Lectures on Quantum Field Theory. Ashok Das

Lectures on Quantum Field Theory

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and we emphasize that the subscript “L” here does not stand for the left-handed particles introduced in the last section. Similarly, in (3.177) we have denoted

It is clear that in the non-relativistic limit, when |p| m, the component u_S(p) is much smaller than (of the order of ) u_L(p) and correspondingly, u_L(p) and u_S(p) are known as the large and the small components of the positive energy Dirac solution. Similarly, v_L(p) and v_S(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| m, and, therefore, E ≈ m (recall that we have set c = 1). Furthermore, if we identify the non-relativistic energy (without the rest mass term) as

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 particle).

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 A_µ represents the four vector potential of the associated electromagnetic field. Since the coordinate representation of p_µ 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| m in the non-relativistic limit. Substituting this back into the first equation in (3.191), we obtain

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 u_L(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

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