The Man Who Saved Quantum Physics When the Schrodinger Equation Failed
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Published 2023-08-15
The Schrodinger Equation is famous, and rightly so. It's the governing equation of a theory called quantum mechanics. It can very accurately predict how quantum systems (i.e. very small systems) will behave through space and over time. The basic premise of it is that it adds together a system's kinetic and potential energies and equates this to the system's total energy. This is seemingly pretty common sense, but the Schrodinger Equation is "quantized", meaning measurements on the system only give very specific results. We can also never predict exactly which measurement outcome we will get, but only the probabilities of each possible outcome. The Schrodinger equation also has "measurement operators", which are the math equivalent of making a measurement on the system.
Importantly, the Schrodinger Equation is not relativistic. In other words, it does not account for the strange effects we see when relativity is accounted for. We know that when objects move at high speeds relative to each other, that they noticeably measure distances and times differently to each other. Because these effects are not accounted for, the Schrodinger Equation does not always accurately predict the behaviour of small systems that may be moving at high speeds. It also treats time as a universal variable (i.e. everybody measures time in the same way), which is not how relativity deals with what it calls "the fourth dimension".
To save quantum mechanics in these high-speed scenarios, we need to look at some other equations that are both quantised and relativistic. The first equation of this sort that we'll look at is known as the Klein-Gordon Equation. To get this equation we start with Einstein's famous mass-energy relation (E = mc^2). But in reality, we start with the full version of this equation which also involves momentum. Taking this full mass-energy equivalence relation, we can then quantise it and derive the Klein-Gordon Equation.
The Klein-Gordon Equation accurately predicts the behaviour of spin-0 particles. In other words, it does not account for spin. But it is quantum and relativistic. It also has a "psi" quantity in it just like the Schrodinger equation, but here "psi" is charge density, not probability density. This is because the Klein-Gordon Equation allows negative solutions for the square modulus of psi, which previously we interpreted as probability. It makes no sense to have negative probabilities, and instead this equation deals with the behaviours of particles with positive, negative, and zero charge.
To account for spin, then, we need to look at yet another equation. Remember, spin is angular momentum that is inherent to a particle (without it moving along a curved path or rotating). The equation that starts to account for spin is the very famous Dirac Equation. It's highly complicated, but can be essentially thought of as the square root of the Klein-Gordon Equation. It has four complex degrees of freedom in its "psi" quantity. The first two of these look like the quantum wave function "psi", but the remaining two encode details for systems that are quantum and also relativistic.
When Dirac came up with his equation, he realized that some potential solutions allowed for particles similar to the ones we know, but with the exact opposite charge. For example, electron-like particles with +1 unit of charge rather than -1 were allowed. Dirac thought this was initially a mistake, but we eventually found particles like this to exist! We now call them antiparticles, which make up antimatter. In other words, what was initially thought to be an accident of under-constrained mathematics, actually provided a wonderful prediction for phenomena never seen before!
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Timestamps:
0:00 - Understanding the Schrodinger Equation
3:50 - Relativistic Quantum Mechanics
5:05 - The Klein-Gordon Equation
7:42 - The Dirac Equation
Videos in Cards:
1) • Schrodinger Equation Explained - Phys...
2) • Spin in Quantum Mechanics: What Is It...
All Comments (21)
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Dirac was known for the scarcity, brevity, and precision of his speech. Dude was not a guy you'd go out with to have a beer. He was once lecturing at a convention and had written several equations at a blackboard. When he was finished, someone raised his hand and said, "I don't understand your equation in the upper right of the board." Dirac sat down. After moments of awkward silence, the moderator of the convention asked Dirac, "Are you going to answer the man's question?" Dirac: "It wasn't a question. It was a statement."
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How are you only at a little over 200k subs? This channel is great and deserves much more! The quality of video production to the information being given out and it's clarity, very much props! Thanks for another vid!
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Parth been watching for a long time you are one of the best physics communicators I have ever seen I really love your videos keep it up :)
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Hi, Parth!I have been studying physics for a while now. Thank you for providing these comprehensive videos. Your teaching style is brilliant. May your channel prosper. <3
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You’re “wave of probability” description at 1:30 was a REVELATION for me
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a very good explanation in the discription box about the subject. Have been watching you since 2020 :) 👍👍
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Fantastic! Superb! I'm running out of words! I'm just going to seriously study QP and this video has equipped me with so much clarity!! Looking forward for much more!!
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I really appreciate the PRECISE wording, even though this is all hand-wavy area. Very informative!
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Excellent as usual. A pity we have to wait so long between vids Parth. Just finished The Strangest Man by Graham Farmelo on Dirac - brilliant book.
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Brilliant video, very clear and amazingly simple. Well done.
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Just found the channel. Very sweet to have you explaining the meat of the matter so well.
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Parth, thank you deeply for your efforts — this presentation really shows how sincerely you thought about Dirac's groundbreaking achievements, and how to transport the message to a broader audience. Highly respected! 👏🤓😎
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This is a huge drill down of the most complicated formulas in science. Well done man (I am interested in bose Einstein condensation atm - on yt level.)
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Excellent video as always! 😊
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Hey Parth, Great job on your video. You have a continuous bass frequency in your audio! take a look at your mic for the futrue
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This is tough Please recommend a book for this topic
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I'd emphasize the brillian idea of Dirac of eliminating square root. In this approach matrices appeared naturally, not because Dirac was a super fan of them.
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The clue about the DIRAC equation is that square rootsvof differential operators apparently make no sense, so DIRAC had to find a way to fix that problem. His idea was genius: He looked for coefficients that anticommutate so that mixed terms would cancel out, and each coefficient squared would yield unity. And there are such coefficients, it's matrices. Needing 4 of them, it had to be complex-valued 4×4 matrices so that Ψ had to be a 4 component vector which leads to the question what each component actually means. So, the DIRAC equation predicts both spin and antimatter.
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the timing just could not be more perfect, I'm currently learning this for my final exams 🙏
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Thanks a lot! 👍1:58 ... then some interesting effects start to show up such as time passing at different rates from one object's perspective to the other and also distances being measured differently from each object the Schrodinger equation doesn't account for these effects it also treats time very differently to how it treats space whereas in relativity time and space are treated on a fairly level footing 2:18 in the sense that with some maths time can be treated as a fourth dimension to go alongside the three dimensions of space that we work with now 2:25 ... 6:31 ... that it doesn't account for a particle property called spin spin is a quantity that particles can possess just like charge and mass and it's a measure of how much angular momentum a particle naturally has 6:42 8:00 ... so how does the direct equation account for spin when it's square at the Klein Gordon equation doesn't and again remember I'm ... equation doesn't account for spin when the Dirac equation does is because the equivalent of squaring actually loses information this is the same way that the square of negative 2 and the square of positive 2 are the same value. 8:24
