Below are the postulates of quantum physics. Postulates are assumed to be true and doesn’t need proof. Usually in classical mechanics, the postulates are obvious, fits in our common sense and thus we accept them without question. In quantum physics, the postulates are mostly mathematical in nature, not intuitive and not easy to digest. Thus the suggestion that the postulates are not irreducible (not fundamental), and not really complete. When I saw this in my quantum mechanics classes in University, I indeed do not understand quantum mechanics at all. It’s just a system of rules to do calculations and then we somehow get the answers to explain or predict experimental results. So I will attempt to remove the mathematical side as much as possible and explain it with much comparison with the classical physics we are intuitively familiar with.
The state of a quantized system: The state of a quantum mechanical system is completely defined by its wavefunction. There is this mathematical thing we call the wavefunction, which exist in a mathematical space called complex Hilbert space which can represent all states of a system. If you know the states or wavefunction of the system, you can answer any questions asked about the system. Where is the electron, how fast is it moving, etc, all these information are contained in its wavefunction. Those who do not believe that wavefunction completely captures the information of the state considers that quantum physics is incomplete. In classical physics, we can just directly observe the position, momentum of the object in question, but in quantum physics, they are encoded in the wavefunction. The following postulates explains how to read those values from the wavefunction.
Physical Observables: Observables are represented in quantum theory by a specific class of mathematical operators. Following Jim Baggott’s introduction in his book, quantum reality, this is like having the right sets of keys. Apply the right key to the wavefunction, we get to read the value of the properties we want to measure. For example, if we want to know the position of an electron, we apply the position operator onto the electron wavefunction and outcomes the expectation values of where we might find the electrons. This is a bit going ahead for it’s in postulate no. 3. In classical physics, we don’t need to have such troublesome mechanism, we can directly see the things we want to observe in the equations of motions of a classical ball. The quantum difference is also that different sets of quantum operators can be non-commutable. This means that the order of measuring one thing or another for non-commutable things matter. Example of some non-commutable operators are: position-momentum, energy-time, spin in x-axis vs spin in y-axis vs spin in z-axis. So in classical physics, everything is commutable, the order of which we measure this or that first doesn’t matter, but in quantum, if we measure first one thing or another, we don’t get the same results if we switch the order. The act of measuring itself seems to change the wavefunction to give different answers to the second part which is not commutable.
Expectation values: The average value of an observable is given by the expectation value of its corresponding operator. This is open the box. With the key above, we can get the average value of the things we want to measure. A key difference with classical physics is that classical physics directly gives us the value exactly, or to as much precision as we want. In quantum, the individual results of each time we see where an electron is, we cannot predict the exact position for each time we see the electron. If we measure identically prepared electrons (same wavefunction) in the same way on their position (position operator), we can get an average value for their position, which is capable of been calculated via this math procedure.
Born’s Rule: The probability that a measurement will yield a particular outcome is derived from the square of the corresponding wavefunction. As hinted above, the individual measurement outcomes are probabilistic in nature, Born’s rule allows us to calculate the exact probabilities for each possible results. That’s the best we can do for quantum. In classical, any probabilities is due to ignorance, and if we gathered enough data, we can predict anything to exact values. This seems not to be so in quantum, depending on the interpretation. The Born’s rule is also regarded as unsatisfactory as it’s added in to bridge the quantum calculation to what we directly observe in experiments. So wavefunction collapses to the results we get, but before measurement, we don’t know which results we will get. Jim Baggott calls it: what we get.
Evolution of wavefunction: In a closed system with no external influences, the wavefunction evolves in time according to the time-dependent Schrödinger equation. How we get from here to there. Without measurement, the wavefunction evolves deterministically based on their past in accordance to the time-dependent Schrödinger equation. This smooth evolution of wavefunction when meeting measurement, abruptly changes the wavefunction to correspond to the results we get, we call this collapse of wavefunction. Some people don’t like this collapse and thus came out with the quantum many-worlds interpretation. In classical physics, we have a similar evolution of the states of classical objects being deterministic, but we lack the sudden collapse and probabilistic results from Born’s rule.
Let’s use a simple test case to just illustrate how quantum works. Say using the double-slit experiment for electrons.
We have an electron gun, shooting electrons at the same velocity, thus the same momentum, and thus same wavelength to the double-slit on the order of the wavelength of the electron. Behind the slit, we place phosphor screen which emits lights when electrons hit them. So we can directly see the results of the electrons passing through the slit.
First, the wavefunction of the electron identically prepared by the electron gun behaves like a wave from the gun all the way to the screen, interfere with itself, producing interference patterns, we see many lines on the screen, not just two lines from the slit. The physical observable measured here is position of electron as they hit the screen. The screen acts as the measuring device. The expectation values depend on the wavefunction, and we do see interference pattern because we didn’t try to see which holes do the electrons go through to make it exhibit particle behaviour. Born’s rule comes in when we reduce the intensity of the electron beam to only one electron coming out at a time. So for each electron, the exact location where it hits the screen is unknown, but we can calculate the probability of it hitting the screen.
For the evolution of the wavefunction, just picture a wave from the gun, through the double-slit, interfere with itself, and hits the screen. The amplitudes of the wave when squared shows the probability of individual electrons hitting the screen and leave it long enough, we get the interference pattern nicely imprinted upon the screen. Nothing too complicated right?
When you try to look at which slit did an individual electron goes through, you’re introducing measurement at the slits area. So due to the measurement, we should apply collapse of wavefunction of the electron just after the slits. Either the electron goes through the left or the right slit. For those which are blocked by the plate which contains the double-slit, we can just ignore those electrons as not within the area of interest. So when you try to ask the electron to reveal their position way before they hit the screen, the picture on the screen changes into just two lines, corresponding to the two slits. The interference effect due to the wave property is gone. The act of measurement changes the nature of the electron from wave to particle.
Wait, I am sorry, this is not an unbiased view of what happened. What I had just described was in accordance with the Copenhagen interpretation. It’s basically very bare-bones, what’s in the mathematical structure is all that is to quantum. We have collapse of wavefunction, the wave and particle nature of the electrons are complimentary, etc. It’s the first interpretation which got popular and because the religion of quantum, of nothing interesting beneath the maths, so just shut up and calculate. For back in 1920-30s quantum is still yet to be applied to the nuclear, atomic, subatomic physics, particle physics, molecular bondings in chemistry and so on. A lot of work of calculations was to be done by the physicists of that time instead of worrying about the philosophical implications of quantum theory. What it means, is there a reality beneath? Why is nature so weird and not classical? Which classical assumptions must we abandon?
I must apologise again, for now, we shall turn to the front of the theatre, to see the experiments, to see what empirical reality tells us before we venture into the stories and metaphysics behind the interpretations. During this trip to the experiments, there will be detailed analysis for some of the experiments and what classical assumption might you need to throw out of the door when you are faced with the results of the experiments. Those analyses are essential to understand why certain classical intuition cannot be applied and quantum interpretations have the job of choosing which ones to retain and which ones to throw out. The results may be very surprising to you if you use classical expectations to anticipate the results. This is presented first in hopes of you not using your first interpretations to interpret the results and then be attached to the first one. Just see this as how nature works. We shall revisit these experiments in each of the interpretations later on to give the story of how this particular interpretation makes sense of the experiment. For now, enjoy the theatre show or if you like the magic show, not the backstage or how does the magician do it?
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