Quantum Mechanics - Part 3

Note:The images in this article have been excluded for the time being as they were unavailable at press time. Any inconvienence is sincerely regretted.

By: Josette

Wave Functions, a mainstay in the
quantum state of matter.(Part 1)

In the previous episode, we?d look at some of the events that led to the formalism of quantum state of matter. For this essay, I will dwell on one of the most fundamental area when striving to understand quantum physics, which is the wave function.

The study of wave function has begun even in classical physics, when the properties of light were studied in classical optics, that of refraction and diffraction. Then Maxwell formulated his four laws of electromagnetism that could only be explained by using wave functions. But why is there a need for wave function?

I will begin by answering that wave functions came into play when rules of quantum theory disallow a particle or point of object from being arbitrarily determined. Take Heisenberg?s Uncertainty Principle for instance when one can?t determine the exact coordinate of an object and its momentum simultaneously. Hence one is only able to take the probability of its position at a given time. The same goes with the orbits of electrons around the nuclei. The probability is given in the form of wave functions, with the amplitude being the highest probability that a particular incident will happen. Now in a three dimensional system, coordinates will be given in the form of x, y and z. But if we were interested in the relativistic account (see Einstein Special Relativity), we would take t as the fourth coordinate. Hence a wave function could be given as (Image 4). But the probability is given as:

which is the normalized graph you see in statistical distribution. If we were to take each of these coordinates and divide them into elements of the length dx, dy and dz, so that the probability of the particle being in the nth element is (Image 5), where x,y and z are the values of the coordinate belonging to the that element.

Therefore in lieu with the concept of expectation (where the expectation is equal to the sum of probability of each event N happening divided by the events N), we would expect to get

This formula allows us to find the expectation value of any dynamical variable.

A single particle wave function can be regarded as a superposition of wave function corresponding to physically distinguishable states. For example, let us take an apparatus similar to a magnetic spectrometer or Nicol prism, which will physically split a beam into separate beams corresponding to different values of some parameter or parameters characterizing the state. If it is impossible to separate the two states, we will consider to two states indistinguishable. We can take the incident beam on the left as a superposition of the form

Where is the normalized wave function and are constants. We can assume the same superposition to be given to the wave function on the right side of the apparatus. Therefore by skipping through the steps, we arrive at the conclusion that:-

A one-particle wave function is written as a superposition of the form above involving the states with distinguishable physical properties , and the probability of finding the ith state.

What exactly is a wave function?

If a particle were localized at a point, where else the wave function is spread out in space, how would it be able to describe the state of the particle? The answer will lie in Born?s statistical interpretation of the wave function, which gives the probability of finding a particle within the vicinity of x+dx, y +dy and z+dz. But this interpretation is also the one that gives indeterminacy to quantum theory and hence allowed for determining the undeterminable. What would we do if we want to determine the condition of the particle before the measurement is made? Three answers could be given in answer to the question above:-

1. The realist position. This is the position advocated by Einstein, that the particle measured was at the same point prior to measurement. But to advocate this position is to acknowledge the incompleteness of quantum mechanics. That (Image 9) is not the whole story and that there are still hidden variables waiting to be discovered.

2. The orthodox position. The particle was never anywhere in the first place. Hence measuring it has forced us to allocate a place in a line to the particle, hence disturbing its natural habitat. This view, also known as the Copenhagen Interpretation, is associated with Bohr and his disciples. It was also a widely accepted theory by physicist and remains a food for thought. The philosophical question would be, has measurement in anyway interfered with the way we see the world?

3. The agnostic position. It means a refusal to answer with the argument that one cant really determine anything before the measurement is made. It?s a position popular with scientists when they can?t properly answer your questions. (: Quantum mechanically speaking, this position is no longer credible as there is an observable difference if the particle has a precise position prior to measurement, as had been determined by John Bell in 1964. I will talk more about Bell?s theorem when we reach a more advance stage of our survey.

If a second measurement were to be made, how could we be certain of the answer? The first measurement had given us the peak, or amplitude, at a certain point. Therefore we say that the wave function collapses upon measurement to a spike(peak) and needs to be quickly measured as soon as the spike spread out again to a curve.

There are two distinct physical processes involving the wave functions, the first being the ?ordinary? ones utilizing the Schroedinger equation and ?measurement?, in which the function suddenly collapses. The one that would interest us is the one utilized by the Schroedinger equation that we would look at in the next installment.

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