Magnetism: Part 3 - Canonical Momentum
Hello again, magnetism enthusiasts! In the last post, we introduced the Bohr magneton, which, as we saw, is a rather natural and intuitive measure of the magnetic moment in atomic systems. We also defined the often confused terms; magnetisation and magnetic field, and we considered the special case in which the magnetisation and the magnetic field are linearly related.
In this post, we will be focusing our attention on the effect of an applied magnetic field on a system of charges, using classical mechanics. There's going to be a little bit of maths, but I'll try my best to put it into words.
Canonical momentum
Let's imagine a particle, for a moment, that is so small, it can be considered as just a point. It has a charge, q, mass, m and is moving with a velocity, v. Now suppose that our particle is moving within an electric field E, and also within a magnetic field, B. Our particle is going to experience a force, F, given by
This is called the Lorentz force. Now, lets try and determine the variation of momentum of a charged particle in a magnetic field. We will start with the force, writing it as the rate of change of momentum,
We can also write the magnetic field in a different, rather helpful form: we introduce the notion of a magnetic vector potential, A, which allows us to write
For those of you who haven't seen the 'upside down triangle' before, it is the Greek letter nabla and in this context, where we have the cross product of nabla and our magnetic vector potential, it means we are finding the curl of the vector field, or the infinitesimal rotation at every point in the field. The great thing about the magnetic vector potential is that it is gauge invariant, which means we can add or subtract any curl-free components to the vector field, without changing the magnetic field: we have gauge freedom.
Now, we introduce the electric potential, phi, which is a scalar field, and coupled with our previous definition of magnetic vector potential, we can write
In words; the electric field can be written as 'minus the gradient of the electric potential, minus the time (partial) derivative (rate of change) of the magnetic vector potential'.
So we have set up the problem nicely, at this point. What we're going to do now is rewrite equation (1) as
Now through application of vector identities, we can simplify our expression to the form
where dA/dt is the connective derivative of A, which measures the rate of change of A at the location of the moving particle. We notice that equation (6) looks a lot like Newton's second law which motivates the definition of the canonical momentum
In the case where we have no magnetic field (A = 0), the familiar momentum, mv is recovered, nicely. The concept of canonical momentum will be incredibly useful to us when we begin to consider the quantised kinetic energy operator, in later posts.
In the next post, we are going to think about the Bohr-van Leeuwen theorem before we shift our attention to the quantum mechanics of spin.
wao