value of the total Hamiltonian with the interaction term included. (You should convince yourself that this is equivalent to calculating the first-order perturbative correction). The result was

[pic] (13.2)

In fact, we can use this as two examples of perturbation theory. If we assume Z=1 (hydrogen nucleus) and sum both the coulomb field of the second proton and the ee-interactions as [pic], then we obtain the formula (13.2) for Z=1. If we use Z=2 and treat only the ee interaction as [pic], we get (13.2) for Z=2. Finally, if we minimize (13.2) with respect to Z, then we get Z=1.69. This is the same as using this value of Z for the nuclear charge, and treating the remaining charge of 0.31 e plus the ee interaction as a perturbation. The results of each of these three perturbative calculations are:

[pic] (13.3)

Each of these can be regarded as a first-order perturbative calculation; the variational method tells us that the 3rd is the best one, since it gives the lowest limit on the ground state.

The Zeeman Effect (A) [Strong-field case]

Back in Ph234 last spring we derived the Zeeman Effect by considering a spin ½ particle in a magnetic field. Let’s revisit that, now in terms of a perturbative correction to the hydrogen energy levels for a hydrogen atom in a strong magnetic field. We take the Coulomb Hamiltonian to be [pic], and treat the interaction with the B-field as a perturbation. Since each n-shell has an n2-fold degeneracy, we want to choose a basis in which [pic] is diagonal. Since the electron has its own intrinsic magnetic moment, and the orbit has an orientation energy of [pic] (see Shankar, page 287-289),

[pic] (13.4)

where

[pic] (13.5)

(Recall that g=2 for the electron.) Then

[pic] (13.6)

where [pic], [pic], and [pic]. Then [pic] is diagonal in the [pic] eigenbasis, and

[pic] (13.7)

The n=1 subshell (which has only s-waves) is split into two components, one for each of the 2 values of [pic]. One of these will increase with increasing B, and the other will decrease.

But let us focus on the n=2 shell, which contains 8 states (2 s-wave and 6 p-wave). For B large, these will be split into 5 components, 3 of which are doubly degenerate. If we define [pic] in equation (13.7), these 5 levels correspond to [pic]. The doubly degenerate levels are those with [pic]. If we include the fin-structure splitting, then each of these 3 levels will be split into 2 components corresponding to the 2 possible values of [pic].

The Zeeman Effect (B) [Weak-field case]

But we have already seen, in deriving the fine structure of hydrogen, that in the absence of external fields the relativistic and spin-orbit terms split the n=2 subshell into 2 levels, corresponding to [pic] and [pic]. Moreover, these are not basis vectors in the [pic] basis. Instead, they are basis vectors of the [pic] basis.

When we add a small magnetic field, then we would expect j to remain a good quantum number, with the vector [pic] precessing about [pic]. More formally, we can use the [pic] basis as the eigenbasis of

[pic] (13.8)

and treat

[pic] (13.9)

as a perturbation.

(to be continued)

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