Ch 8: Why is probability equal to amplitude squared? | Maths of Quantum Mechanics

Hello!

This is the eighth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll dive into how we calculate probabilities in quantum mechanics. We'll derive the Born rule for calculating probability as amplitude squared, then put together our fundamental framework for how we mathematically model a quantum particle.

If you have any questions or comments, shoot me an email at:
Thanks!

Animations:
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Music:
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♪ Galatea by HOME
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[gustavoalejandromorletavil]

It is very interesting how this subject is imparted in different fields. For instance, I study ChemE and we skipped many steps in order to reach the Schrödinger equation in like two hours. Since our interest is to describe atomic orbitals, the rest of the semester was just that. I'm really satisfied with your explanation videos so far!

[daniellowell]

Fantastic series. Extremely clear and methodical and, in addition, the pacing of the narration allowed me to absorb what you were saying with out the next concept overlapping and clobbering it.

[Agustin-mijy]

I honestly cannot be more grateful for this series. Huge props man! Congrats.

[shashwatbhatnagar]

I owe you a lot . Your vedioes are the work of art . Thanks a lot

[yellowflower]

How can we generalize this argument for basis of infinite dimension?

[anywallsocket]

Seems to me, that despite your explanation, the Born Rule can be assumed simply because it makes the result both real and positive.

I.e., why can you not use the same methods to show the Born Rule to be |x|^k where k is positive and even?

[JDY]

I am studying quantum computing right now, and your explanations about quantum state (particle) and observable (physical quantities) are so much better than any other textbook. I truly appreciate your year and a half of effort. Thank you so much!

I agree with your point that most textbooks present most of these properties as axioms. Studying through textbooks feels like I am just pushing through math, rather than understanding the true physics. Do you mind sharing how you went about studying this field and how you got these intuitions? Your videos help me a lot, but I also want to train myself to also gain these intuitions as well.

[charliekirkland]

This channel has already seen massive growth, wouldn't be surprised if you reached 100k subs by the end of 2023!

[randomcodestuff]

This series is amazing! In three years of lectures I got only a very murky idea of why the maths of QM is the way it is. Keep up the good work!

[TekCroach]

Today I just happened to land on this video series by chance. Now I see I have commented a year ago too. Let me say again this video series is the best. If you unveil the rigorous mathematical jargons/constructs, and a few bizarre quantum phenomena (like wave-particle duality, tunneling effect, and entanglement), QM is just as normal classical physics. Just digest (believe; take by faith) the few weird QM phenomena, and the rest is just School physics. But, only this sort of teaching is possible to convince that. Thank you again. Fantastic!

[redouaneoulaouaina]

Thank you for explanation ..plz tell me why did you expand ket phi on two basis one is continous (for momentum) and the other in discrete basis ( for energy) and you said it is the same ket phi .. in other way can we associate the same ket phi to more than one observable

[hyoukaa]

Hi I'm having trouble derivating the problem shown at 20:13
Should the operator E(hat) be written as ΣEi|Ei⟩⟨Ei| ?

This is the only expression I could think of, I know Σ|Ei⟩⟨Ei| is the identity operator, but can any operator other than the identity operator just be written as the eigen value times the identity operator eg. ΣEi|Ei⟩⟨Ei|？

[theemathas]

Based on the sigma/integral difference at 1:30, I am assuming that the energy eigenbasis is a countably infinite set, but the position (and momentum) eigenbasis is uncountably infinite (having the cardinality of the continuum). How can this be possible for the same single vector space?

The possible coefficient combinations (countably many coefficients) for the energy basis can be represented as a function from integers to reals. The possible coefficient combinations (uncountably many coefficients) for the position basis can be represented as a function from reals to reals.

Since this is a single vector space, this gives a one-to-one mapping between “functions from integers to reals” and “functions from reals to reals”. I don’t know how cardinal arithmetic works, but surely, such a mapping can’t exist, right?

[gollygaming]

Amazing video! This was the best one yet! Such a clear explanation of a really cool derivation :)

[yuminti]

Smart, very cleaver~ it’s not my first time being amazed by how concrete the logic of math is and being amazed by people who come up with them! Nevertheless, being amazed by you for explaining it so intuitively and easy to understand. Thank you. Fabulous ~

[observer]

Question around 1:30: how come the same quantum state can be expressed by both discrete superposition and continuous superposition? Former is infinite dimension but countable and latter is infinite dimension but uncountable. So is the dimension of the Hilbert space countable or uncountable?

[stevenschilizzi]

This series has very high pedagogical value. It will likely become a must-watch for all budding physicists and anyone interested in this topic. Kudos to the author and much gratitude from everyone else!

[johnsmith]

Anyone know how to do that last expansion with the expected value the way he intedned?

[diegopg]

Simply brilliant. This series clearly shows where the mathematics of quantum mechanics comes from, not only from mathematical treatment, but also from physical intuition. I'm speechless. Thank you and congratulations for these videos!

[andreashon]

Instantly subscribe, after I've just seen the first video of your playlist in my youtube feed. Love structurized courses, like yours. Hope you'll keep going!