The SECOND Most Important Equation in Quantum Mechanics: Eigenvalue Equation Explained for BEGINNERS

The second most important equation in quantum mechanics (in my opinion) is known as the Eigenvalue equation. Originally, it's found in a branch of mathematics known as linear algebra, but in this video we see how it can be used to represent the measurement of quantum systems.

We start by understanding the mathematical meaning of terms in the eigenvalue equation. We see that when certain matrices are applied to any general vector, they result in another vector being formed. This can be thought of as the original vector being transformed.

But sometimes, we can find certain vectors that we can apply our matrix to, which does NOT result in the vector being transformed. The resultant vector still points in the same direction, but may be stretched or squashed by a certain amount. If this is the case, our eigenvalue equation applies. The vector that does not get transformed is known as the "eigenvector", and the factor by which this vector stretches by is known as the "eigenvalue". This is why our equation is known as the eigenvalue equation. The eigenvalues are often represented by the greek letter lambda, though there are various conventions currently in use.

It turns out that when we make a measurement on a system, say finding the spin of an electron in a particular direction, the states in which our system could be found are the eigenstates of the measurement operator. In this instance, that's the spin "clockwise" and spin "anticlockwise" states, otherwise known as spin up and spin down. So when we measure a system that's already in an eigenstate, the state of the system does not change, and the numerical value we end up measuring is the eigenvalue. In this case, the measured value would be the size of the spin angular momentum of the particle.

However when a measurement operator is applied to a system that's not in an eigenstate, the state does change when we measure it. It's worth noting that every eigenstate behaves like a vector perpendicular to every other eigenstate. So any system that's not in an eigenstate can be described by a superposition of different eigenstates. And when we make a measurement, the system collapses to one of the possible eigenstates - we can even calculate the probability of this happening. This collapse is known as the "Collapse of the Wave Function".

This collapse is often quoted as the reason "consciousness" controls the quantum world, and that our measurements affect the universe as a whole. However this process is not well understood, and could even involve interactions between inanimate objects without a conscious being necessary.

There are also small differences between the linear algebra treatment of matrices and vectors, and how they work in quantum mechanics for systems not initially in an eigenstate. These are discussed in the video.

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Timestamps:
0:00 - The Second Most Important Equation of Quantum Mechanics (in my opinion)
0:48 - The Mathematical Meaning of the Eigenvalue Equation (Vectors and Matrices)
2:47 - The Eigenvalue Equation in Quantum Mechanics (Measurement Operators, Electron Spin)
6:25 - Collapse of the Wave Function (Measuring States that are Not Eigenstates)
8:45 - The Slight Differences Between Matrices in Linear Algebra, and Quantum Theory
9:17 - The Schrodinger Equation as an Eigenvalue Equation!
10:00 - Why the Eigenvalue Equation is the Second Most Important Quantum Equation

[fahadkhan]

A random recommendation by YT long ago and not disappointed till 🙌

[garciansmith]

This concept was made much clearer to me after watching your video. Making the connection between this abstract idea of eigenvectors and eigenvalues from linear algebra to measurements of a physically real quantum system is very important for students to understand, but I feel like it is never adequately explained in physics classes which push the idea of "shut up and calculate!"

[scraps]

Man, I wish this channel was around back in my undergrad days lol. I love that you make sure that people that aren't familiar with matrix algebra can follow the entire video, great work dude

[JagdishCVyas]

Short, crisp yet clear presentation of basic concepts. Thanks.

[Seriouslyfunny]

I studied this a long time ago in my engineering chemistry course....
Brings back the hostel memories, the late nightouts and the cramming up of weird theories😅
Great work though 👍

[johncgibson]

I was a EE student 20 years ago. And this spin measurement immediately cleared up my understanding of eigen. This is great.

[DrDeuteron]

note that the time evolution operator U(t) = exp(-iHt), so eigenstates of H are stationary, their evolution is an unobservable global phase (with frequency proportional to energy). This is why we like them; that, and they form a natural basis for the space of all quantum states.

[RaphaelCastro]

Hi! Congratulations!!! It's the first time I've seen someone explain eigenvalues and eigenvectors in a very simple and understandable way. And most of your other videos go in the same direction. Have you ever considered transforming these material to some sort of book or guide? I think that it would be very helpful to undergrad physics students.

[varunrmallya]

Random recommendation I got long ago has turned into a source of excitement now. Never clicked so fast

[mintakan]

Great explanation, esp. for those who have a linear algebra background. I think the only thing left, is explaining the "wave function" (which has sin's and cosines). A video on eigenfunctions, Hibert space, ... would be a nice thing to do next (?)

[vinvinn]

I'm currently taking quantum mechanics course and your videos helped me so much. Thank you!

[yatinannam]

I am 15 yrs old and Ik the QUANTUM MECHANICS better by ur videos
Love from INDIA

[Mk-mepm]

This guy is astonishing when it comes to physics and transferring his knowledge + he also plays the guitar wow

[EricKolotyluk]

Wow! Really great explanation. Quantum mechanics seems much less mysterious now. I have noticed a number of video bloggers have been giving increasingly clear and less mysterious explanations of quantum physics, and yes, the math really helps. Please keep up the good work.

[TerranIV]

Thanks for the great video on eigenvalues! I would love to see a video on electron spin. It seems like the best explanation for why they "don't really spin" is more a reason to not believe the measured values, than any actual reason to think they don't really spin. I'd love to hear what you think about it though!!!

[tn]

This is actually really helpful in helping me understand matrix operations and how those operations are used

[truecerium]

Hello Parth, I really like your videos. As a (former) physicist it´s like a walk on memory lane and quite frankly I understand some concepts presented by you better compared to my studies 30 years ago. Would it be possible for you to explain coherence (i.e. in lasers or accelerators)? Much obliged and keep up your important work

[ragtradeyt]

I'd like to echo other peoples comments: Having studied all this stuff 30 year ago with the Open University (& got the degree! I know not how!!) I was and until now unsure of the meaning of these things. PG - you nailed it! to use a US expression. So much clearer in my mind now - Many many thanks!

[sudhanshuscorner]

Very good bro keep it up! sharing knowledge makes others knowledgeable and one does not regress by sharing knowledge, people don't actually understand this, but you do. Thanks for sharing your valuable knowledge.

[abstract_machine]

really good job on this. i feel like understanding the relationship between what math represents what theory is very important in many subjects. thank you