Schrodinger equation for three dimention?

[budafeet57]

I have learned time-independent Schrödinger equation only from my textbook.
I know Eψ(x) = - hbar^2 / 2m ψ''(x) + Uψ(x)
and ψ(x) = Asinkx + B coskx

what if it's three dimension?
do I do Eψ(x, y, z) = - hbar^2 / 2m ψ''(x, y, z) + Uψ(x, y, z) ?
and what is the wave equation supposed to be?

 

[capandbells]

Close. In three dimensions the time-independent Schrödinger equation for the wavefunction (in Cartesian coordinates) is

-\frac{\hbar^2}{2m}\nabla^2\psi (x,y,z) + U(x,y,z)\psi(x,y,z) = E\psi(x,y,z)

where \nabla^2 is the Laplacian. You could also express this in coordinate systems other than Cartesian (spherical, cylindrical, etc.)

What the solutions are depends on what the potential is and what the boundary conditions are.

 

[budafeet57]

Thanks.
I was doing a problem: An electron is confined within a three-dimensional cubic region the
size of an atom where L = 200 pm.

and I remembered somehow, my teacher gave me these equation

do they work in such condition?

 

[Mark M]

That's the solution to the infinite square well problem in three dimensions. Here is a derivation for the solution in one dimension, you can generalize it to three.

The first equation is the time-independent portion of the wavefunction, and the second line contains the boundary conditions. Since the outside of the box is an infinite potential, no particle may be found there. So, $\psi$ must take a value of zero at the edges (0, and L). The third line is the time-independent SE in three dimensions. Finally, the last line gives you the energy levels, which you should notice are quantized by the principle quantum number (n).