- A.39.1 Introduction
- A.39.2 Fine structure
- A.39.3 Weak and intermediate Zeeman effect
- A.39.4 Lamb shift
- A.39.5 Hyperfine splitting

A.39 The relativistic hydrogen atom

The description of the hydrogen atom given earlier in chapter 4.3 is very accurate by engineering standards. However, it is not exact. This addendum examines various relativistic effects that were ignored in the analysis.

The approach will be to take the results of chapter 4.3 as the starting point. Then corrections are applied to them using perturbation theory as described in addendum {A.38}.

A.39.1 Introduction

According to the description of the hydrogen atom given in chapter
4.3, all energy eigenfunctions hydrogen atom fine structure.

It means that
eigenfunctions that all should have exactly the same energy,
don’t.

To explain why, the solution of chapter 4.3 must be
corrected for a variety of relativistic effects. Before doing so, it
is helpful to express the nonrelativistic energy levels of that
chapter in terms of the rest mass energy

Nobody knows why it has the value that it has. Still, obviously it is
a measurable value, so, following the stated ideas of quantum
mechanics, maybe the universe measured

this value
during its early formation by a process that we may never understand,
(since we do not have other measured values for

In any case, for engineering purposes it is a small number, less than
1%. That makes the hydrogen energy levels really small compared to
the rest mass energy of the electron, because they are proportional to
the square of

And that in turn means that the relativistic errors in the hydrogen energy levels are small. Still, even small errors can sometimes be very important. The required corrections are listed below in order of decreasing magnitude.

- Fine structure.
The electron should really be described relativistically using the Dirac equation instead of classically. In classical terms, that will introduce three corrections to the energy levels:

- Einstein’s relativistic correction of the classical
kinetic energy
of the electron. Spin-orbit interaction

, due to the fact that the spin of the moving electron changes the energy levels. The spin of the electron makes it act like a little electromagnet. It can be seen from classical electrodynamics that a moving magnet will interact with the electric field of the nucleus, and that changes the energy levels. Note that the name spin-orbit interaction is a well chosen one, a rarity in physics.- There is a third correction for states of zero angular momentum, the Darwin term. It is a crude fix for the fundamental problem that the relativistic wave function is not just a modified classical one, but also involves interaction with the anti-particle of the electron, the positron.

, and the error they introduce is on the order of 0.001%. So theexact

solution of chapter 4.3 is, by engineering standards, pretty exact after all. - Einstein’s relativistic correction of the classical
kinetic energy
- Lamb shift. Relativistically, the electron is affected by
virtual photons and virtual electron-positron pairs. It adds a
correction of relative magnitude
to the energy levels, one or two orders of magnitude smaller still than the fine structure corrections. To understand the correction properly requires quantum electrodynamics. - Hyperfine splitting. Like the electron, the proton acts
as a little electromagnet too. Therefore the energy depends on how
it aligns with the magnetic field generated by the electron. This
effect is a factor
smaller still than the fine structure corrections, making the associated energy changes about two orders of magnitude smaller. Hyperfine splitting couples the spins of proton and electron, and in the ground state, they combine in the singlet state. A slightly higher energy level occurs when they are in a spin-one triplet state; transitions between these states radiate very low energy photons with a wave length of 21 cm. This is the source of the

21 centimeter line” or “hydrogen line

radiation that is of great importance in cosmology. For example, it has been used to analyze the spiral arms of the galaxy, and the hope at the time of this writing is that it can shed light on the so calleddark ages

that the universe went through. Since so little energy is released, the transition is very slow, chapter 7.6.1. It takes on the order of 10 million years, but that is a small time on the scale of the universe.The message to take away from that is that even errors in the ground state energy of hydrogen that are two million times smaller than the energy itself can be of critical importance under the right conditions.

The following subsections discuss each correction in more detail.

A.39.2 Fine structure

From the Dirac equation, it can be seen that three terms need to be added to the nonrelativistic Hamiltonian of chapter 4.3 to correct the energy levels for relativistic effects. The three terms are worked out in derivation {D.81}. But that mathematics really provides very little insight. It is much more instructive to try to understand the corrections from a more physical point of view.

The first term is relatively easy to understand. Consider
Einstein’s famous relation

(A.248) |

The first term corresponds to the kinetic energy operator used in the nonrelativistic quantum solution of chapter 4.3. (It may be noted that the relativistic momentum

The second correction that must be added to the nonrelativistic
Hamiltonian is the so-called spin-orbit interaction.

In classical terms, it is due to the spin of the electron, which makes
it into a magnetic dipole.

Think of it as a magnet of
infinitesimally small size, but with infinitely strong north and
south poles to make up for it. The product of the infinitesimal
vector from south to north pole times the infinite strength of the
poles is finite, and defines the magnetic dipole moment

So how big is this effect? Well, the energy of an electric dipole

As you might guess, the electric dipole generated by the magnetic poles of the moving electron is proportional to the speed of the electron

Now the order of the vectors in this triple product can be changed, and the dipole strength

The expression between the parentheses is the angular momentum

The final correction that must be added to the nonrelativistic
Hamiltonian is the so-called “Darwin term:”

If that is not very satisfactory, the following much more detailed
derivation can be found on the web. It does succeed in
explaining the Darwin term fully within the nonrelativistic picture
alone. First assume that the electric potential of the nucleus does
not really become infinite as see

this potential sharply, but perceives of its
features a bit vaguely, as diffused out symmetrically over a typical
distance equal to the so-called Compton wave length
Zitterbewegung

; (2) the
electron has decided to move at the speed of light, which is quite
possible nonrelativistically, so its uncertainty in position is of
the order of the Compton wave length, and it just cannot figure out
where the right potential is with all that uncertainty in position and
light that fails to reach it; (3) the electron needs glasses. Further
assume that the Compton wave length is much smaller than the size over
which the nuclear potential is smoothed out. In that case, the
potential within a Compton wave length can be approximated by a second
order Taylor series, and the diffusion of it over the Compton wave
length will produce an error proportional to the Laplacian of the
potential (the only fully symmetric combination of derivatives in the
second order Taylor series.). Now if the potential is smoothed over
the nuclear region, its Laplacian, giving the charge density, is known
to produce a nonzero spike only within that smoothed nuclear region,
figure 13.7 or (13.30). Since the nuclear size
is small compared to the electron wave functions, that spike can then
be approximated as a delta function. Tell all your friends you heard
it here first.

The key question is now what are the changes in the hydrogen energy
levels due to the three perturbations discussed above. That can be
answered by perturbation theory as soon as the good eigenfunctions
have been identified. Recall that the usual hydrogen energy
eigenfunctions

The radial factor is no problem; it commutes with every orbital angular momentum component, since these are purely angular derivatives, chapter 4.2.2. It also commutes with every component of spin because all spatial functions and operators do, chapter 5.5.3. As far as the dot product is concerned, it commutes with

The quantum numbers

Fortunately,

and adding it all up, you get

Such good eigenfunctions can be constructed from the

As far as the other two contributions to the fine structure are
concerned, according to chapter 4.3.1

To get the energy changes, the Hamiltonian perturbation coefficients

must be found. Starting with the Einstein term, it is

Unlike what you may have read elsewhere,

Noting from chapter 4.3 that

The spin-orbit energy correction is

For states with no orbital angular momentum, all components of

to give

That leaves only the expectation value of

or zero if

Finally the Darwin term,

Now a delta function at the origin has the property to pick out the value at the origin of whatever function it is in an integral with, compare chapter 7.9.1. Derivation {D.15}, (D.9), implies that the value of the wave functions at the origin is zero unless

To get the total energy change due to fine structure, the three
contributions must be added together. For

In the ground state

Woof.

A.39.3 Weak and intermediate Zeeman effect

The weak Zeeman effect is the effect of a magnetic field that is
sufficiently weak that it leaves the fine structure energy
eigenfunctions almost unchanged. The Zeeman effect is then a small
perturbation on a problem in which the unperturbed

(by
the Zeeman effect) eigenfunctions

The Zeeman Hamiltonian

commutes with both

To evaluate this rigorously would require that the

However, the following simplistic derivation is usually given instead,
including in this book. First get rid of

For the final inner product, make a semi-classical argument that only the component of

and the dot product in it can be found from expanding

to give

For a given eigenfunction

If the

In the intermediate Zeeman effect, the fine structure and Zeeman
effects are comparable in size. The dominant perturbation Hamiltonian
is now the combination of the fine structure and Zeeman ones. Since
the Zeeman part does not commute with

A.39.4 Lamb shift

A famous experiment by Lamb & Retherford in 1947 showed that the
hydrogen atom state

The cause of the unexpected energy difference is called Lamb shift. To explain why it occurs would require quantum electrodynamics, and that is well beyond the scope of this book. Roughly speaking, the effect is due to a variety of interactions with virtual photons and electron/positron pairs. A good qualitative discussion on a nontechnical level is given by Feynman [19].

Here it must suffice to list the approximate energy corrections
involved. For states with zero orbital angular momentum, the energy
change due to Lamb shift is

where

It follows that the energy change is really small for states with
nonzero orbital angular momentum, which includes the 2

Qualitatively, the reason that the Lamb shift is small for states with
nonzero angular momentum has to do with distance from the nucleus.
The nontrivial effects of the cloud of virtual particles around the
electron are most pronounced in the strong electric field very close
to the nucleus. In states of nonzero angular momentum, the wave
function is zero at the nucleus, (D.9). So in those states
the electron is unlikely to be found very close to the nucleus. In
states of zero angular momentum, the square magnitude of the wave
function is

A.39.5 Hyperfine splitting

Hyperfine splitting of the hydrogen atom energy levels is due to the
fact that the nucleus acts as a little magnet just like the electron.
The single-proton nucleus and electron have magnetic dipole moments
due to their spin equal to

in which the

This discussion will restrict itself to the ground state, which is by
far the most important case. For the ground state, there is no
orbital contribution to the magnetic field of the electron. There is
only a spin-spin coupling

between the magnetic moments
of the electron and proton, The energy involved can be thought of most
simply as the energy

The good states are not immediately self-evident, so the four
unperturbed ground states will just be taken to be the ones which the
electron and proton spins combine into the triplet or singlet states
of chapter 5.5.6:

or

using these states.

Now the first term in the spin-spin Hamiltonian does not produce a
contribution to the perturbation coefficients. The reason is that the
inner product of the perturbation coefficients written in spherical
coordinates involves an integration over the surfaces of constant

So only the second term in the Hamiltonian survives, and
the Hamiltonian perturbation coefficients become

The spatial integration in this inner product merely picks out the
value

Since the triplet and singlet spin states are orthonormal, only the Hamiltonian perturbation coefficients for which

Plugging it all in and rewriting in terms of the Bohr energy and fine
structure constant, the energy changes are:

(A.256) |