Normalizable wavefunctions and the question of time evolution

MIT 8.04 Quantum Physics I, Spring 2016
View the complete course: ocw.mit.edu/8-0...
Instructor: Barton Zwiebach
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@santos6418]

I am really enjoying these classes, because I am remembering about many subjects of this course. Professor Zwiebach is brilliant!

[@jacobvandijk6525]

He is really good, but there are only a few people who are/were brilliant. Don't forget that he is only reproducing well-known ideas. Newton, Faraday, Maxwell, Einstein, etc. they created/produced new ideas (but not out of the blue)! That's quite a different story, Daniel.

[@ladyvader4358]

@@jacobvandijk6525 Maybe he teaches future Newton Einstein etc. A good teacher is everything!

[@jacobvandijk6525]

@@ladyvader4358 You are absolutely right, my friend.

[@TheKundan11]

Brilliant explanation Prof. Cleared my doubt about what is Normalisation. Thanks.

[@zacharythatcher7328]

The easiest way to interpret the issue is to imagine actually plugging a solved solution to the equation into the RHS at a given time and position. In order to then know exactly how that point will evolve infinitesimally in time, you simply have to multiply it by the potential at the point in space at that point in time, and add that to the second derivative of the solution at the point and space in time with respect to space (aka evaluate the RHS). What you get will, by definition, be the infinitesimal change in time at that point in time and space.

[@willam.willamswong7330]

Great course, dear professor!

[@jacobvandijk6525]

@ 2:08 When he says "All over space", he means "along the entire x-axis"! The integral contains dx, not dV(olume). Only 90 videos to go! :-)

[@babakshiri7270]

Nice, clear teaching

[@andreipangi]

I can't imagine an example where lim[x->inf] d/dx(psi(x, t)) = a, a != 0 and lim[x->inf] psi(x, t) = 0, can someone help me out? Doesn't the limit imply that from a xn > 0, d/dx(psi(xn + x, t)) in interval a +/- delta, with delta as small as needed and that would result in psi(xn + x, t) increasing from that point onward and as such lim[x->inf] psi(x, t) = inf?

[@not_amanullah]

Thanks ❤🤍

[@not_amanullah]

This is helpful ❤️🤍

[@pocojoyo]

Why should the derivative of the wave function go to 0 too ,as x approaches + or - infinity ?

[@jlae111]

The limit has to be 0 to ensure that the integral of the wave function = 1 to be probabilistic.

[@surojpaul14]

@13:33 he used psi(prime) which is equal to psi/root of N,,but I can't get it how can he found this relation between them

[@David-mm6nx]

psi prime is a normalized wavefunction. Psi is a normalizable wavefunction because its integral of square from -inf to inf is a constant N. For all normalizable wavefunctions (psi in this case), there exists a normalized wavefunction (psi prime in this case) that is a constant times the normalizable wavefunction (in this case, the constant is 1/sqrt(N)). This constant is determined by the normalization condition (integral that has to equal 1), so psi prime in this case must satisfy this condition because it is a normalized wavefunction. Hence, the 2 conditions for psi prime are: 1. be some constant times psi, a normalizable wavefunction 2. conform to the normalization condition. (integral of square from -inf to inf is 1) The only psi prime that satisfies this is psi*1/sqrt(N) with N being the integral of the square of psi from -inf to inf. We can see this directly: integral of psi from -inf to inf is N, and N*(1/N) = 1, so our normalization constant squared must be 1/N. Taking square roots: sqrt(1/N)=1/sqrt(N).

[@leoteng1640]

I would conjecture that proper interpretation of scientific facts is akin to the proper interpretation of biblical text. Wrong interpretation leads to no further and constructive discoveries, while right interpretation does. Right interpretation leads to scientific inventions while wrong interpretation leads to naught. Biblical interpretation has the same outcome, and certainly I would posit and assert that proper interpretation is not up to an individual as most Protestantism asserts, but must suscept ourselves to authority. The problem is biblical notions are much harder to prove. Due to this fact, Christ has given this authority to Peter, and his college of apostles to take on this responsibility. We must obey this authority of the church in its interpretation, but at the same time discourse to further reveal more about its truth.

[@ashishkumarsharma1323]

I am still not able to comprehend why the psi goes to 0 when x tends to infinity? can anyone give me an answer in a more simplistic way? please.

[@urasgungor3461]

It is a necessary condition for the wavefunction to be normalizable, as if it is not satisfied the integral of psi^2 won't have a finite value. But it doesn't have to be true for all psi, they would be non-normalizable if they diverge!

[@jacobvandijk6525]

I wonder, has "normalization" anything to do with "renormalization" in quantum field theory?

[@danielcastillo2299]

No, when you normalize a wave function you use the normalization condition, which is that the integral of the modulus squared over all space is equal to 1, to, in a way, figure out the amplitude. Renormalization in QFT, is a set of mathematical tools that are used to treat infinities that come up in calculations.

[@1eV]

@@danielcastillo2299 I had a question. I can't really hear clearly what prof Zwiebach pronounces d(psi)/dx as. Does he say "dip psi d x"?

[@kemalm9383]

Awesome job. Why didn't he use linear algebra to explain it?

[@handhdhd6522]

LINEAR ALGEBRA SUCKS! no just kidding, I don’t think its a prereq for the course @MIT so it’s taught as you go along which a lot of schools do

[@wulphstein]

If wave functions are actual things that exist, then the time evolution of the wave function is what time itself is made of.

[@gabe2237]

Recall from a recent episode in the series and also this episode that the wave function is not a physical, or tangible thing. The wave function represents probabilities of where the particle might be at a given time

[@kemalm9383]

It's similar to the convolution theory. 😂