Modern physics is dominated by the concepts of Quantum Mechanics. This page aims to give a brief introduction to some of these ideas.

Until the closing decades of the last century the physical world, as studied by experiment, could be explained according to the principles of classical (or Newtonian) mechanics: the physics of everyday life. By the turn of the century, however, the cracks were beginning to show and the disciplines of Relativity and Quantum Mechanics were developed to account for them. Relativity came first, and described the physics of very massive and very fast objects, then came Quantum Mechanics in the 1920's to describe the physics of very small objects.

Neither of these theories provide an easy intuitive picture of the
world, since they contradict the predictions of familiar Newtonian
Mechanics in the regimes for which they were developed. Nevertheless,
both schemes reproduce the Newtonian results when applied to the
everyday world. In seeking to understand the physics of semiconductors
at an atomic level we *must* start from a Quantum Mechanical
viewpoint, since the entities with which we will be dealing
(electrons, atoms, etc) are so very small....

In Quantum Mechanics this neat distinction is blurred. Entities which we
would normally think of as particles (e.g. electrons) can behave like
waves
in certain situations, while entities which we would normally think of as
waves (e.g. electromagnetic radiation: light) can behave like
particles. Thus electrons can create wave-like diffraction patterns
upon passing through narrow slits, just like water waves do as they pass
through the entrance to a harbour. Conversely, the photoelectric effect
(i.e. the absorption of light by electrons in solids) can only be explained
if the light has a particulate nature (leading to the concept of
**photons**).

Such ideas led DeBroglie to the
conclusion that all entities had both wave and particle aspects, and that
different aspects were manifested by the entity according to what type of
process it was undergoing. This became known as the **Principle of Wave-Particle
Duality**. Furthermore, DeBroglie was able to relate the momentum of a "particle"
to the **wavelength** (i.e. the peak-to-peak distance) of the
corresponding "wave". The DeBroglie relation tells us that p=h/lambda, where
p is the particle's momentum, lambda is its wavelength and h is Planck's
constant. Thus it is possible to calculate the quantum wavelength of a
particle through knowledge of its momentum.

This was important because wave phenomena, such as diffraction,
are generally only important when waves interact with objects of a size
comparable to their wavelength. Fortunately for the theory, the wavelength
of everyday objects moving at everyday speeds turns out to be incredibly small. So small in fact that
no Quantum Mechanical effects should be noticeable at the macroscopic level, confirming that Newtonian
Mechanics
is perfectly acceptable for everyday applications (as required by the **Correspondence
Principle**). Conversely, small
objects like electrons have wavelengths comparable to the
microscopic
atomic structures they encounter in solids. Thus a Quantum Mechanical
description, which includes their wave-like aspects, is essential to their understanding.

Hopefully the foregoing discussion provides a convincing enough argument to use Quantum Mechanical ideas when dealing with electrons in solids. Next we must address the question of how exactly one describes electrons in a wave-like manner....

The approach suggested by Schrodinger was to postulate a function which
would vary in both time and space in a wave-like manner (the so-called
**wavefunction**) and which would carry within it information about
a particle or system. The time-dependent Schrodinger equation allows us to
deterministically predict the behaviour of the wavefunction over time, once
we know its environment. The information concerning environment is in the
form of the **potential** which would be experienced by the particle
according to classical mechanics (if you are unfamiliar with the classical
concept of potential an explanation is available).

Whenever we make a measurement on a Quantum system, the results are
dictated
by the wavefunction at the time at which the measurement is made. It turns
out that for each possible quantity we might want to measure (an **observable**) there is a set
of special wavefunctions (known as **eigenfunctions**) which will
always return the same value (an **eigenvalue**) for the observable.
e.g.....

EIGENFUNCTION always returns EIGENVALUE psi_1(x,t) a_1 psi_2(x,t) a_2 psi_3(x,t) a_3 psi_4(x,t) a_4 etc.... etc.... where (x,t) is standard notation to remind us that the eigenfunctions psi_n(x,t) are dependent upon position (x) and time (t).

Even if the wavefunction happens not to be one of these eigenfunctions,
it is always possible to think of it as a unique **superposition** of two or
more of the
eigenfunctions, e.g....

psi(x,t) = c_1*psi_1(x,t) + c_2*psi_2(x,t) + c_3*psi_3(x,t) + .... where c_1, c_2,.... are coefficients which define the composition of the state.

If a measurement is made on such a state, then the following two things will happen:

- The wavefunction will suddenly change into one or other of the eigenfunctions making it up. This is known as the collapse of the wavefunction and the probability of the wavefunction collapsing into a particular eigenfunction depends on how much that eigenfunction contributed to the original superposition. More precisely, the probability that a given eigenfunction will be chosen is proportional to the square of the coefficient of that eigenfunction in the superposition, normalised so that the overall probability of collapse is unity (i.e. the sum of the squares of all the coefficients is 1).
- The measurement will return the eigenvalue associated with the
eigenfunction into which the wavefunction has collapsed. Clearly therefore
the measurement can only ever yield an eigenvalue (even though the original
state was not an eigenfunction), and it will do so with a probability
determined by the composition of the original superposition. There are
clearly only a limited number of discrete values which the
observable can take. We say that the system is
**quantised**(which means essentially the same as discretised).

Once the wavefunction has collapsed into one particular eigenfunction it
will stay in that state until it is perturbed by the outside world. The fundamental limitation of Quantum Mechanics lies in the **Heisenberg Uncertainty Principle** which tells us that certain quantum measurements disturb the system and
push the wavefunction back into a superposed state once again.

For example, consider a measurement of the position of a particle. Before the measurement is made the particle wavefunction is a superposition of several position eigenfunctions, each corresponding to a different possible position for the particle. When the measurement is made the wavefunction collapses into one of these eigenfunctions, with a probability determined by the composition of the original superposition. One particular position will be recorded by the measurement: the one corresponding to the eigenfunction chosen by the particle.

If a further position measurement is made shortly
afterwards the wavefunction will still be the same as when the first
measurement was made (because nothing has happened to change it), and so
the same position will be recorded. However, if a measurement of the
momentum of the particle is now made, the particle wavefunction will
change to one of the momentum eigenfunctions (which are not the same as the
position eigenfunctions). Thus, if a still later measurement of the position
is made, the particle will once again be in a superposition of possible
position eigenfunctions, so the position recorded by the measurement will
once again come down to probability. What all this means is that one
cannot know both the position *and* the momentum of a particle *at
the same time* because when you measure one quantity you randomise the
value of the other. See below....

notation: x=position, p=momentum action | wavefunction after action -----------------|----------------------------------------------------- start | superposition of x and/or p eigenfunctions measure x | x eigenfunction = superposition of p eigenfunctions measure x again | same x eigenfunction measure p | p eigenfunction = superposition of x eigenfunctions measure x again | x eigenfunction (not necessarily same one as before)

Precisely what constitutes a measurement and the process by which the wavefunction collapses are two issues I am not even going to touch on. Suffice to say they are still matters for vigorous debate !

At any rate, in a macroscopic system the wavefunctions of the many component particles are constantly being disturbed by measurement-like processes, so a macroscopic measurement on the system only ever yields a time- and particle- averaged value for an observable. This averaged value need not, of course, be an eigenvalue, so we do not generally observe quantisation at the macroscopic level (the correspondence principle again). If we are to investigate the microscopic behaviour of particles we would (in an ideal world) like to know the wavefunctions of any individual particles at any given instant in time....

The time-dependent Schrodinger equation allows us to calculate the
wavefunctions of particles, given the potential in which they move.
Importantly, all the solutions of this equation will vary over time in some
kind of wave-like manner, but only certain solutions will vary in a
predictable pure sinusoidal manner. These special solutions of the time-dependent Schrodinger equation turn out to be the energy eigenfunctions,
and can be written as a time-independent factor multiplied by a
sinusoidal time-dependent factor related to the energy (in fact the frequency of the sine wave is given by the relation E=h*frequency).
Because of the simple time-dependence of these functions the time-dependent Schrodinger equation reduces to the time-independent
Schrodinger equation *for the time-independent part of the energy
eigenfunctions*.
That is to say that we can find the energy eigenfunctions simply by solving
the time-independent Schrodinger equation and multiplying the solutions
by a simple sinusoidal factor related to the energy. It should therefore
always be remembered that the solutions to the time-independent
Schrodinger equation are simply the amplitudes of the solutions to the
full time-dependent equation.

The bottom line is that we can use the time-dependent Schrodinger
equation (or often the simpler time-independent version) to tell us what
the wavefunctions of a quantum system are, entirely deterministically.
That is,
we do not have to resort to the language of probability. Once we try to
apply this knowledge to the real world (i.e. to predict the outcome of
measurements, etc) *then* we have to speak in terms of probabilities.

As a last point, it is important to realise that there is no real
physical interpretation
for the wavefunction. It simply contains information regarding the system to
which it refers. However, one of the most important characteristics of a
wavefunction is that the square of its magnitude is a measure of the
probability of finding a particle described by the wavefunction at a given
point in space. That is, in regions where the square of the magnitude of the
wavefunction is large, the probability of finding the particle in that
region is also large, and *vice versa*.

This is not intended to be an exhaustive description of what is a very subtle and complex subject, indeed it cannot be so, given my intention to avoid equations wherever possible. The interested reader is urged to consult one of the large number of textbooks on the subject, some of which are listed in the reading list on the contents page. We shall, however, expand greatly upon the basic framework of Quantum Mechanics in later chapters....

Stephen Jenkins Index. Directory Tree.

Last modified: Mon Nov 4 12:10:21 GMT 1996