QED
Quantum electrodynamics (QED) combines the waveparticle nature of light into a
single theory. It does this chiefly by combining two concepts: uncertainty and
quanta.
Heisenberg's Uncertainty Principle
Heisenberg's uncertainty principle appears to be a defining property of nature
on an atomic level. There are several equivalent ways of stating this principle.
One of these is
which states that the product of the uncertainties in position Dx and momentum Dp of a particle in the xdirection cannot be less than where and h is Planck's constant (= 6.63x10^{34} kg m^{2 }s^{1}).
The consequence of this relation is that the exact location and speed of a
particle cannot be known at the same time.
To see why we are not aware of this in everyday life, consider a ball with a
mass of 100 g thrown at a speed of 100 km/hr (=27.8 m/s). Assume
we can measure its speed to an accuracy of 0.1 m/s. The uncertainty
principle then states that we are limited in how accurate we can know the
position of the ball at any moment. The uncertainty limit is 7x10^{32} m,
too small to measure, considering a typical atom in the ball is about 3x10^{10} m
across. But since momentum varies with the mass of an object, for a particle
like an electron with a mass 10^{29} smaller than that of the ball,
the uncertainty in position is 10^{29} larger, and no longer
insignificant.
Another, equivalent form of the Uncertainty relation is
where the product of the uncertainty in the particle's energy and the time
period of the measurement cannot be less than .
Since any event must take a finite time, the particle's energy cannot
be perfectly known. Eg, if an atomic event takes 10^{8} s,
then the minimum uncertainty in energy is only about 10^{26} J.
Schrödinger's Equation
The ultimate assumption, in applying QED to atoms and atomic events, is that in
describing all observed phenomena we are dealing only with the interactions
between charged particles (electrons, protons) and the electromagnetic field
they generate.
The motions of particles in a field are described by a special functions called
wavefunctions, Y.
These wavefunctions are solutions of Schrödinger’s equation
where m is the reduced mass of the particle, a squared second derivative operator called the Laplace Operator, V(r) the field, and E energy.
The Laplace operator is defined as
which means "the spatial rate of change in the spatial rate of
change",
and in this it is like acceleration.
The two terms inside the brackets are called the
'Hamiltonian'. The first
term of the Hamiltonian represents the kinetic energy and the second term the
potential energy.
Comment: Strictly speaking Schrödinger’s equation
is nonrelativistic and we should use Dirac's relativistic equation. This has
important consequences for understanding atomic structure and spectra,
principally electron spin, but we will deal with relativistic effects separately
as most phenomena of interest can be explained with the nonrelativistic
equation. For consistency, in future versions of this page, I am tempted to
begin with Dirac's equation.
The only possible wavefunctions Y that satisfy Schrödinger’s equation represent stationary waves with definite energy.
One way to imagine this is to consider a guitar string. The guitar string is
fixed at either end and can only produce standing waves (resonance notes) by
plucking the sting at fixed locations. For example, plucking the string in the centre
produces the first harmonic; plucking onequarter way along produces the second
harmonic, etc.
Because only certain solutions (states) are possible and each has a definite energy, Schrödinger’s equation
embodies 'quantisation':
 only certain states are possible;
 only discrete (quantum) steps from one allowed state to another allowed state are
possible;
 changes of state involve a definite change in energy.
Schrödinger’s equation also embodies 'probability'. The square of the
electron wavefunction, i.e. Y^{2},
at a particular point is the probability that the electron will be found near
that point.
First published on the web:
15 February 2000.
Author: Richard Payling
