Infinite square well in quantum mechanics 

Glad to hear this! :)

Thanks for the encouragement!

Glad you found it useful!

Glad that you found the video helpful! :)

We haven't yet covered that in a video... hope to do that at some later point!

Glad you liked it! We are hoping to have more code in some of our future videos :)

Most welcome!

I agree entirely, what other approach were you familiar with?

@@ProfessorMdoesScience Basically same method, but for V(x)=0 in 0

@@juanluna2361 For some reason setting the potential to zero in the range 0

Glad you liked it!

It is great to hear, we never know whether people take a look at it! :)

@@ProfessorMdoesScience For me, the Python code provides additional insights to the lecture content, especially the part of control buttons. It helps me to build "images" into my mind.

Noted! We are hoping to have some more, for example when we work through approximation methods in quantum mechanics.

Glad you like it! :)

Unfortunately we don't currently have a video doing this explicitly. We do hope to make more videos focusing on the position representation, so we'll eventually get there. What have you tried in terms of the derivation?

We cover some of these topics in our series on wave mechanics, check it out here: ru-vid.com/group/PL8W2boV7eVfnHHCwSB7Y0jtvyWkN49UaZ Hope you like it!

Thanks for the feedback!

Yes, in three dimensions the momentum operator is proportional to the gradient, and the kinetic energy to the Laplacian. We very briefly go over it in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Yw2YrTLSq5U.html

@@ProfessorMdoesScience thank you very much

Not quite: you can still get bound states for a finite well which are quantized, and these would correspond to eigenvalues below the well walls. For higher energies above the finite well walls, you would then a continuum of states that are no longer quantized. We hope to cover the finite square well in the future, but I hope this helps for now!

@@ProfessorMdoesScience please, if you can. I prefer to grip myself to your youtube lectures more than others because seemingly you never distance yourself from bra-ket notation, Hilbert vector space properties (orthogonality and orthonormality, inner and outer products, eigenfunctions and eigenvalues, ext.) I think it’s key to understand how complex-valued wavefunction works in quantum mechanics and how we extract physically meaningful information from it. There are not a lot of great lectures on quantum mechanics out there.

Not sure if there is a specific name for this. But note that acting with the Hamiltonian on a state is not the same as measuring the probability amplitude of a given energy eigenstate. You can expand any state (whether you've acted on it with the Hamiltonian or not) in the energy basis, to figure out the probability amplitude of any given energy eigenstate in that particular state. I hope this helps!

@@ProfessorMdoesScience My thought process was like this. I'm trying to understand what acting on a general state does to it and how that is useful. Does the state even live in our state space anymore? I can expand any state in the energy basis. Then the S.E tells me what happens to that state when I act on it with the hamiltonian. If the state is an Eigenstate ui i just get Ei|ui> . But if that state is a more general linear combination Ψ (sum ui>)I get sum ciEi|ui> as the result. (which, because I know the eigenstates of the (time independent)Hamiltonian are eigenstates of the d/dt operator, must be the same as d/dt Ψ up to a constant ihbar) So if I am given this state, i can still figure out the measurement probabilities from the prefactors. But they are not the same as before acting on it with the Hamiltonian, because acting this adds the Ei prefactors. So this would be the collection of the coefficients ui> multiplied by the spectrum of the Hamiltonian?

@@narfwhals7843 A few thoughts: 1. Acting with a general operator A on a state |psi> changes it to another state |phi>=A|psi>. Both states still live in the same state space. 2. I would not think of the Schrödinger equation as an eigenvalue equation. It is the equation of motion of quantum mechanics, which tells you the time dependence of the state. But do not identify ihbar*d/dt with an operator H (otherwise the equation would become a tautology). 3. As to measurement probabilities, they will be different between states |psi> and |phi> after applying any operator A to |psi>. How they differ will depend on both A and the initial state |psi>. I hope this helps!

@@ProfessorMdoesScience Thank you :) My reasoning for this "identification" is that the only functions which satisfy the S.E for eigenstates of the Hamiltionian must be those that are multiplied by a constant when taking the time derivative. So the exponentials. This makes them eigentfunctions of the d/dt operator. I don't think this is actually an identification, though, because not all eigenfunctions of the d/dt operator will satisfy the S.E. Only those with eigenvalues that are a proportional to i/t.

The wave function in momentum space is the Fourier transform of the wave function in position space, you can find the details of why in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-2lr3aA4vaBs.html For an infinite square well, if you calculate the Fourier transform of an eigenstate in position space, you will find that the momentum space wave function is made of two sinc functions centered at momenta of +hbar*n*pi/a and -hbar*n*pi/a, corresponding to motion to the right and to the left inside the well.

The state of a quantum system can in principle be anything, and this is not a probabilisitc feature. For example, you could in principle prepare a quantum system to be in the ground state (state 1). Probability comes into quantum mechanics when you perform a measurement. If you had prepared the system in state 1, it is an eigenstate of the energy operator, which means that, if you measured energy, you would get the ground state energy with probability 1. However, you could also prepare the state in a "superposition" of multiple energy eigenstates. In this case, the probability of getting the ground state energy after a measurement would be smaller than 1, and its precise value is given by the measurement postulates in quantum mechanics. We have a few videos covering this, the first one of which is this one: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-u1R3kRWh1ek.html If you follow the "background" and "what next?" sections in the description of that video, you should find all the relevant information to understand these concepts. Alternatively, we build these ideas from the ground up by following the playlist on the postulates of quantum mechanics: ru-vid.com/group/PL8W2boV7eVfmMcKF-ljTvAJQ2z-vILSxb I hope this helps!

Thanks for the suggestion!

Thanks for the feedback!

The borders of the potential in the video *are* defined between -a/2 and a/2, see around minute 02:35. If you displaced the potential to, for example, the range (0,a), then your solutions would only be sines.

If we have an operator A, then an eigenstate |psi> is simply a quantum state such that when we act with A on it, we get the same state back (up to a constant lambda, called the eigenvalue): A|psi>=lambda |psi> This notation is the bra-ket notation, but, as you hint in your question, we can represent these states in different ways. Wave functions are simply one of many possible representations, corresponding to the position representation. In this language, the equation above becomes (in 1D): A psi(x) = lambda psi(x), where psi(x) = is the wave function. This equation shows how psi(x) is the eigenstate (also called eigenfunction) of the opreator A. You can learn more about eigenvalues and eigenstates here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-p1zg-c1nvwQ.html And about wave functions here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-2lr3aA4vaBs.html I hope this helps!

@@ProfessorMdoesScienceThanks！ it is clear!

We don't yet have any videos on tunneling, but hoping to publish some in the future!

@@ProfessorMdoesScience Thanks, please keep us update.

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