Solving Schrödinger Equation: Step-by-Step Tutorial

[Superposed_Cat]

Hi all, I am a 10th grade student who has a basic grasp on matrices, differential calculus and can integrate. I am having trouble with the Schrödinger equation (which part? where to start and just all of it)I was wondering where I could find a detailed stepwise walkthrough to the Schrödinger equation. It would be nice to have one in both one and 2 dimensions. Thanks for ANY help that I receive.

I also don't get the iH part what are you supposed to do there? and by the way I live in a place where it's virtually impossible to get an advanced physics textbook.

 

[dauto]

Do you have any experience with wave equations? Or any partial differential equations? What about complex numbers?

Also, there is more than one way to write the Schrödinger equation so it would be helpful if you let us know the exact equation you're talking about.

 

[jtbell]

The first example of solving the Schrödinger equation, that just about everybody does, is the "infinite square well" a.k.a. "particle in a box." Have you seen that yet?

A Google search for "infinite square well" turns up lots of lecture notes, many of which unfortunately seem to assume you have a book handy, or are supplements to an in-class lecture. This page, however, seems to be mostly self-contained, and does it pretty much the way I used to do it in a second-year intro modern physics course:

http://physicspages.com/2011/01/26/the-infinite-square-well-particle-in-a-box/

It does make reference to another page on the time-independent SE:

http://physicspages.com/2011/01/22/the-time-independent-Schrödinger-equation/

 

[Astrum]

Hi all, I am a 10th grade student who has a basic grasp on matrices, differential calculus and can integrate. I am having trouble with the Schrödinger equation (which part? where to start and just all of it)I was wondering where I could find a detailed stepwise walkthrough to the Schrödinger equation. It would be nice to have one in both one and 2 dimensions. Thanks for ANY help that I receive.

I suspect that you don't have any real experience with physics, and are trying to start off with QM. You can't fully appreciate QM without having studied (bare-minimum) classical mechanics. It's very much like trying to watch a movie in a foreign language without even knowing the alphabet.

If you're really dead-set on trying to learn QM, you could try a book like Griffiths' "Introduction to Quantum Mechanics", but I don't think anybody here would advise this, especially since you haven't told us about your physics background.

I also don't get the iH part what are you supposed to do there?

$\hat{H} = -\frac{\hbar ^2}{2m}\nabla ^2 +V$ is the Hamiltonian operator, and $i$ is the imaginary number.

 

[Superposed_Cat]

the only thing I have experience with is calculus, any website that teaches the higher math in a format that's easy to understand? Plus what's the inverted triangle?

 

[jtbell]

The "inverted triangle" is the nabla operator. In Schrödinger's equation, you see its square:
$$\nabla^2 \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2}$$

Have you studied partial derivatives yet? (e.g. $\partial \psi / \partial x$)

In the USA, people usually learn these in the third semester of an undergraduate calculus course. It often goes by the name "vector calculus." Even introductory QM books and web pages assume that you have studied that much calculus, and are at least acquainted with the basic ideas of differential equations.

 

[Superposed_Cat]

I can do differential equations, I have not done pde's yet though...any links? sorry to bother.

 

[jtbell]

You really don't need to know anything special about PDEs in order to start learning about solving Schrödinger's equation. If you know what partial derivatives are, you'll be OK.

In the standard examples that all introductory QM textbooks use, the first thing they do is use the "separation of variables" technique to convert the partial differential equation into two or more ordinary differential equations. The second page that I linked to above describes that.

 

[Superposed_Cat]

If I can solve Schrödinger eqn can I then solve any physics equation? aside from ones in general relativity.

 

[craigi]

If I can solve Schrödinger eqn can I then solve any physics equation? aside from ones in general relativity.

No.

It's just one equation. There are inumerous more to solve too.

Some equations even have no analytical solution.

 

[Superposed_Cat]

No.



Some equations even have no analytical solution.

Like what?

 

[craigi]

Like what?

http://en.wikipedia.org/wiki/Three-body_problem

 

[jtbell]

Even Schrödinger's equation has analytical solutions (ones that you can write down formulas for) for only a few highly symmetric situations. In general, you have to use numerical methods to generate the solution point by point and e.g. make a graph of it.

 

[Astrum]

If I can solve Schrödinger eqn can I then solve any physics equation? aside from ones in general relativity.

I don't mean to be rude, but what are you talking about? I seriously can't see a point in studying QM without knowing any physics beforehand. Asking a question like this is ridiculous (and also extremely vague). The answer is a big "NO". As others have said, most ODEs,PDEs can't be solved analytically, and Schrödinger's equation is not "One equation to rule them all"

We have countless people coming on the forums, both kids and adults, asking to "learn" quantum mechanics when they haven't studied any prior physics/math. There's a reason QM is not taught (generally) as a first year course. Physics is way more than mathematics. Just because you have experience with calculus (and I suspect little experience, too) doesn't mean you understand the physics!

I appreciate that you are eager to study physics, but please DO IT THE RIGHT WAY. A great first textbook is An Introduction to Mechanics. The math is simple, but the physics is fairly advanced for a first year text. It doesn't cover Lagrangian mechanics, but you can use another book for that.

 

[Superposed_Cat]

True astrum, but what I meant was assuming you learned math in the same sequence taught by schools up until integral calculus could you solve the vast majority of physics equations aside form topological ones or ones that require tensors? i.e. does it get harder than pde's and tensors?

Introduction to Mechanics. The math is simple, but the physics is fairly advanced for a first year text. It doesn't cover Lagrangian mechanics, but you can use another book for that.

I really don't have money for textbooks, or in general. Hence me asking for links.

 

[Astrum]

I really don't have money for textbooks, or in general. Hence me asking for links.

Yeah, the answer is still no. Linear Algebra is a very important field that is used all the time in physics, you'll encounter a lot in QM (although not too much in Griffiths QM).

I'm sure you can use "other means" for getting textbooks. Linking illegal content will get your post removed, but I'm sure you're resourceful enough to do it on your own.

 

[king vitamin]

I really don't have money for textbooks, or in general. Hence me asking for links.

Many professors and universities post excellent lectures and lecture notes online, and being good at googling for these is also a good method for learning. A good way to filter for these is to put "site:.edu" in with your google search, automatically returning results from an .edu domain so that you're more likely to return lecture notes.

My undergrad quantum mechanics professor posted all of his course books for free online:
http://farside.ph.utexas.edu/teaching.html

 

[Superposed_Cat]

Hey, I can do it now thanks to YouTube! :) Thanks for all your help

 

[meBigGuy]

This site gives an enlightning (for me, anyway) view of the math required for physics.
http://superstringtheory.com/math/ Don't forget to go to pages II and III.

Susskind's lecture on entanglement at theortical minimum.com is simplified mathematically and may give you interesting starting points for further learning. The site also has many other courses.
http://theoreticalminimum.com/courses/quantum-entanglement/2006/fall