Electrostatic "Force", the Lagrangian and Schrodinger's Equation

Electrostatic Potential and Electrostatic Force

We want to understand the electrostatic forces between charges.  We really don't know WHY there is a force between charged particles, but we can understand, with the help of the Klein-Gordon equation, how an electrostatic potential can change the motion of a charged particle.  For reasons that will become evident later, I am choosing to work with the electrostatic potential instead of an electric field.

We will start with the simplest possible case, just two stationary charges.  One we call the source charge, q_s, and the other the test charge, q_t.  We will calculate the force on the test charge due to the potential field around the source charge.  The most fundamental interaction between these charges is based on the electrostatic potential.  The test charge will respond to the gradient of the potential due to the source charge.  This potential is a function of the separation between the charges:

Because the electromagnetic interaction propagates at the speed of light, the distance r is between the test charge at time “now” and the source charge at a retarded time delta_t in the past.  This distance is also c*delta_t and so the potential could also be expressed as:

We will come back to this form much later, but for a while we will only consider the r version of this potential function.  

The force on the test charge causes an acceleration of the charge.  This is what we mean by force: something that causes acceleration.  It is well known that the electrostatic force here is the gradient of this potential.  We will derive this from first principles, however, starting with the version of the Klein-Gordon equation derived earlier:

Consider first a particle at rest, i.e. with k = 0.  It has charge q_t but is in a location far from any source charge, and thus is at zero electrostatic potential.  The Klein-Gordon equation then gives:

Now consider what happens when the observer moves this particle close to another charge, q_s.  To make this move the observer has to do some work on the test charge.  As a result the particle now represents a higher (or lower) energy.  This means the h_bar omega term is now larger than it was previously.  We are going to make the assumption that somehow the Klein-Gordon equation must continue to be satisfied.  To do this we can subtract the electrostatic potential from the particle total energy:

If this particle had been allowed to fall freely to this position (assuming the potential energy at this location is negative), then the particle would have maintained a constant total energy (h_bar omega ), but it would have acquired some momentum along the way.  In the Klein-Gordon equation momentum subtracts from the energy term.  By subtracting the potential energy change from the total energy, it is now possible for the change in the momentum term to balance the equation.  The momentum gains while accelerating down the potential gradient cancels the extra energy from the potential change.
  

Based on this conceptual understanding, we can now calculate the electric force on the particle.  Consider the particle to be at rest at some location with potential phi.  We will assume that the potential varies over space and time.  First we solve for the total energy:

Now we do a Taylor expansion of the potential, and just keep the first terms in delta_t and delta_r.

We can now put this frequency into a wave function for the particle:

This can be rearranged a little bit to give:

We can now identify a momentum term in addition to a modified energy term:

Thus we see that the gradient of the potential results in a momentum term that is increasing with time, i.e. an acceleration.  The rate of change of the potential gives a frequency(energy) that changes with time. 

The “force” is the rate of change of momentum:

The Lagrangian

(This section deleted!  The original statements here were not well thought out and incorrect.  I need to revisit how the Lagrangian is related to this derivation.)

Schrödinger’s Equation

We are now ready to derive Schrödinger’s Equation as a non-relativistic approximation.  We start with the Klein-Gordon Equation with the electrostatic potential:

First solve for the total energy, h_bar omega:

Next pull the rest mass energy out of the radical:

Now we make the non-relativistic approximation that the second term in the radical is <<1 (i.e. that h_bar kc is small compared to m_0 c^2) and expand the radical in a Taylor Series about k=0:

If we now replace the energy and momentum by their respective operators we get:

This is very similar to Schrodinger’s Equation, except that the momentum term is not the second derivative but is instead the square of the first derivative.  For wave functions expanded in orthogonal momentum eigen states this gives exactly the same result, however.

Also this equation has the extra rest mass energy term.  Since this is a constant, it has no effect on the dynamics predicted, just like a constant potential does not affect the dynamics.

Finally we must also recognize that this is a scalar wave equation.  So far we haven't addressed spin at all, nor the possibility that the wave function  might sometimes be a vector, tensor or more complex thing.  We have made no assumptions about what the wave function is at all yet, except that it shows a frequency and wave number that we recognize as the total energy and momentum.