- A.38.1 Basic perturbation theory
- A.38.2 Ionization energy of helium
- A.38.3 Degenerate perturbation theory
- A.38.4 The Zeeman effect
- A.38.5 The Stark effect

A.38 Perturbation Theory

Most of the time in quantum mechanics, exact solution of the Hamiltonian eigenvalue problem of interest is not possible. To deal with that, approximations are made.

Perturbation theory can be used when the Hamiltonian unperturbed Hamiltonian

This addendum explains how perturbation theory works. It also gives a few simple but important examples: the helium atom and the Zeeman and Stark effects. Addendum,{A.39} will use the approach to study relativistic effects on the hydrogen atom.

A.38.1 Basic perturbation theory

To use perturbation theory, the eigenfunctions and eigenvalues of the
unperturbed Hamiltonian

The key to perturbation theory are the “Hamiltonian perturbation coefficients” defined as

In the application of perturbation theory, the idea is to pick one
unperturbed eigenfunction good

eigenfunction to correct the energy. How to do
that will be discussed in subsection A.38.3.

For now just assume that the energy is not degenerate or that you
picked a good eigenfunction

Unfortunately, it does happen quite a lot that the above correction

Sometimes you may also be interested in what happens to the energy
eigenfunctions, not just the energy eigenvalues. The corresponding
formula is

In some cases, instead of using second order theory as above, it may
be simpler to compute the first order wave function perturbation and
the second order energy change from

One application of perturbation theory is the “Hellmann-Feynman theorem.” Here the perturbation Hamiltonian is
an infinitesimal change

A.38.2 Ionization energy of helium

One prominent deficiency in the approximate analysis of the heavier atoms in chapter 5.9 was the poor ionization energy that it gave for helium. The purpose of this example is to derive a much more reasonable value using perturbation theory.

Exactly speaking, the ionization energy is the difference between the
energy of the helium atom with both its electrons in the ground state
and the helium ion with its second electron removed. Now the energy
of the helium ion with electron 2 removed is easy; the Hamiltonian for
the remaining electron 1 is

where the first term represents the kinetic energy of the electron and the second its attraction to the two-proton nucleus. The helium nucleus normally also contains two neutrons, but they do not attract the electron.

This Hamiltonian is exactly the same as the one for the hydrogen atom
in chapter 4.3, except that it has

and so its energy and wave function become

where

It is interesting to see that the helium ion has four times the energy of the hydrogen atom. The reasons for this much higher energy are both that the nucleus is twice as strong, and that the electron is twice as close to it: the Bohr radius is half the size. More generally, in heavy atoms the electrons that are poorly shielded from the nucleus, which means the inner electrons, have energies that scale with the square of the nuclear strength. For such electrons, relativistic effects are much more important than they are for the electron in a hydrogen atom.

The neutral helium atom is not by far as easy to analyze as the ion.
Its Hamiltonian is, from (5.34):

The first two terms are the kinetic energy and nuclear attraction of electron 1, and the next two the same for electron 2. The final term is the electron to electron repulsion, the curse of quantum mechanics. This final term is the reason that the ground state of helium cannot be found analytically.

Note however that the repulsion term is qualitatively similar to the
nuclear attraction terms, except that there are four of these nuclear
attraction terms versus a single repulsion term. So maybe then, it
may work to treat the repulsion term as a small perturbation, call it

The solution of the eigenvalue problem

According to this result, the energy of the atom is

To get a better ionization energy, try perturbation theory. According
to first order perturbation theory, a better value for the energy of
the hydrogen atom should be

or substituting in from above,

The inner product of the final term can be written out as

This integral can be done analytically. Try it, if you are so inclined; integrate

The result of the integration is

Therefore, the helium atom energy increases by

The second order perturbation result should give a much more accurate result still. However, if you did the integral above, you may feel little inclination to try the ones involving all possible products of hydrogen energy eigenfunctions.

Instead, the result can be improved using a variational approach, like
the ones that were used earlier for the hydrogen molecule and
molecular ion, and this requires almost no additional work. The idea
is to accept the hint from perturbation theory that the wave function
of helium can be approximated as

However, instead of accepting the perturbation theory result that

as small as possible. This will produce the most accurate ground state energy possible for a ground state wave function of this form, guaranteed no worse than assuming that

No new integrals need to be done to evaluate the inner product above.
Instead, noting that for the hydrogen atom according to the virial
theorem of chapter 7.2 the expectation kinetic energy
equals

and this subsection added

Using these results with the helium Hamiltonian, the expectation energy of the helium atom can be written out to be

Setting the derivative with respect to

A.38.3 Degenerate perturbation theory

Energy eigenvalues are degenerate if there is more than one
independent eigenfunction with that energy. Now, if you try to use
perturbation theory to correct a degenerate eigenvalue of a
Hamiltonian

Then as far as

with arbitrary coefficients

Unfortunately, the full Hamiltonian good

eigenfunctions. It is said that the perturbation
breaks the degeneracy

of the energy eigenvalue.
The single energy eigenvalue splits into several eigenvalues of
different energy. Only good combinations will show up these changed
energies; the bad ones will pick up uncertainty in energy that hides
the effect of the perturbation.

The various ways of ensuring good eigenfunctions are illustrated in
the following subsections for example perturbations of the energy
levels of the hydrogen atom. Recall that the unperturbed energy
eigenfunctions of the hydrogen atom electron, as derived in chapter
4.3, and also including spin, are given as

There are two important rules to identify the good eigenfunctions, {D.79}:

- 1.
- Look for good quantum numbers. The quantum numbers that make
the energy eigenfunctions of the unperturbed Hamiltonian
unique correspond to the eigenvalues of additional operators besides the Hamiltonian. If the perturbation Hamiltonian commutes with one of these additional operators, the corresponding quantum number is good. You do not need to combine eigenfunctions with different values of that quantum number. In particular, if the perturbation Hamiltonian commutes with all additional operators that make the eigenfunctions of

unique, stop worrying: every eigenfunction is good already. For example, for the usual hydrogen energy eigenfunctions

, the quantum numbers, , andmake the eigenfunctions at a given unperturbed energy level unique. They correspond to the operators , , and. If the perturbation Hamiltoniancommutes with any one of these operators, the corresponding quantum number is good. If the perturbation commutes with all three, all eigenfunctions are good already. - 2.
- Even if some quantum numbers are bad because the perturbation
does not commute with that operator, eigenfunctions are still good
if there are no other eigenfunctions with the same unperturbed
energy and the same good quantum numbers.
Otherwise linear algebra is required. For each set of energy eigenfunctions

with the same unperturbed energy and the same good quantum numbers, but different bad ones, form the matrix of Hamiltonian perturbation coefficients

The eigenvalues of this matrix are the first order energy corrections. Also, the coefficientsof each good eigenfunction

must be an eigenvector of the matrix.Unfortunately, if the eigenvalues of this matrix are not all different, the eigenvectors are not unique, so you remain unsure about what are the good eigenfunctions. In that case, if the second order energy corrections are needed, the detailed analysis of derivation {D.79} will need to be followed.

If you are not familiar with linear algebra at all, in all cases mentioned here the matrices are just two by two, and you can find that solution spelled out in the notations under

eigenvector.

The following, related, practical observation can also be made:

Hamiltonian perturbation coefficients can only be nonzero if all the good quantum numbers are the same.

A.38.4 The Zeeman effect

If you put an hydrogen atom in an external magnetic field
Zeeman effect.

If for simplicity a coordinate system is used with its

(A.245) |

For this perturbation, the

Actually, this is not approximate at all; it is the exact eigenvalue of

The Zeeman effect can be seen in an experimental spectrum. Consider
first the ground state. If there is no electromagnetic field, the two
ground states

so the single line splits into two! Do note that the energy change due to even an extremely strong magnetic field of 100 Tesla is only 0.006 eV or so, chapter 13.4, so it is not like the spectrum would become unrecognizable. The single spectral line of the eight

Lshell states will similarly split in five closely spaced but separate lines, corresponding to the five possible values

Some disclaimers should be given here. First of all, the 2 in
strong

Zeeman effect, in which the magnetic
field is strong enough to swamp the relativistic errors.

A.38.5 The Stark effect

If an hydrogen atom is placed in an external electric field
Stark effect.

Of course a Zeeman, Dutch for sea-man,
would be most interested in magnetic fields. A Stark, maybe in a
spark? (Apologies.)

If the

(A.246) |

Since the typical magnitude of

And additionally, it turns out that for many eigenfunctions, including
the ground state, the first order correction to the energy is zero.
To get the energy change in that case, you need to compute the second
order term, which is a pain. And that term will be much smaller still
than even

Now first suppose that you ignore the warnings on good eigenfunctions,
and just compute the energy changes using the inner product

The reason is that negative

So, since all first order energy changes that you compute are zero, you would naturally conclude that to first order approximation none of the energy levels of a hydrogen atom changes due to the electric field. But that conclusion is wrong for anything but the ground state energy. And the reason it is wrong is because the good eigenfunctions have not been used.

Consider the operators

Still, the two states

But now consider the eight-fold degenerate

However, the remaining two

It suffices to just analyze the spin up states, because the spin down
ones go exactly the same way. The coefficients

The

diagonalelements of this matrix (top left corner and bottom right corner) are zero because of cancellation of negative and positive

The eigenvectors of this matrix are simple enough to guess; they have
either equal or opposite coefficients

If you want to check these expressions, note that the product of a matrix times a vector is found by taking dot products between the rows of the matrix and the vector. It follows that the good combination

Remarkably, the good combinations of sp

hybrids of carbon fame, as described in chapter
5.11.4. Note from figure 5.13 in that section
that these hybrids do not have the same magnitude at opposite
sides of the nucleus. They have an intrinsic “electric dipole
moment,” with the charge shifted towards one side of the atom,
and the electron then wants to align this dipole moment with the
ambient electric field. That is much like in Zeeman splitting, where
electron wants to align its orbital and spin magnetic dipole moments
with the ambient magnetic field.

The crucial thing to take away from all this is: always, always, check whether the eigenfunction is good before applying perturbation theory.

It is obviously somewhat disappointing that perturbation theory did
not give any information about the energy change of the ground state
beyond the fact that it is second order, i.e. very small compared to

It is however possible to find the perturbation in the wave function
from the alternate approach (A.243), {D.80}.
In that way the second order ground state energy is found to be

Note that the atom likes an electric field: it lowers its ground state energy. Also note that the energy change is indeed second order; it is proportional to the square of the electric field strength. You can think of the attraction of the atom to the electric field as a two-stage process: first the electric field polarizes the atom by distorting its initially symmetric charge distribution. Then it interacts with this polarized atom in much the same way that it interacts with the sp hybrids. But since the polarization is now only proportional to the field strength, the net energy drop is proportional to the square of the field strength.

Finally, note that the typical value of 0.000 5 eV or so for

A weird prediction of quantum mechanics is that the electron will
eventually escape from the atom, leaving it ionized. The reason is
that the potential is linear in tunneling out

of the atom through the energy barrier,
chapter 7.12.2. Realistically, though, for even strong
experimental fields like the one mentioned above, the “life
time” of the electron in the atom before it has a decent chance
of being found outside it far exceeds the age of the universe.