 Now, we turn our attention to solving the Dirac equation. We start with the simplest case, an electronic rest, that is, with zero momentum. Here's the formal Dirac equation. Our wave function has four components. Plugging the wave function 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. Once we're interested in zero momentum, these go away. This leaves the energy operator applied to a wave function component equals plus or minus the mass times that component. Since each of these four equations has the form energy operator applied to the wave function component equals the 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 one 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 one. So this corresponds to e equals mc squared. That makes sense. The energy of a particle at rest is just the intrinsic energy mc squared. We get a similar result if we have only a psi two component. For a wave function with only a psi three or psi four 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 mc squared, zero mass corresponds to zero energy. That is, zero energy corresponds to nothing. So what could negative energy possibly correspond to? We're tempted to just ignore the third and fourth components of the wave function 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 wave function components relate to spin. For a two-component spinner, the Pauli matrix sigma x hat times h bar over two forms the operator for the z-component of spin. But we have a four-component wave function. Let's guess that the corresponding four-by-four matrix contains sigma hat z repeated twice. Applying this to the wave function, we see that psi one and psi three would correspond to spin up and psi two and psi four 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 quantity such as momentum. Now consider the Lz hat operator corresponding to the z-component of orbital angular momentum. We can show that h hat and Lz hat do not commute. The same is true for the x and y components of orbital angular momentum. This means that unlike the Schroedinger equation, the Dirac equation does not allow a state of definite energy to have a state of 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 the orbital and spin angular momentum. 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-way function components as follows. Components psi one and psi two correspond to spin up and spin down positive energy states. In a non-relativistic theory, this two-component spinner would fully specify an electron state. Components psi three and psi four correspond to spin up and spin down negative energy states. It's natural to view this as a second spinner and refer to all four components as a by spinner or a for spinner. However, it's still hard to see what, if anything, the negative energy spinner represents. Now we consider a moving electron. We take the momentum to be in the x direction. Looking at the complete set 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 wave function components. Specifically, the first and fourth equations both involve the first and fourth components, while the second and third equations both involve the second and third components. Therefore, if we have a psi one component, we will necessarily have a psi four component also. Let's call the unknown coefficient of the psi four component a. The first equation reduces to e equals px a plus m. The second and third equations are trivial, and the fourth equation reduces to ea equals px minus m a. Solving the first and fourth equations, we get a equals e minus m over px, and a equals px 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 indeed have a solution. Here's our result. When px 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 wave functions. 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 will 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's an observer in a frame moving with the electron. You'll 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. And 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 Schrodinger 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 equations 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 wave function. 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 momentum are not. This form of the Dirac equation can be solved analytically although it's a more complicated process than the corresponding Schrodinger case. The solutions are described by three quantum numbers n, j and m sub j. One largely specifies the energy level and takes on the values 1, 2, 3 and so on. J specifies the total angular momentum and also slightly affects energy. It takes on values 1 half, 3 halves and so on up to n minus 1 half. M sub j specifies the z component of angular momentum. It takes on values minus j minus j plus 1 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 discuss this in detail in the video on spin. The Schrodinger 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. It doesn't look anything like the Schrodinger equation result. In addition to the principal 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. Here that in our units the speed of light, c, is 1. So this corresponds to mc squared. The second term is identical to the Schrodinger equation energy level and depends only on the principal 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 3 halves or 1 half is the tiny amount .0004535 electron volts. This is in excellent agreement with the observed fine structure split. So the Dirac equation accurately predicts the energy difference between a 2p orbital with a line orbital and spin angular momenta and a 2p 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 Schrodinger equation. Just as the Dirac and Schrodinger 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 1s wave function. The Schrodinger solution is e to the minus r. There are two Dirac solutions. These have the same energy but different spinners. 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 Schrodinger solution with a spinner for either spin up or spin down. So the Dirac equation predicts to 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.