You've Heard of SPIN - But How Is it Encoded in the Math of Quantum Physics? Parth G

The concept of Spin is hard, but the mathematics is actually quite simple!

In this video I wanted to take a look at how we build up our mathematical representation (or at least one of them) of quantum mechanical spin. To do this, we'll start by looking at the spin of an electron, and understanding what it is.

In quantum mechanics, spin is the inherent angular momentum a particle / system has. It does not gain this angular momentum by moving along an angular (curved) path or spinning in some way - the particle just behaves as if it has angular momentum! Any extra angular momentum it gains as a result of its motion is added to the spin of the particle. Spin is a particle property, just like charge or mass.

With electrons, which are "spin-(1/2)" particles, we know that a measurement of its spin along a particular direction (e.g. z-direction) will result in us finding the electron in a "spin up" or "spin down" state. What this actually means is that the size of the electron's spin angular momentum is the same in both cases (i.e. same spin speed). But for spin up the electron behaves as if it's rotating counterclockwise around the axis, and for spin down it's clockwise. We just represent these spins with arrows pointing in the direction (up) or against (down) the axis for simplicity.

Any quantum system, like our electron, can be represented by a wave function. This wave function contains all the information we can know about the electron, such as what state it's in and the probability of finding a given spin state when we next make a measurement on it.

If we want to find out any information about a system, we have to make a measurement on it. One such example is trying to find the spin of our electron along the z direction. Another example is trying to find the particle's momentum in a given direction.

Taking a measurement is mathematically represented by a "measurement operator" being applied to the system's wave function. If the system is already in a nice "eigenstate", or a state that is one of the possible measurement results of our measurement, then making the measurement will not change the system state. In addition to this, the eigenvalue equation tells us the actual value we will measure in the experiment - in this case, the size of the spin of the electron.

If the system is not in an eigenstate, then a measurement will cause the wave function to "collapse" into one of the possible measurement results. The probability of the system collapsing into a particular state can be calculated from the wave function as it was before we made the measurement. This also links to the concept of superposition, since any quantum state can be written as some superposition of the measurement results of any measurement.

As we see in this video, a quantum state (such as the spin up state we could find our particle in) can be easily represented with a vector. And measurement operators can be represented by matrices. Then we can use the rules of linear algebra to see how measurement operators can be applied to a quantum system. We can also use the usual rules of matrix transformations to work out measurement operators in other directions (e.g. x- and y-directions).

We also see how the measurement matrices used to represent the spin measurements in x-, y-, and z-directions are very close to the Pauli matrices that crop up often when discussing spin-(1/2) particles. Lastly, we see how to construct bigger vectors and matrices for systems where there are more than two possible measurement results - it's just easiest to start with two-state systems like the spin up and spin down states of an electron.

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

Many of you have asked about the stuff I use to make my videos, so I'm posting some affiliate links here! I make a small commission if you make a purchase through these links.

Timestamps:
0:00 - Spin: Conceptually Hard, Mathematically Easy(ish)
2:50 - Measurement Operators (i.e. the Math Showing How to Measure a System)
3:50 - Mathematical Representation of Spin Wave Functions (as Vectors)
5:29 - Representing Measurement Operators as Matrices in Linear Algebra
6:40 - The Wave Function Collapses Depending on Our Chosen Measurement!
8:17 - Quantum Superposition (Blend) of Different States
9:29 - The Pauli Matrices
9:55 - Constructing Bigger Vectors and Matrices

[ParthGChannel]

And of course, let me know what other topics I should cover in the future!

[churchboy]

Yes, I want to see you derive the Pauli Matrices, and even higher order Matrices. That sounds fun, I love the math!

[thomasruddy]

Yes, please show the derivation of the Pauli matrices - I see this everywhere but have never seen the derivation. Thanks, great vids.

[susmitislam]

Hey man, great video! A small issue: I think the σ_y matrix has got its signs flipped. (Also, a request: more fluid mechanics!)

[fmdhvmw]

These videos are always great, and really easy to comprehend :) If you're ever looking for video ideas, have you considered creating a series of videos to run through something like the basics of quantum mechanics or general relativity, like the physics version of 3Blue1Brown's "Essence of Calculus/Linear Algebra" series? I know most of that information is already in videos you've made, but it'd be super interesting to see it all laid out in order as a definite series!

[tom-lukaslubbeke]

Hey Parth,
Thank you so much for your content. It isn't easy to get along channels on youtube that give mathematical intuition about the physics discussed but still in a fun and interesting way. You really make that work. Keep it going and have a great end of the year and a great start into the new year
Cheers

[jlpsinde]

Amazing video, thanks a lot. Yes, please do more about this mathematics.

[pyro.monkey]

I would love to see a video on the derivation of the S operator great video I like when you explain the math and not just the concept

[ytpanda]

Would love for all those extra derivations in another video :)

[johncrwarner]

I am fascinated currently with Geometric Algebra
and was thinking that some ideas in that
have been used to encode quantum spin states.
Have you explored those ideas?

[giu_mal]

Great video! If someone is more interested in how the spin was discovered you can read about the Stern-Gerlach experiment.

[graymars]

Dude 😎 dude 😎 I'm learning....that's HUGE
thank you so much

[ELIAS-fzir]

One problem for me is, if one says spin is a type of angular momentum shouldn’t there be an associated energy coming with this? But the energy of a particle at rest in vacuum with spin, is just it‘s rest mass(neglecting vacuum fluctuations). I know that the dimension of Spin is the same dimension as angular momentum and it builds the same algebra as angular momentum, but are these the only reasons why one can say it‘s a type of angular momentum?

[artemdruzhinin]

Hi Parth! Could you please clarify the following. Before measurement, the spin is in some superposition state in 3D space along all 3 axes, right? We can only measure it along one axis at a time? Why is the spin along y-axis is a superposition of two states along the z-axis (not y-axis itself)? Can the spin along y-axis be a superposition of two states along x-axis? Thank you!

[shutupimlearning]

as always Parth fantastic video! I was wondering if you did a thesis and was it on QM? Cheers

[nintendoswitchfan]

I loved your video, But please make a video on the Dirac equation.

[UsamaThakurr]

Is there any work done on this which will not describe this in probabilistic way or finding the exact position of a particle before measurement? I mean it does sound a little impossible but still have you come acrossed something like this?

[markholm]

I have had a QM question for several years and have not yet gotten a physicist to give me a satisfactory answer. The usual first example of Feynman diagrams is two electrons scattering as the result of exchanging one photon. We know from electrostatics that two electrons, having like charge, must repel and the Feynman diagram is drawn to show this. The example is easily extended to two positively charged particles, such as protons. But what about the case of a positively charged and a negatively charged particle exchanging one photon. We know this must result in attraction, the direction of the momentum change vector must be reversed. The photon is the only means of transferring anything between the particles. How does QM explain that a photon emitted by a particle in one instance causes repulsion and presumably an identical photon in the other instance causes attraction?

[esorse]

You could argue that a unique one sylllable sound corresponding to the symbol 0 is a necessary condition for evaluating the proposition, 0 is the number "0", as either true, or false, like A is the letter "A", otherwise you're left with an undecideable, Kantian synthetic apriori proposition, where the subject '0' contains the predicate 'is the number zero'.

[FlaminTubbyToast]

So I’m getting into physics calculus and proofs this next semester. I was wondering if you could do a video on LaTeX and where to get it?