The state of a physical system is described by a "wave function", usually denoted by the symbol.. In particular cases, we will know more or less exactly what function this is - e.g a sine or a cosine, a quadratic expression, etc. This wave function can depend on time, spatial coordinates, etc. Quantum Theory tells us that to make calculations about real measurements that could be made on the system, we must take the square of the wave function. The value so obtained will give us the probability of obtaining, through measurement on the system, a particular value of the quantity we are interested in. We would expect a theory from Classical Physics to give us an exact value of the quantity we were interested in; here, however, the best we can do is calculate a probability of obtaining the value. The wave function is also called a Probability Amplitude, for this reason.
For example, if, on the basis of our knowledge of conditions in which a particle might find itself (in a box, with a magnetic field, for example) we knew how to write down the particle's wave function; and let's say this wave function depended on its position (call that x) and the time measured from some starting time (call that t). In that case, we would write its wave function as (x,t). x could take any values of position that the particle could reach. If then, we wanted to know the chance, or probability, of finding the particle at a particular value of x, say x = 45 cm, at a particular time of, say 7 seconds, Quantum Theory tells us that the answer is(45,7) 2. Note that this is very different from Classical Physics; there, we might know that the "equation of motion" of the particle was, e.g. x = 6t; then the answer to our question would be that the position of the particle at time 7 seconds, would be the exactly x = 42 cm.
Now let us look at the odd way in which Quantum Theory does its calculations about the world. Suppose we have an experiment about a physical process which can happen in more than one way, and we know the Probability Amplitude (or wave function) for each way. To calculate what results we would expect in an experiment which does not distinguish which way actually happens, we have to first add the Probability Amplitudes; then we square the result of this addition to get the answer to compare to measurement. If, on the other hand, the experiment does distinguish which way actually happens, we square the Probability Amplitudes before adding them. To see how this works, let's look at the Double Slit Experiment for electrons.
Suppose we have a set-up which has equal size slits, located at the
same distance from the source of electrons, then the probability that the
electron goes through slit number 1 is equal to the probability that it
goes through slit number 2. We express this fact by writing 12
= 0.5 (or 50%).
and the result is 1 (or 100%);
OR1 = -2, and the result is 0 (or 0%).
For the Double Slit Experiment, this is obviously (??) a calculation of the interference pattern, with its maxima ( 1, in some arbitrary units) and minima ( 0) which we observe. However, if our experiment has some means for detecting, even in principle, which hole the electron goes through (Experiment II), the result of this experiment must be written as II2 =12+22. This is clearly (??) the case in which no interference is observed.
Thus the Quantum Theory has managed to come up with a recipe to give calculations which agree with the observations we make on this weird world in which we live.
What can we say about the wave function (Probability Amplitude) of the electron after it has gone through the slit system, but just before we look at it to decide which slit it went through? In this case, Nature tells us we must write its wave function as . =1 +2 , as explained above. But if we make a measurement to determine which slit the electron did go through, we know we must get the result 1 (if it went through slit number 1) OR2 (if it went through slit number 2). Then we say that the wavefunction has collapsed on to its final value.
According to Schrödinger, the electron can be represented by a wave-function, which contains all the information we can know about the particle. If an electron looks like anything we are familiar with (and it doesn't!!), it comes closest to a small "packet" of waves confined to a region of space x. This wavefunction obeys a wave equation first written down by Schrödinger. The square of the wavefunct ion gives the probability of finding it at a given place (and time).
(MATH NOTE: To represent such a function, we need a superposition of many wave forms, with a "spread" of wavelengths. Since p = h/ this implies a corresponding spread in momentum; this can be calculated to be p = h/x - as we might have expected from Heisenberg's Uncertainty Principle).