Recall from the section on Heisenberg's Uncertainty Principle
that particles at the microscopic scale can be described by a quantum
wave function .
This isn't necessarily a function of space. It can be a function of momentum
instead, and either case will provide an adequate description of the function.
The state vector is a concept used in Dirac notation that represents the
"constant" behind the wave. The advantage of using vectors is
that one can apply operators to them through matrix algebra. In other
words, the state vector completely describes the state of a physical system.
The Schrodinger Equations are partial differential equations that can
be solved in order to find the form of a particle's wave function under
certain conditions. So, if the particle experiences a known potential,
and is limited to a certain region of space, then we can determine the
nature of it's wave function.
The Time Dependent Schrodinger Equation governs the form of the state
vector as a particle's physical system changes over time. It is as follows:
This is obviously just for the one-dimensional case were
the particle may only move along the one axis, but can be adapted to higher
dimensions by substituting the laplacian operator for (d^2/dx^2) - this
represents the second-order derivatives with respect to y and z. Here
"V" represents the potential experienced by the particle as
a function of position. The function of on
the left hand side is also known as the Hamiltonian operator. The total
energy of any system involving quantum particles can be determined by
applying this operator to the wavefunction of that particle. The time-dependant
function on the right hand side tells us that the energy of the system
will vary with time. Various other operators maybe applied to find the
momentum of the particle - given by i(d/dx)
and it's kinetic energy - given by -
- among other quantities.
The Time Independent Schrodinger Equation governs the form of the state
vector when it doesn't change with time. It is as follows:
One consequence of using the TISE is that all the systems
it describes have constant energy. We can see this because the Hamiltonian
operation on the state vector results in an eigenvalue E appearing on
the left hand side.
One other condition on wavefunctions is that they must
be normalised over the space in which the particle may appear, in order
that the probability of the particle appearing somewhere within the required
space is always 1. Usually this is over all space, but sometimes (as in
the case of an infinite square potential well) it may be less. A wavefunction
must be normalised over the required space in order for the results of
operators to make sense.
The Schrodinger Equation isn't the only one that makes
use of wavefunctions. The Klein-Gordon Equation (in which space and time
are both second-order derivatives) was actually formulated before hand.
Schrodinger didn't like it because it only worked for spinless (spin =
0), charge-neutral particles. However in changing it Schrodinger lost
it's relativistic correctness - this is because it treats time and space
differently (by giving them different derivatives).
The next "evolution" in quantum equations was the Dirac equation,
which aside from being linear (containing only first order derivatives),
did account for spin-1/2 particles (such as electrons). It was almost
delightfully simple (asked "How did you find it?" Dirac responded,
"Beautiful"). It is as follows:
Another advantage of the Dirac equation is that it accounts for atomic
light-emission spectra in ways that non-relativistic quantum mechanics
(such as the Schrodinger Equation) was unable to. A third consequence
of the Dirac equation (one that we won't derive here) is particularly
compelling. It was known by this time that electrons possessed "spin",
or angular momentum. In fact, this had to be assumed for most other quantum
mechanical theories to work. This was not required of the Dirac equation
- in fact spin (and as an added bonus, the magnetism) of electrons became
a natural consequence of the combination of quantum and relativistic properties.
Thus Dirac explained the reason why such properties existed.
Spin and Magnetism
Unlike the wavefunctions input into the Schrodinger equations, the Dirac
equation only accepts input from state vectors with four components, one
of which was the "spin" mentioned earlier. But the intriguing
thing was this: for every solution of the Dirac equation for a particle
with positive energy there was a counterpart solution requiring the existence
of negative energy. In the case of the electron (+ve energy) Dirac called
it's -ve energy equivalent the "anti-electron", or as it is
more widely known, the positron. Thus Dirac predicted the existence of
anti-matter, and the positron itself was discovered experimentally in
But what of ground-states (minimum energy)? These are known to exist and
be positive non-zero. Dirac, after much puzzling, put forward the theory
that there existed a "sea" of particles with negative energy
that "filled up" all the negative energy states. Electrons then
"floated" on this sea, moving towards the lowest energy states
when possible. This is what happens when a positron and electron collide:
the electron loses energy as it moves to a negative-energy state, and
this is emitted in the form of photons. So when a positron and electron
meet, they "annihilate" each other as the electron disappears
into the Dirac sea.