Here's What a Quantum Wave Function REALLY Represents

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.

Thanks for watching, please do check out my socials here:
Instagram - @parthvlogs
Music Chanel - Parth G's Shenanigans

Here are some affiliate links for things I use! I make a small commission if you make a purchase through these links.

Videos Linked in Cards:

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

[ParthGChannel]

Hi friends, thanks for watching as always! Please let me know what other topics to cover in future videos :)

[HilarionWon]

This is brilliant. I have heard so many other experts try to explain this concept. But this version really spoke to me. Thank you clearing this up for me in such a clear way. You are the best!

[kkumar]

The special theory of relativity series is incomplete Bro🤨🤨

[silverspin]

I have watched dozens of videos on this topic from very renowned channels, but this made the most sense to me with the "WHY" of the wave function. This and the video by science asylum are great, else feel incomplete.

[vivek_adi]

Nicely illustrated, which has tremendous power in simplifying complex phenomenon

[silverspin]

Im baffled by the lack of views for i think this deserves more - such a succint explanation

[lambda]

Good video! Thank you. When is it important to know a particle’s exact location? A video on this topic would be great.

[mimmyjau]

Thank you for explaining the motivation for the wave function in comparison to the probability density. I was confused as to why we bothered with the wave function when the probability density seemed like the more "important" function.

[elendor]

Great video! I'm having to redo this stuff due to lack of funding so this was a nice refresher. Thanks!

[Tim-Kaa]

Thank you. Excellent video, keep them coming!

[Dark-tkxu]

Hi Parth, thank you for these videos. Can you create a playlist on solid state physics, electronic materials. While there are many videos on quantum physics and particle physics on youtube, solid-state physics related videos are rare.

[SaadRqAziz]

amazing!! you have a wonderful ability in simplifying topics, thank you

[PenandPaperScience]

Again a very concise and enjoyable video! Keep it up! :)

[geoffreyzziwambazza]

Great explanation… also your hair is on point 😂

[schmetterling]

The wave function doesn't represent any probability on its own. We always need to apply the Born rule. In case of a spatial distribution (in a spatial representation) the projection operator is simply a trivial unity operator. There is no physical equivalent for that, to begin with.

[vgzvusr]

4:46 Wrong probability density graph for hydrogen. Isn't it a Gauss curve?

[anattasunnata]

Hi! Thanks for your wonderful videos.
Have you read about some criticism of the Copenhagen intepretation of QM?
I'd like to recommend to you a fantastic book that explains some of the problems of notions such as the wave function telling us the probability of 'finding' something (which involves an observer that finds something), and proposes the alternative of the wave function informing us about the probability of a particle being present after the decoherence of the extended and blurry state in which the quantum object is.

The book is called "Scientific Philosophy", and it's written by the astrophysicist and philosopher, Gustavo Romero.
The main argument for his position comes from the fact that nowhere in the theory from where the formulation of the wave function comes we can find a reference to observers or measurements; such references are added ad hoc from outside the theory, and Romero suggests that they are the product of a) not having an axiomatized QM theory, and b) having theoretical physicist with poor training and knowledge of philosophical issues, from where the notions of the importance of measurements and observers come from, ideas that are too close to idealism, which tend to give to the human mind an important role in the determination properties and states of the external world.

Hope this may be useful to you.
Kind regards!

[yousofmehrabi]

How about probability of particle being found in a point?

[wulphstein]

It seems to me that a wave function says something about the way a particle exists in space(time). What happens if we assume that spacetime is somehow made of wave functions?

[thelastfrontier]

and this is alll assuming the coppenhagen convention to be correct? i mean what if its incorrect.