: Introduction to total angular : lecture_7 : lecture_7

# Solution of the Dirac equation for a free particle

The Dirac Hamiltonian takes the form

where

Using , in the coordinate basis, the Dirac equation for a free particle reads

Since the operator on the left side is a 44 matrix, the wave function is actually a four-component vector of functions of and :

which is called a four-component Dirac spinor. In order to generate an eigenvalue problem, we look for a solution of the form

which, when substituted into the Dirac equation gives the eigenvalue equation

Note that, since is only a function of , then so that the eigenvalues of can be used to characterize the states. In particular, we look for free-particle (plane-wave) solutions of the form:

where is a four-component vector which satisfies

Since the matrix on the left is expressible in terms of 22 blocks, we look for in the form of a vector composed of two two-component vectors:

Therefore, writing the equation in matrix form, we find

or

which yields two equations

From the second equation:

Note, one could also solve the first for and obtain

Using the first of these, then a single equation for can be obtained

However,

Hence, we have the condition

Since , the equation is only satisfied if the quantity in the brackets vanishes, which yields the eigenvalues

We see that the eigenvalues can be positive or negative. A plot of the energy levels is shown below:

There is a continuum for (turquoise) and for (periwinkle). There is also a gap between and .

We will show that for , an appropriate solution is to take

If this is the case, then

However,

so that

so that the full solution is

Note that when , the third and fourth components of vanish. In this case, energy is just and the full time-dependent wave function becomes

which are both forward propagating solutions. These correspond to particle solutions, in particular, a spin-1/2 particle propagating forward in time with an energy equal to the rest mass energy.

When , we take

so that

By the same reasoning, the solution for is

so that in the limit , and ,

which describes particles moving backward in times. Thus, the interpretation is that the negative energy solutions correspond to anti-particles, the the components, and of correspond to the particle and anti-particle components, respectively. Thus, the Dirac equation no only describes spin but it also includes particle and the corresponding anti-particle solutions!

In the non-relativistic limit, for , we have

so that

since , it follows that

Neglecting it, and recalling that for ,

the eigenfunctions reduce to

The lower component has become reduntant, and the eigenfunctions just correspond to those of a free particle with an attached spin eigenfunction or for or , respectively. For , the lower component, is called the minor component and the upper component is called the major component.

: Introduction to total angular : lecture_7 : lecture_7
Mark Tuckerman 平成17年2月18日