Going to a Relativistic Equation

1
Going to a Relativistic Equation
Principles of Quantum Mechanics

• There exists a state function F that summarizes
  all we know about the state.
• Every physical observable is represented by a
  hermitian operator.
• Physical systems an be classified as eigenstates
  of operators.
• The eigenfunction for a complete set.
• The time development of the system is given by

Transition from Classical to Quantum Mechanics
2
Relativistic System
This is just the classical wave equation
(Klein-Gordon Equation)
where
Try to find a probability density a la
Schrödinger
Want to identify this as Probability density
-- Problem Not positive definite.
3
Dirac Try a first order equation
Square matrices
Column vector
Want to recover Klein-Gordon equation if we take
another derivative
0
1
Defining equations for matrices
if
4
Probability Current
(-)
Probability Current Term with
Probability Density Term
5
Stationary Solution
Stationary
Assume ? is an eigenvector of ß with eigenvalue ?
and that ?f(t) ? then
Negative eigenvalues Problem?
6
EM Force in Lagrangian
If
Then
Lorentz Force
7
EM in Dirac Equation
Here
More things to show that we are on the right
track
8
Non-Relativistic Limit
Break up column vectors into upper and lower
halves
First term has Pauli matrices. Inversion because
as are off-diagonal.
Assume that we are close to a state where, if we
were not moving
Further, factor out the time dependence of the
stationary state
9
Non-Relativistic Limit (continued)
Assume Small.
Bottom Half
Top Half
Use Pauli Identity
10
Non-Relativistic Limit (continued)
Square of Field assume small
Assume Coulomb Gauge
Assume constant field
11
Non-Relativistic Limit (continued)
Paulis equation for an electron with right
gyromagnetic ratio (2).