Quantum Mechanics/Meaning of Quantum Wave Function
In the classical picture, we usually work with the position and momentum of a particle or particles. From these, we can generate all the physical information about the system.
However, it is easy to see how such an approach will run into serious problems for quantum mechanics. If we consider, for the moment, the Uncertainty Principle to be our definition of Quantum Mechanics then, there is the position of a particle having no simple meaning, as there is an inherent uncertainty in it.
Talking about a particular position and momentum is wrong, in quantum mechanics.
To get round this problem, instead of having a particular value of position (or momentum) of a particle, we instead assign a probability to each point in space, of finding the particle there. We could similarly define the probability of it having a particular momentum but, while equally valid, that approach is not initially useful.
Thus, in quantum mechanics, we have a wavefunction, which, as we will see, contains all the information about the system. In general, this wavefunction will be complex. This wavefunction's modulus squared is the probability distribution of the system.
Position (or momentum for that matter) now, is not a variable. Rather, it is an operator. Before we see how it works, let us deviate a little, in order to understand operators better.
Take a two state system for example. By two states, we mean that a particular property (say the color) can have only two values (say Red and Blue). Now, a wavefunction of the system will be a mixture of these states. Let us say R is the wavefunction when the particle is "definitely" Red, and B when it is definitely Blue. So, the general wavefunction, W of the system is a linear combination of the two.
where is the probability that the particle is Red. Since this is a two state system, . Be aware that, in general, and may be complex numbers, which is why we must use rather than in order to get a real probability.
Now, the big news. The fundamental assumption of quantum mechanics is that any measurement will send the system into one of that operator's eigenstates. Eigenstates are just the states of a system unchanged by an operator.
In this example, when we measure the color of the system, we multiply it by a color operator, which turns a wavefunction into a color, Red or Blue.
So, Quantum Mechanics says that if you make a measurement of the colour, you will never find the system to be a mixture of Red and Blue! It is only that you will sometimes find it Red, sometimes Blue, with the corresponding probabilities.
If you were to measure some other property of the system, you might find it in a state which you could deduce had mixed color, but you wouldn't be able to observe this mixture. The uncertainty principle stems from this phenomenon.
Measurement is nothing but acting the corresponding operator on our wavefunction. Remember, once we have acted an operator on our wavefunction, the system has already "collapsed" into one of its eigen-states (allowed states). Any subsequent measurements will yield the same result, because the system isn't anymore a mixture of all those eigen-states.
To see the probability effect as stated above, you'll have to prepare the system from scratch. In other words, before making a measurement, prepare infinite mental copies of the system (an ensemble) and then perform the same measurement on each of them.
Coming back to position; this is a special operator, in that its spectrum (the range of eigen-values) is continuous. Our example had a discrete (2 level) system. The details of working out stuff become more complicated for such an operator, but the basic premise remains exactly the same.
What happens now? We know that Quantum Mechanics should reduce to Classical Mechanics at a large enough scale. For a single particle, this wavefunction is peaked around the "Classical Position" and the standard deviation is ~ h. So, on normal scales, we won't see this "fuzziness" of the particle.
To see how a result matches with Classical Mechanics, we can use the concept of an "Expectation Value". An expectation value of an operator is just the average value of its eigenvalues, weighted with the corresponding probabilities. It can be shown that the expectation values of position and momentum are related like the classical position and momentum.