The Lorentz Force from the Classical Hamiltonian
In this section, we wish to verify that the Hamiltonian
gives the correct Lorentz Force law in classical physics.
We will then proceed to use this Hamiltonian in Quantum Mechanics.
Hamilton's equations are
and the conjugate momentum is already identified
Remember that these are applied assuming q and p are independent variables.
, we have
. The momentum conjugate to
includes momentum in the field.
We now time differentiate this equation and write it in terms of the components of a vector.
Similarly for the other Hamilton equation (in each vector component)
, we have
We now have two equations for
derived from the two
Hamilton equations. We equate the two right hand sides yielding
The total time derivative of
has one part from
changing with time and another
from the particle moving and
changing in space.
We notice the electric field term in this equation.
Let's work with the other two terms to see if they give us the rest of the Lorentz Force.
We need only prove that
To prove this, we will expand the expression using the totally antisymmetric tensor.
So we have
which is the Lorentz force law.
So this is the right Hamiltonian for an electron in a electromagnetic field.
We now need to quantize it.