When done in discrete form like this it reminds me of solving a system of lots of masses connected by springs. You get an almost identical tradiagonal matrix
@@josephjones2211 interesting. I'm fairly certain that a symmetric tridiagonal matrix corresponds to a differential equation in the limit N->infinity. In the case of infinitely many infinitely small masses, this probably corresponds to a particular differential equation, like the wave equation or something. Worth looking into in a future video
@@MrPSolver It exactly does correspond to the wave equation as N->infinity. I have just coded it in to your notebook now. I am going to put it on my github jjones2000
So awesome! Watching your Python videos is just so immensely intellectually satisfying, inspiring, and galvanizing. Love your videos. The value of your videos cannot be overstated.
This is an absurdly cool video! I have two TH-cam accounts and went on both to like this video. Thanks for the great content. You're fantastic at explaining things.
Dude, I saw your video about the skydiving differential equation on reddit and was like "meeeh, not again, we don't need this another time". But this, this was brilliant! Great video, nice explanation of a scary problem. Keep it up. Could you also share the jupyter notebooks in the end? Also, I miss the link to your blog in the video description ;) Keep promoting yourself.
Thank you for your comment! :) I can certainly share the notebooks in the end: I'll make a git repository and upload the notebooks there. They should be easily accessible after that.
Hi Luke, Just a courtesy to let you know that I cited this video in a recent paper. Can't put any details because it gets deleted for containing a link but you can probably find them in your comments box. Great channel!
The d variable is just the main diagonal of the main matrix, which has length N-1. The vector mL2V(y) has length N+1 (it's got 2001 points here), and we can ignore the first and last point of a numpy array by taking mL2V(y)[1:-1] (this is just an indexing trick). Then we add a scalar constant to out numpy array: namely 1/dy**2
I was trying to solve woods-saxon potential. I was messed up myself by numerov's algorithm. Then I found your video. Finally I solved the problem using this tridiagonal eigenvalues. But I want to make a request. Can you make a video on how to find all the eigenvalues in a certain range, for any 1-d potential using numerov's algorithm and shooting method.
16:20 aren't energy levels in sho supposed to be equally spaced? In there m and L both are constants so your bar should be increasing linearly right? It seems a little off
very interesting. I have one simple doubt about the number of eigen states. The number of eigen states is equal to the dimension of the matrix, and the dimension of the matrix depends on the number of points we have chosen. So, the number of eigen states depends on the method of solutions!?
Really Great Video! Maybe next time you could compare your results with the analytical ones so in this case the qm harmonic oscillator model. I did it and the results are really close! So again thanks for the video
very nice video! I have a lot of fun using python again! One little question, maybe it is a conventional thing. Then you divide by Δy^2, do you actually mean (Δy)^2 ? (since it is the 2nd (discrete) derivative?
If the wave function was unbounded, you'd have to make the grid very dense, say 100000 points between 0 and 1 and make the potential very localized (maybe between like 0.5 and 0.51). This would be an approximation of a particle that was unbounded, since the infinite boundaries would be very far from the localized potential. (Sort of like how an air molecule is approximately "unbounded" inside a giant warehouse, even though the warehouse has walls)
You are completely right with this. The difference betwen two eigenvalues of the harmonic oscillator has to be constant, so something is not right here. I'm not sure what it is though. Interestingly, the energy output isn't exactly quadratic, since np.sqrt(eigenvalues) also doesn't give a linear result.
@@TueteKaffee Man. I've got the answer. In Numerical Analysis, methods based in finite differences to solve Sturm-Loiuville equations (The Schroedinger equations is a Sturm-Liouville like problem) approximate well the first eigenvalues. For practical purposes, this is what we want: the first energy levels. So, higher energies will not be well approximated.
Can you make your seriers of videos as in a particular playlist? Or anyway to group some of your videos, so that we can play all vidoes in a list ? ^_^
Hi, it is interesting video. learn a lot from it. but there is a question about the demensionless step. in term of potenial, why directly become mL^2V(y)phi? why not mL^2V(Ly)phi?
Hi, amazing video. I wold like to ask I tried to model hydrogen atom with potential in a form of 1000/y, the problem of dividing by zero I solved by starting at y=0.0001 and since energy is not 0 at the right end- in infinity I extended the matrix by one element. The wavefunction looks quit similar to the hydrogen atom but I am not able to see the "infinity" in the graph even if I set N to 10000. Any suggestions will be highly appreciated, have a nice day.
A very small tip. The topics you're representing are good and cool and all that. But I didn't learn them this way. Your videos aren't educational if that's what you want them to be. They are more like "look I can do cool stuff, and you know what, just take the code from the description, because I know that you can't do it yourself". Try to focus on the problem and discribe the code slowly. At the end we both KNOW, that these problems aren't a big deal. Peace.
You sir, you are a freaking beast! Keep it up and I'll be here every single time!
When done in discrete form like this it reminds me of solving a system of lots of masses connected by springs. You get an almost identical tradiagonal matrix
Is the tridiagonal matrix for the system of masses symmetric?
The system of masses would be a great idea for a future video!
@@MrPSolver yes I did the problem just yesterday in my dynamical systems class
@@josephjones2211 interesting. I'm fairly certain that a symmetric tridiagonal matrix corresponds to a differential equation in the limit N->infinity. In the case of infinitely many infinitely small masses, this probably corresponds to a particular differential equation, like the wave equation or something. Worth looking into in a future video
@@MrPSolver It exactly does correspond to the wave equation as N->infinity. I have just coded it in to your notebook now. I am going to put it on my github jjones2000
So awesome! Watching your Python videos is just so immensely intellectually satisfying, inspiring, and galvanizing. Love your videos. The value of your videos cannot be overstated.
This is an absurdly cool video! I have two TH-cam accounts and went on both to like this video. Thanks for the great content. You're fantastic at explaining things.
Dude, I saw your video about the skydiving differential equation on reddit and was like "meeeh, not again, we don't need this another time".
But this, this was brilliant! Great video, nice explanation of a scary problem. Keep it up.
Could you also share the jupyter notebooks in the end?
Also, I miss the link to your blog in the video description ;) Keep promoting yourself.
Thank you for your comment! :) I can certainly share the notebooks in the end: I'll make a git repository and upload the notebooks there. They should be easily accessible after that.
This just became my favourite youtube channel!
this video was really clear and fun, and i love how excited you get. thanks for this!
Road to Reality in the Background. You know the good stuff ! !
Hi Luke, Just a courtesy to let you know that I cited this video in a recent paper. Can't put any details because it gets deleted for containing a link but you can probably find them in your comments box. Great channel!
This is ridiculously high-quality content
that was so cool! saving this.
can you show how to do that for the 3D equation?
Also can you explain the way you wrote the d variable in your code?
The d variable is just the main diagonal of the main matrix, which has length N-1. The vector mL2V(y) has length N+1 (it's got 2001 points here), and we can ignore the first and last point of a numpy array by taking mL2V(y)[1:-1] (this is just an indexing trick). Then we add a scalar constant to out numpy array: namely 1/dy**2
Reminds me of the finite difference method that i used once to simulate a 2D Situation with the Navier Stokes Equations.
YOU are a great teacher (from a professor).
However you can't use this method for scattering states, as you can't set the boundary conditions as psi(0)=psi(L)=0.
I was trying to solve woods-saxon potential. I was messed up myself by numerov's algorithm. Then I found your video. Finally I solved the problem using this tridiagonal eigenvalues. But I want to make a request. Can you make a video on how to find all the eigenvalues in a certain range, for any 1-d potential using numerov's algorithm and shooting method.
16:20 aren't energy levels in sho supposed to be equally spaced? In there m and L both are constants so your bar should be increasing linearly right? It seems a little off
very interesting. I have one simple doubt about the number of eigen states. The number of eigen states is equal to the dimension of the matrix, and the dimension of the matrix depends on the number of points we have chosen.
So, the number of eigen states depends on the method of solutions!?
Really Great Video! Maybe next time you could compare your results with the analytical ones so in this case the qm harmonic oscillator model. I did it and the results are really close! So again thanks for the video
MAKE DIRAC EQUATION
Does it work for a non-linear potential that depends on Psi (my example is: Laplacian of V = - Psi^2) ?
Can I get the link for the Wordpress paper ?
very nice video! I have a lot of fun using python again!
One little question, maybe it is a conventional thing. Then you divide by Δy^2, do you actually mean (Δy)^2 ? (since it is the 2nd (discrete) derivative?
Yes (∆y)^2. Thank you for mentioning this: I will be more careful with my notation in the future :)
how does this extend for the case of a 2 dimensional schrodinger equation?
This is pure gold!
How would one solve for energies if the wavefunction is leaking instead of being zero at 0 and L?
If the wave function was unbounded, you'd have to make the grid very dense, say 100000 points between 0 and 1 and make the potential very localized (maybe between like 0.5 and 0.51). This would be an approximation of a particle that was unbounded, since the infinite boundaries would be very far from the localized potential. (Sort of like how an air molecule is approximately "unbounded" inside a giant warehouse, even though the warehouse has walls)
@@MrPSolver Oh, makes sense. Thank you!
Really enojoy this type of content, a good way in my optinion to learn Python if you know some physics
素晴らしいビデオです！
Great tutorial, thanks...
Life saviour ❤️
This is similar to the problem 2.61 of Griffiths 3rd Edition
Hello. I've got one doubt. The energy output is quadratic. I think that it would be linear, since E = (n + 1/2)hw? Am I wrong?
You are completely right with this. The difference betwen two eigenvalues of the harmonic oscillator has to be constant, so something is not right here. I'm not sure what it is though. Interestingly, the energy output isn't exactly quadratic, since np.sqrt(eigenvalues) also doesn't give a linear result.
@@TueteKaffee Man. I've got the answer. In Numerical Analysis, methods based in finite differences to solve Sturm-Loiuville equations (The Schroedinger equations is a Sturm-Liouville like problem) approximate well the first eigenvalues. For practical purposes, this is what we want: the first energy levels. So, higher energies will not be well approximated.
@@diegoquantico Thanks a lot :)
Can you make your seriers of videos as in a particular playlist? Or anyway to group some of your videos, so that we can play all vidoes in a list ? ^_^
You are awesome too good
What happened to reduced planck constant in the non dimensionalization step?
omg this is awesome thanks a lot!
Hi, it is interesting video. learn a lot from it. but there is a question about the demensionless step. in term of potenial, why directly become mL^2V(y)phi? why not mL^2V(Ly)phi?
I think that it's because V is linear. Then, we get one L from the Potential V and another L from the eigenfinction Psi
Why V(x) = L^2V(y)? What if the V(x) = x^4 or something else...?
Hi, amazing video.
I wold like to ask I tried to model hydrogen atom with potential in a form of 1000/y, the problem of dividing by zero I solved by starting at y=0.0001 and since energy is not 0 at the right end- in infinity I extended the matrix by one element. The wavefunction looks quit similar to the hydrogen atom but I am not able to see the "infinity" in the graph even if I set N to 10000. Any suggestions will be highly appreciated, have a nice day.
I have liked and subscribed. Please show me how to make a chemistry simulator. I want the orbitals to dance
damm looks similar to the euler lagrange equations we studied in aeroelasticity
Wonderful content 🔥
There is one thing that I do not understand. psi(x) is a complex number, but according to this video, it seems to be always real number?
I think psi(x) can be defined as real for the time-independent case.
Sir,can you post the code here
Literally smarter than that guy on your wall
guy on the wall didn't need python
137th like
A very small tip.
The topics you're representing are good and cool and all that. But I didn't learn them this way. Your videos aren't educational if that's what you want them to be. They are more like "look I can do cool stuff, and you know what, just take the code from the description, because I know that you can't do it yourself". Try to focus on the problem and discribe the code slowly. At the end we both KNOW, that these problems aren't a big deal. Peace.
Thanks for the very small tip!
Not true. This videos are top quality educational content, what are you talking about?
You expect him to teach you QM from the ground or what?