large and the small components of the positive energy Dirac solution. Similarly, vL(p) and vS(p) are also known as the large and the small components of the negative energy solution. In the non-relativistic limit, we expect the large components to give the dominant contribution to the wave function.
Let us next look at the positive energy solutions in (3.176), which satisfy the equation
This would lead to the two (2-component) equations
We note that the second equation in (3.181) gives the relation
while, with the substitution of this, the first equation in (3.181) takes the form
where we have used the fact that for a non-relativistic system, |p|
then, equation (3.183) has the form
Namely, the Dirac equation in this case reduces to the Schrödinger equation for a two component spinor which we are familiar with. This is, of course, what we know for a free non-relativistic electron (spin
3.9Electron in an external magnetic field
The coupling of a charged particle to an external electromagnetic field can be achieved through what is conventionally known as the minimal coupling. This preserves the gauge invariance associated with the Maxwell’s equations and corresponds to defining
where e denotes the charge of the particle and Aµ represents the four vector potential of the associated electromagnetic field. Since the coordinate representation of pµ is given by (see (1.33) and remember that we are choosing ħ = 1)
the minimal coupling prescription also corresponds to defining (in the coordinate representation)
Let us next consider an electron interacting with a time independent external magnetic field. In this case, we have
where we are assuming that A = A(x). The Dirac equation for the positive energy electrons, in this case, takes the form
Explicitly, we can write the two (2-component) equations as
In this case, the second equation in (3.191) leads to
where in the last relation, we have used |p|
Let us simplify the expression on the left hand side of (3.193) using the following identity for the Pauli matrices
Note that (here, we are going to use purely three dimensional notation for simplicity)
We can use this in (3.194) to write
Consequently, in the non-relativistic limit, when we can approximate the Dirac equation by that satisfied by the two component spinor uL(p), equation (3.193) takes the form
where we have identified (as before)
We recognize (3.197) to be the Schrödinger equation for a charged electron with a minimal coupling to an external vector field along with a magnetic dipole interaction with the external magnetic field. Namely, a minimally coupled Dirac particle automatically leads, in the non-relativistic limit, to a magnetic dipole interaction (recall that in the non-relativistic theory, we have to add such an interaction by hand) and we can identify the magnetic moment operator associated with the electron to correspond to
Of course, this shows that a point Dirac particle has a magnetic moment corresponding to a gyro-magnetic ratio
Let
