Thursday, March 11, 2010

It's art, OK?

Schrödinger's wanted to express the phase and plane wave (in complex vector form):

\Psi(\mathbf{x},t) = Ae^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}

and to realize that since

\frac{\partial}{\partial t} \Psi = -i\omega \Psi

then

E \Psi = \hbar \omega \Psi = i\hbar\frac{\partial}{\partial t} \Psi

leading to...

\frac{\partial}{\partial x} \Psi = i k_x \Psi

and

\frac{\partial^2}{\partial x^2} \Psi = - k_x^2 \Psi

we find:

p_x^2 \Psi = (\hbar k_x)^2 \Psi = -\hbar^2\frac{\partial^2}{\partial x^2} \Psi

(very important!)

for a plane wave we obtain:

p^2 \Psi = (p_x^2 + p_y^2 + p_z^2) \Psi = -\hbar^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) \Psi = -\hbar^2\nabla^2 \Psi

And, by inserting these expressions for the energy and momentum into the classical formula we started with, we get Schrödinger's equation, for a single particle in the 3-dimensional case in the presence of a potential V:

i\hbar\frac{\partial}{\partial t}\Psi=-\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi


this was sort of semi, totally ripped from wikipedia.... i don't have the little fancy psi, gradient, or h-bar keys handy.
i love quantum physics. love love love love love it. Feynmann once said, "If you're not afraid of quantum mechanics it's because you don't know enough about it."
I say quantum physics is God's loophole. seriously, some weird stuff happens here people. But i love it. it's magic. it's beautiful. and it's AWESOME!
science is art to me. it is love. it is passion. it is reason. it is....
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