Now we turn our attention to solving the Dirac equation. We start with the simplest case,
an electron at rest, that is, with zero momentum. Here's the formal Dirac equation.
Our wavefunction has four-components. Plugging the wavefunction into the Dirac equation
and doing the matrix multiplications we get four equations in the four unknown components.
The terms with slopes in x, y or z are momentum terms.
Since we're interested in zero momentum, these go away.
This leaves the energy operator applied to a wavefunction component equals plus or minus
the mass times that component. Since each of these four equations has the
form energy operator applied to the wavefunction component equals a constant times the component,
these must correspond to states of definite energy.
A state of definite energy has time dependence of the form e to the minus i E t where E is
the energy. Let's try a wave function with only a psi
1 term. Plugging into our set of four equations it's
clear that this is a solution if E equals m. Recall that we're using units in which
the speed of light c equals 1. So this corresponds to E equals m c squared. That makes sense.
The energy of a particle at rest is just the intrinsic energy m c squared. We get a similar
result if we have only a psi 2 component. For a wavefunction with only a psi 3 or psi
4 component, however, we end up with E equals minus m.
We have two solutions with E equals m, corresponding to the first two components.
And, we have two solutions with E equals minus m, corresponding to the last two components.
Clearly Dirac's approach did not avoid the problem of negative energy solutions. Since
E equals m c squared, zero mass corresponds to zero energy. That is, zero energy corresponds
to nothing. So what could negative energy possibly correspond to? We are tempted to
just ignore the third and fourth components of the wavefunction as being some sort of
mathematical artifact that doesn't correspond to anything in the real world.
Now let's try to figure out how the four wavefunction components relate to spin.
For a two-component spinor the Pauli matrix sigma-x-hat times h-bar over 2 forms the operator
for the z component of spin. But we have a four-component wavefunction.
Let's guess that the corresponding four-by-four matrix contains sigma-hat-z repeated twice.
Applying this to the wavefunction we see that psi-1 and psi-3 would correspond to spin up
and psi-2 and psi-4 to spin down. Then the s-hat-z operator is one-half times
this matrix. Remember that in our system of units h-bar is one.
Dirac's equation has the form E-hat psi equals H-hat psi where E-hat is the energy operator
and H-hat is called the Hamiltonian operator. It expresses energy explicitly in terms of
dynamical quantities such as momentum. Now consider the L-z-hat operator corresponding
to the z component of orbital angular momentum. As shown in an appendix video, H-hat and L-z-hat
do not commute. The same is true for the x and y components of orbital angular momentum.
This means that unlike the Schr�dinger equation, the Dirac equation does not allow a state
definite energy to have a definite orbital angular momentum. Put another way, in the
Dirac equation orbital angular momentum is not conserved for a closed system.
The total angular momentum, J, equals the sum of orbital and spin angular momenta.
This does commute with the Hamiltonian. So, total angular momentum is conserved by
the Dirac equation. And, our guess for the spin operator is consistent
with this. We can now interpret the four wavefunction
components as follows. Components psi-1 and psi-2 correspond to spin-up and spin-down
positive energy states. In a non-relativistic theory this two-component spinor would fully
specify an electron state. Components psi-3 and psi-4 correspond to spin-up and spin-down
negative energy states. It's natural to view this as a second spinor and refer to all four
components as a bispinor or a four-spinor. However, it's hard to see what, if anything,
the negative-energy spinor represents. Now we consider a moving electron. We take
the momentum to be in the x direction. Looking at the complete system of equations
and recalling that the slopes in y and z correspond to the y and z momenta
we can drop those terms since those momenta are zero.
This leaves a somewhat reduced set of equations. Notice that, unlike the electron at rest case,
these equations mix up the wavefunction components. Specifically, the 1st and 4th equations both
involve the 1st and 4th components while the 2nd and 3rd equations both involve the 2nd
and 3rd components. Therefore, if we have a psi-1 component we
will necessarily have a psi-4 component also. Let's call the unknown coefficient of the
psi-4 component a . The 1st equation reduces to E equals p-x a
plus m . The 2nd and 3rd equations are trivial. The 4th equation reduces to E a equals p-x
minus m a . Solving the 1st and 4th equations we get a
equals E minus m over p-x and a equals p-x over E plus m . For there to be a solution
these expressions have to be equal. Setting them equal we find that the condition
for a solution is simply that E-squared equals p-squared plus m-squared. This is always true
so we do indeed have a solution. Here's our result. When p-x is zero this reduces to our
previous solution for a positive energy, spin-up electron at rest. But for non-zero momentum
the negative energy, spin-down component is necessarily non-zero.
Here's the solution that reduces to a positive-energy, spin-down electron at rest. For non-zero momentum
it must have a negative energy, spin-up component. We see that for positive-energy electrons,
it's not generally possible to be in a pure spin-up or spin-down state while moving. And,
they necessarily have non-zero negative-energy components in their wavefunctions.
Here are the corresponding negative-energy solutions. They contain non-zero, positive-energy
components and also cannot be pure spin-up or spin-down states.
Notice that for the positive-energy solutions, as momentum increases, energy increases. This
is the way we expect physics to work. A free particle is at the lowest possible energy
when it's at rest. To make it move requires that an external source of energy impart kinetic
energy to the particle. But, for the negative-energy solutions, as
momentum increases, energy decreases! So, it seems that for a negative-energy electron
to go faster, that is, to accelerate, simply requires it to give off energy. And, since
accelerating charges radiate energy as electromagnetic waves, it appears that all negative-energy
electrons would naturally tend to accelerate without end, ultimately radiating infinite
energy as they approach the speed of light. This is an obviously ridiculous prediction,
which, for the moment we'll ignore. As an interesting aside, imagine you're in
a laboratory reference frame and along comes an electron. You measure its state and find
our positive-energy, mostly-spin-up solution. Now imagine there is an observer in a frame
moving with the electron. She will see the electron at rest and so measure it to be in
the at-rest, pure spin-up state. So, different observers perceive the electron to have different
momentum and energy. That's true in non-relativistic quantum mechanics too. But, in relativistic
quantum mechanics they also perceive the electron to be in a different spin state.
A triumph of the non-relativistic Schr�dinger equation is that it can be solved analytically
for the hydrogen atom, and this predicts energy levels that closely agree with what's observed.
Naturally physicists were motivated to look at the Dirac equation's predictions for hydrogen.
The addition of a potential energy term describes the electron moving in the presence of a proton.
In our natural units this takes the form minus alpha over r times the wavefunction. Here
r is the distance between proton and electron and alpha is the so-called fine-structure
constant which is approximately 1 over 137 .
As for the free electron, total angular momentum J is conserved but orbital and spin angular
momenta are not. This form of the Dirac equation can be solved analytically, although it's
a more complicated process than the corresponding Schr�dinger case.
The solutions are described by three quantum numbers: n , j and m-sub-j.
n largely specifies the energy level and takes on values 1, 2, 3, and so on
j specifies the total angular momentum and also slightly affects energy. It takes on
values one-half, three-halves, and so on up to n minus one-half.
m-sub-j specifies the z component of angular momentum. It takes on values minus j, minus
j plus one, and so on up to j . Before we look at the Dirac equation predictions
for the hydrogen atom, recall it had long been known that the hydrogen spectrum has
a so-called fine structure. We discussed this in detail in the video on spin.
The Schr�dinger equation predicts the same energy levels for all orbital types, s, p,
d, etc. corresponding to the same n value. But all orbital types except s are observed
to actually split into two very close energy levels.
The exact hydrogen-atom energy expression that comes out of the Dirac equation solution
is fairly complicated and doesn't look anything like the Schr�dinger equation result.
In addition to the principle quantum number n, the energy explicitly depends on the angular
momentum quantum number j . It's helpful to expand this into a series
of decreasing terms. The first term, m, is simply the intrinsic energy of the electron
mass. Remember that in our units the speed of light c is one, so this corresponds to
m c squared. The second term is identical to the Schr�dinger
equation energy level and depends only on the principle quantum number n .
The next term is where the angular momentum quantum number j plays a role and corresponds
to the fine structure. The predicted energy split for n equals 2
and j equals three-halves or one-half is the tiny amount point 0 0 0 0 4 5 3 5 electron
volts. This is in excellent agreement with the observed
fine-structure split. So, the Dirac equation accurately predicts
the energy difference between a 2-p orbital with aligned orbital and spin angular momenta,
and a 2-p orbital with opposed orbital and spin angular momenta.
In the video on spin we pointed out that the fine structure could be explained by spin-orbit
coupling. The electron orbit and electron spin each effectively create tiny magnets.
The higher energy state corresponds to these, and the corresponding angular momenta, being
aligned while the lower energy state corresponds to these being opposed. The Dirac equation
accurately predicts this effect without us having to explicitly put it into the equation.
In the video on hydrogen we looked at the structure of the hydrogen orbitals as predicted
by the Schr�dinger equation. Just as the Dirac and Schr�dinger equations predict
energy levels differing by only tiny amounts, they also predict electron probability distributions
that are practically indistinguishable.
Let's look at the details of the ground state, 1-s wavefunction. The Schr�dinger solution
is e to the minus r . There are two Dirac solutions.
These have the same energy but different spinors. Note that for compactness we don't show the
time dependence factor. The radius r is measured in units of the Bohr
radius, alpha is the fine-structure constant and gamma is square-root 1 minus alpha squared.
This appears in several places as 1 minus gamma which is a very small number.
If it were zero then these would be identical to the Schr�dinger solution with a spinor
for either spin up or spin down. So, the Dirac equation:
predicts the spin one-half properties of electrons, and, accurately describes the fine structure
of hydrogen. These are monumental accomplishments that strongly suggest the Dirac equation is
on the proverbial right track. But, there are those negative energy states.
These are a deal breaker, a show stopper. They not only seem non-physical, they lead
to impossible predictions.