Ch 2: What are kets and wavefunctions? | Maths of Quantum Mechanics 

ikr!

That would be pretty sweet

Great suggestion, I would love to learn why it’s natural and useful to describe the ‘symmetries’ of particles etc

Andrew the Fairy, you didn't mention your grade so we can safely assume a D- or C- at best. 🤣🙋 Later dweeb

@@ToriKo_ figure it out dolt

@@mikevaldez7684 were you having a bad day or do you always make comments like this?

I agrée

A lot of smart people are saying that QM needs to be re-formalized

My man

Same here bro

If I was teaching a course on am I would use this and ask students to watch this

Are you familiar with linear algebra? This is a change of basis. When we write |E1> we have chosen to represent our state in the "Energy basis" and when we write |p1> we chose the "momentum basis". These are both valid choices to _represent_ our general state vector |psi> and there are many more. In any basis |psi> will be a superposition of basis vectors. |psi>=c1|E1>+c2|E2>+c3... or |psi>=C1|p1>+C2|p2>+C3... Where the c's and C's are the coefficients for that particular basis. Any basis that spans the entire space will contain the full information, but some(like spin) only span a subspace.

@@narfwhals7843 wow okay that’s super interesting. I’m not really familiar with linear algebra, but I’ve seen quite a few videos explaining basis vectors. Your explanation makes sense to me but I imagine there are a bunch of subtleties and inner workings to the explanation that I’m failing to grasp. Thanks for ur time and explanation

Exactly the question that popped into my mind after watching - thanks for asking this!

@@ToriKo_ Well, the entire point of Chapter 1 in the series was precisely the point that you *need* linear algebra to have a solid grasp on these subjects, because ultimately, quantum mechanics is just one particular way of doing linear algebra. In fact, the video explicitly tells you that you need to have at least some minimal education in linear algebra, even if not formal. The video recommended 3b1b's linear algebra series on YT, which I agree with. Having the basics down is absolutely fundamental if you want to have a solid grasp of the intuition behind the mathematics of quantum mechanics.

Hello! Thank you for watching. I think there may be some confusion into what we mean by “vector”. Energy itself is a scalar quantity. However, in the quantum mechanical framework, our particle can be in a state representing a certain energy measurement outcome. This state is represented by a vector, called a ket. The terminology is weird, but the vectors we’re talking about in quantum mechanics are a bit different than the vectors in classical mechanics. So energy is still a scalar quantity when measured. -QuantumSense

@@quantumsensechannel Thank you!

Energy is not a vector, but there are vectors associated with a particular energy. These are called the "eigenstates" for that energy.

Hello, thank you for watching! This is a good clarifying question. You are correct that those two linear combinations describe the same quantum state. So in that quantum state, you are in a superposition of possible angular momenta AND superposition of possible energies. I would be careful in saying “an energy state with momenta”, since we are not in an energy state, we are in a superposition of energy states. And although I showed those two, the particle could also simultaneously be in a superposition for position outcomes, or any other physical quantity. In a later episode, we formalize this a bit by showing that these “outcome states” are the eigenstates of the corresponding observable, which form a basis. So these different linear combinations are just ways to write our quantum state in different bases. So how do we distinguish between the energy and angular momenta linear combinations? You don’t! They exist at the same time, under the same quantum state. They just show up when expanding our quantum state in that respective linear combination. In order to break the superposition, you have to make a measurement, which changes your quantum state (and we’ll also discuss this more in a later episode). Let me know if this doesn’t clear it up! -QuantumSense

Hello, thank you for watching. I have an episode released on hermitian operators, where we define what they are. Also, in general the wavefunction is not hermitian (since it can be complex). -QuantumSense

You are close, but not quite there. A superposition is indeed just a weighted summation of possible "elementary" states, as you suggest, but those states often have nothing to do with position. What these states are ultimately depends on what exactly the system is.

In some systems, some operators will have for each possible value you might measure for it, a 1D space of vectors, and in this case this works as a nice basis for the vector space. In many systems, this will be true for energy. However, not all operators will, by themselves, pick out a good basis.

The plank length is many many orders smaller than the length scales we are operating at.

Hello! Thank you for watching, this is a great question. In truth, it is an axiom of the quantum framework. We haven't derived this fact, since we have nothing to derive it from! But given what we showed in the first episode, hopefully it makes some intuitive sense why we would have such an axiom in our quantum theory. -QuantumSense

@@quantumsensechannel thanks so much for the response! it does make sense why it would be an axiom of the system rather than a consequence of how addition and vectors are defined. i can’t wait to continue exploring your series!

What a vector is is defined earlier in the video. At 1:39. Objects that obey these rules are vectors. If by "plain old vector" you mean arrow, then sort of. Arrows generally are vectors. So you can use the vector addition rules you are used to for an intuition. Energy itself is not a vector. But Energy _states_ are objects in our vector space. The energy of that state is the measurement outcome and just a number, but we can collect the different possibilities of outcomes into a vector. Similar to how a basis vector can basically be represented by a single number because all the other coefficients are 0.

This is why actually taking a linear algebra course, as was explicitly recommended in Chapter 1, is important. This video series is not meant to teach you linear algebra. This video is meant for you to already know linear algebra, and from there, to build on top of those linear-algebraic concepts to achieve an understanding of quantum mechanics.

Hello! The continuity of the coefficient function is actually very important, and in all honesty, I felt kind of bad brushing it off to later in the series. Remember that the coefficient function is the wavefunction, so we're asking how important the continuity of the wavefunction is. If you've ever solved the Schrodinger equation before, you might have seen that continuity is a consequence of solving that equation. More intuitively, we'll show that the momentum operator is proportional to the first derivative of the wavefunction. So if our wavefunction weren't continuous, then the resulting derivative would blow up at a point, which gives us nonsense for the resulting momentum. This is more of a physical interpretation, but I think it gives good intuition regardless. Hopefully this answered some of your question! -QuantumSense

@@quantumsensechannel not going to lie, this is my first introduction to quantum mechanics. I am simply a math major that decided to learn quantum mechanics out of interest, but it is cool to see that there is a proof for why the coefficient is always continuous. hopefully my questions aren’t too annoying, and thank you for the time you take to answer them!

@@kennethhou912 If you're a math major check out Brian Hall's "Quantum Theory for Mathematicians". It's very rigorous and probably much better for a mathematically inclined person than the average QM textbook.