Monday 10 December 2007

More mass = longer

I hope you don't have ambiguous thoughts with this title. But that's the only title I could think about for this topic. And the topic is an excellent piece of math that will draw the crowds [1].

The Schrödinger equation, stripped of constants, looks something like this:

$-\sum_{i=1}^{3N}~{ \frac{1}{2m_i} \frac{\partial^2 \Psi(x_1,...,x_{3N})}{\partial x_i^2}} + V\Psi(x_1,...,x_{3N})=E\Psi(x_1,...,x_{3N})$

Now we switch to mass-weighted coordinates:

$y_i = x_i\sqrt{m_i}$

And we change the wave function to mass-weighted coordinates:

$\Phi(y_1,...,y_{3N}):=\Psi(x_1,...,x_{3N})$

Now we don't need that tedious mass in front of the derivative because:

$-\frac{1}{2}\frac{\partial^2 \Phi(y_1,...,y_{3N})}{\partial y_i^2}= -\frac{1}{2}\frac{\partial^2 \Psi(x_1,...,x_{3N})}{(\partial x_i \sqrt{m_i})^2}=-\frac{1}{2m_i}\frac{\partial^2 \Psi(x_1,...,x_{3N})}{\partial x_i^2}$
(Usually I would complain if someone shows a prove like this. But I think it's easier to see this way. And I also did it the "clean" chain rule way.)

Using this we get Schrödingers equation without the mass:

$-\sum_{i=1}^{3N}~{ \frac{1}{2} \frac{\partial^2 \Phi(y_1,...,y_{3N})}{\partial y_i^2}} + V\Phi(y_1,...,y_{3N})=E\Phi(y_1,...,y_{3N})$

That means: the product of mass1/2 and length is more natural than just length.
It also means: if an elephant walks for one meter it's the same thing as if a mouse walks 1000m (assuming that the squareroot of an elephant's mass is 1000 as much as the squareroot of a mouse's mass).

The idea is from this article. It turns out that the approach is handy if you want to propagate wave packets over a grid you obtained from linear interpolation. And who doesn't want to do that?

[1] Just like my diploma thesis never ceases to catch the interest of everyone I talk to.

Ψ*Ψ said...

If you draw the sigma the right way, it looks like a monster with sharp pointy teeth. At least there is no curly S of death in your post. :)
I swear I liked math when I was younger. But now all I have to do is count to 4 and draw hexagons! It's like being in kindergarten again.

The notion of a quantum elephant is very interesting. Imagine the double slit experiment with such an entity...

db

Noel O'Boyle said...

Indeed, it has earlier been shown that the nature of an elephant depends on how it is measured. :-)

Felix said...

little sigmas and partial d's don't bite people (unless they feel threatened) but of curly S's of death one should stay away :)

I am sure there is a lot of (probably useless) math that can be done with your molecules, for example topologies

-

if a fullerene can go through a double slit, I am sure an elephant can also do that.
and I would like to put an elephant into a 1m cage and see if it behaves the same way as a mouse in a 1km cage (as predicted by quantum theory). if it doesn't, we know there is something wrong with our perception