In the previous sections, we have taken very much a qualitative look at some of the features and characteristics of quantum physics.  However, if any theory is to be useful, then it has to be capable of making predictions.  I spend my days building mathematical models of the movement of radiation through the environment, and unless my models are capable of predicting, for example, how much radiation will be in a certain location in 10 years time, then then the models are not much use.  The same applies to quantum physics.  It has to be able to do something "useful".

Giving a theory some predictive power almost always means putting it into the language of mathematics.  The mathematical theory of quantum physics is very complicated, and in some instances, whole new branches of mathematics had to be invented to explain certain features of quantum physics.  Needless to say, we will not pursue this here!  However, we can at least take a look at some of the issues that arise from the mathematical treatment of quantum physics.  In particular, we will take a look at a mathematical concept called a "wavefunction".

In simple terms, a wavefunction is a mathematical function that satisfies Schrödinger's Equation.  Schrödinger's Equation is a "differential equation" (don't worry about what that means) that describes the energy balance in a quantum mechanical system.  Remember in the first section, we talked about a bowling ball being allowed to fall from a high building.  That bowling ball has an energy balance.  When it is stationary at the start of its fall, it has a certain amount of "potential energy", by virtue of being raised from the ground level.  As it falls, that potential energy is converted to kinetic energy, such that the sum of its potential and kinetic energies is a constant.

Schrödinger's Equation expresses the same type of energy balance, except in the specialised way that is required for quantum objects and systems.  The result is a differential equation, the solutions of which are wavefunctions that describe the quantum system.  Unfortunately, there is no everyday analogue of a wavefunction.  However, wavefunctions have a number of important properties, the key one being:

For every quantum system there exists a wavefunction, from which all possible predictions of the physical system can be obtained.

In this context, "possible predictions" include the energy of the system, the momentum of particles in the system, etc.  Thus, if we know the wavefunction, we can extract from that wavefunction all the information that can possibly be obtained.

If you read the section on the Heisenberg uncertainty principle, you will recall that it is not possible to obtain information about the position and momentum of quantum objects to an arbitrary degree of accuracy.  Instead, we have a limitation on the accuracy to which we can know position and momentum simultaneously.  This begs the rather obvious question, if we can't know these quantities accurately, then what do we do if we want to know, for example, where an electron is likely to be found within an atom?

The wavefunction provides us with the means to answer this question.  By itself, the wavefunction does not provide very much information.  For a start, the wavefunction is not a number in the sense that "numbers" are familiar to us.  In fact, it is what is called a "complex number". Complex numbers can be written down on paper, just as easily as 1.2345 can be written down, but they don't mean much, in that form, in the physical world.  However, complex numbers can be processed mathematically to give meaningful numbers.

The simplest operation is to take the "magnitude" of the wavefunction, and to square it.  This is a rather similar procedure to using Pythagoras' Theorem to find the length of the hypotenuse in  right-angled triangles, but we will not go into this further here.  The derived quantity obtained by squaring the magnitude of the wavefunction has a very important interpretation.  It provides the probability of finding a quantum object in the vicinity of a given point.

In simpler language, this is helping us out with our inability to track the precise position of a quantum object, because of the Heisenberg uncertainty principle.  So, if we have a quantum system and we want to know how likely it is that a quantum object will be located at a particular point, we can take the wavefunction for that system and evaluate it at the coordinates of the position of interest.  The resulting number is then the probability that the quantum object will be located there, in exactly the same way that 0.5 is the probability that a tossed coin will land "heads".

.It's difficult to go much further in this discussion without getting involved in the maths involved with quantum physics.  However, it is interesting to note that the wavefunction is the concept that enables us to make predictions of our quantum systems, in particular the probability that a quantum object will be located at a certain location.  As we hinted in our discussions of the uncertainty principle, it is a basic principle of quantum physics that the best we can do is to evaluate probabilities for quantities such as location.

On to Energy Quantisation