The quantum mechanical wave function can seem a little bit mysterious, but here's what it really represents.
We begin by looking at a particle that can only move along a single direction, between two fixed points. These restrictions are absolutely not necessary in order to understand the wave function, but make visualizing it much easier.
In the Copenhagen Interpretation of quantum mechanics, when we make a measurement on a system, we can get a range of possible results (each with its own probability). So in our experiment with a particle along a line, the particle could be found in many different places with different probabilities.
What this translates to in practice is that if we did the exact same measurement on multiple identical systems, we would get different measurement results each time - but the ratios in which we get each possible measurement result corresponds to the probabilities of getting each result.
This is very different to classical physics, where if we did the same experiment over and over then we would get the same measurement result in a very deterministic manner.
The Copenhagen Interpretation, however, links the quantum wave function with the probability of finding our particle in different regions along our line. Specifically, the square modulus of the wave function gives us a probability density for the possible measurement results.
In other words, if we find the square modulus of the wave function and then integrate this to find the area under our function between two points, then this gives us the probability of finding the particle between those two points in real space.
We need to take the square modulus of the wave function, rather than just squaring it, because the wave function can actually be imaginary. However probabilities can only ever be positive, hence the necessity of the modulus.
In this video we also look at why we care about the wave function at all, if the physically measurable quantity is in fact the probability density, i.e. the square modulus of the wave function, and NOT the wave function itself.
Firstly, two systems may be similar in every way except for a "phase difference" in the wave function of one of them, given by some multiple of the imaginary number i, multiplied by the wave function of the other system. This phase difference ensures that the two systems are slightly different to each other. But their probability density functions are the same, because the phase factor disappears when taking the square modulus.
But again, why does this matter, if it's not directly measurable? Well, it turns out that in some circumstances the phase information has important consequences for things we measure experimentally. For example, the double slit experiment produces different results depending on the phase of the wave function representing particles passing through the slits. And the same is true for the Aharonov Bohm effect, for which I've made a full video (linked below).
And most importantly, the wave function is actually the quantity described by the Schrodinger Equation. This is the most important equation in the theory of quantum mechanics, and looks at how the wave function of a system changes over time (based on the properties of the system). It accounts for different kinetic and potential energies in the system to calculate the value of the wave function at every point in space and in time.
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Videos Linked in Cards:
2:32 - Copenhagen Interpretation youtu.be/qJCh53SdS6s
5:08 - Imaginary Wave Function youtu.be/Ms2Y9g0VC-c
6:44 - Aharonov Bohm Effect youtu.be/YMjD8jevTUw
7:13 - Schrodinger Equation youtu.be/BFTxP03H13k
Timestamps:
0:00 - Measuring a Particle's Position, and Probabilities!
1:27 - Identical Measurements in Classical vs Quantum Physics
3:23 - How Probabilities Relate to the Wave Function
4:39 - The Imaginary Wave Function and Its Phase
7:00 - The Schrodinger Equation
7:33 - What the Wave Function REALLY Represents
