# Classically

Working with a beam of electrons, we can define a pulse as n charges per unit volume, each carrying charge q and moving with speed . Defining to be the time-average speed of the electrons, we can find the number of particles within a beam. If A is the beam cross-sectional area, then l, the length of the pulse , has a volume Al with nAl charges within its bounds. Since each electron carries charge q, the total charge is

Defining the charge density as

The time it takes the pulse to move past a given point

The total current is

The current density

where is the drift velocity.

Beam Power =

# Relativistic Quantum Mechanics

Using the operator relations

Working with

Having this equation operate on a particle's wave function, we can obtain the relativistic Schrödinger equation, also known as the Klein-Gordon equation.

Letting

The Klein-Gordon equation can be written as

Multiplying by the complex conjugate

Subtracting

From the conservation of probabilty, we know that

This can be written in the form

where

This gives

and

The number of particles in a beam passing through a unit area per unit time is

The number of stationary target particles per unit volume is

Initial flux=

where is the relative velocity between the particles in the frame where particle 1 is at rest

Using the relativistic definition of energy

Letting be the energy of particle 2 wiith respect to particle 1, the relativistic energy equation can be rewritten such that

where similarly is defined as the momentum of particle 2 with respect to particle 1.

The relative velocity can be expressed as

The invariant form of F is

where in the center of mass frame and

As shown earlier