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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. In quantum mechanics, it turns out that the state of a quantum system behaves very similarly to a vector in some abstract mathematical space, known as a Hilbert Space. I've discussed this idea in more detail in this video:    • The Language of Quantum Physics is St...   And the act of making a measurement on a quantum system, can be mathematically described by a matrix! Specifically, these matrices are called "measurement operators". More about that in this video:    • Ever heard of Quantum Operators and C...   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. 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. A Quantum Physics Book I Enjoy: https://amzn.to/3sxLlgL My Camera (Sony A6400): https://amzn.to/2SjZzWq ND Filter: https://amzn.to/3qoGwHk Microphone and Stand (Fifine): https://amzn.to/2OwyWvt Gorillapod Tripod: https://amzn.to/3wQ0L2Q Thanks so much for watching - please do check out my socials here: Instagram - @parthvlogs Patreon - patreon.com/parthg Music Chanel - Parth G's Shenanigans Merch - https://parth-gs-merch-stand.creator-... 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