How Can a Secondary School Student Solve Schrödinger's Equation?

[misogynisticfeminist]

Hi, I'm quite new to quantum physics, and I know that the Schrödinger's equation should be at the quantum physics section but I was wondering if anyone can guide me into the mathematical part of it. I'm only a secondary school student, and can do differentiation and integration. (definite integrals, product/quotient rule etc.) the very basic ones.

A Step-by-step guide on the mathematical aspect to solve Schrödinger's equation would be very very much appreciated thanks...

 

[HallsofIvy]

Schrödinger's equation, in general, is a very difficult non-linear differential equation. Look up the "RKB approximation" in a good quantum mechanics text. It is actually a variation of the "perturbation method" which is covered in advanced d.e. texts.

 

[theFuture]

This whole tex thing isn't working out for me. Do a google search for "infinite square well" and you will come up with the classic first example that most students ecounter with the equation. From there, look at the finite square well and simple harmonic oscillator.

 

[gravenewworld]

shrodinger's equation involves solving partial differential equations. It can only be completely solved for the hydrogen and hydrogen like atoms, because the helium atom for example has 2 electons and the nucleus. It would be like trying to solve the differential equation for a 3 bodied system, which has never been solved. Once you go past the hydrogen atom, approximations are used. The time independent shrodinger equation solution for the hydrogen atom is just a function of sines and cosines.

 

[homology]

Go easy guys

Yeah the Schrödinger equation is hard, but there are some simple 1-D situations where its readily solvable. Like theFuture said, look for the infinite square well which is the easiest case. This is also done in probably all introductory quantum texts. Its also done on most texts that deal with "Modern Physics" which is usually a sophomore level course which introduces some ideas from modern physics.

misogynisticfeminist try solving the Schrödinger equation for yourself, in the case of an infinite well, before you look anywhere. So consider the Schrödinger (time-independent, 1-D):

$$\frac{-\hbar^2}{2m}\frac{d^2 \psi}{d x^2}+V\psi=E\psi$$

Where, for the infinite well, the potential is:

$$V(x)= \left \{ \begin{array}{col}<br /> 0, \ if \ 0\leq x \leq a \\<br /> \infty, \ otherwise<br /> \end{array}\right$$

Thus the Schrödinger equation only makes sense inside of the region [0,a], so it simplifies to

$$\frac{-\hbar^2}{2m}\frac{d^2 \psi}{d x^2}=E\psi$$

or

$$\frac{d^2 \psi}{d x^2}=-{k^2}\psi$$

where $$k\equiv\frac{\sqrt{2mE}}{\hbar}$$

Or to make it just a little more familiar to a beginning calc student

$${\psi^''}=-k^2\psi$$

You have probably even seen this form, play with it, try to find solutions (remember you need a linear combination of 2 independent solutions and the two unknowns are solved by boundary conditions. What do you think the boundary conditions should be?).

Good Luck,

Kevin

 

[theFuture]

Thanks, homology. That's what I was trying to get up but the tex wasn't working out for me. This can also be "extended" to a situation where outside of the well there is a finite potential. Another interesting, and solvable case is the simple harmonic oscilator.

 

[homology]

Sure, I didn't mention the others since misogynisticfeminist expressed their rudimentary calculus skills. The harmonic oscillator involves Hermite polynomials or ladder operators both of which can seem a bit strange or daunting at first. The finite well can also be difficult since it involves breaking up the DE into three parts and matching boundary conditions, considering bound and free states and so on. But you're right of course.

Kevin

 

[quantumdude]

Schrödinger's equation, in general, is a very difficult non-linear differential equation.

What's nonlinear about it? Both superposition and homogeneity are satisfied.

Look up the "RKB approximation" in a good quantum mechanics text.

You mean "WKB", right?

 

[Gokul43201]

misogynisticfeminist,

I think you should probably put off the Schrödinger Equation until after you've learned about differential equations and how to solve them given a set of boundary conditions.

Also, please don't look at things like the WKB or Mean Field approximations just yet. You will learn all about those at college/Grad School.

Finally, I should add that a "purely mathematical" knowledge of different solutions to the SE, while useful for showing off to friends, is otherwise quite wasted. To really appreciate what is happening with the SE, it is important to understand what a wavefunction is (preferably in the context of some previous wave mechanics experience) as well as understand basic electromagnetics so things have a context.

 

[Feynman]

The SCHROD-Equation is:
The following code was used to generate this LaTeX image:



$$H\psi=E\psi$$
It depend on The Hamiltonian operator