CPS 100 Lecture Notes - Lecture 3: Positron, Cystic Fibrosis Foundation, Zitterbewegung
Dirac equation
Before Dirac came along, Wave Mechanics made great strides in explaining the
behavior of particles.
However, there were a few problems with the current mechanics. First, they did not
account for relativity, and only applied to particles with lower velocities than that of
light. In addition, particles like the electron were only considered to be electrified
points, and were not classified by other fundamental characteristics, like spin
(angular momentum). Wave mechanics postulated that the wave associated with
electrons was only a scalar function.
The Dirac equation introduces four components (each with their own partial
differential equation) to the wave. Two are in a positive energy state, each with a
spin of 1/2 “up” and “down”, and the other two in negative energy state, each with a
spin of 1/2 “up” and “down”. The Dirac equation assigns the new properties of spin
and magnetic moment. The Dirac magnetic moment is Where S is the spin vector,
q the charge, and m the mass.
In short, the Dirac equation provides us with a complete idea of the electron. It
integrates the corpuscular aspect, with mass, charge, magnetic moment, and spin,
and the wave aspect that explains the behavior of the electron in atomic systems,
crystals, for example, including diffraction. The equation also opens up the idea of
antimatter. If an electron were to pass from a negative energy state to a positive
one, a “hole” would be created in the negative state, and in order to balance this
change, there needs to be a particle with the same properties as an electron that is
positive. In 1932, Carl Anderson discovered the positron through experimentation,
proving Dirac’s theories.
There is a formal procedure known as the Foldy-Wouthuysen transformation to
systematically take the non-relativistic limit and obtain the Pauli-Schroedinger
equation. The Foldy-Wouthuysen transformation lets one show that the magnetic
moment of the electron is one Bohr magneton, twice the naive expectation of half a
Bohr magneton. The spin-orbit coupling also arises as well as the Darwin term.
One of the more remarkable features of the Dirac equation arises because it is
linear in the spatial momenta. This means that the velocity operator
v⃗ =ddtx⃗ =iℏ[H,x]=α⃗ c
Thus the possible values for the velocity are the spectrum of eigenvalues of αc
which is ⟨v⟩=±c. This means that a Dirac particle is always instantaneously
traveling at the speed of light. It only appears to be moving slower because it is
fluctuating back-and-forth on a short time scale. This is known as zitterbewegung
("trembling motion" in German) and is responsible for the Darwin term which can
be interpreted as a lessening of the Coulombic attraction when the electron is
within Compton wavelength of the proton.
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Before dirac came along, wave mechanics made great strides in explaining the behavior of particles. However, there were a few problems with the current mechanics. First, they did not account for relativity, and only applied to particles with lower velocities than that of light. In addition, particles like the electron were only considered to be electrified points, and were not classified by other fundamental characteristics, like spin (angular momentum). Wave mechanics postulated that the wave associated with electrons was only a scalar function. The dirac equation introduces four components (each with their own partial differential equation) to the wave. Two are in a positive energy state, each with a spin of 1/2 up and down , and the other two in negative energy state, each with a spin of 1/2 up and down . The dirac equation assigns the new properties of spin and magnetic moment.
