Schrodinger equation for three dimention?

[budafeet57]

I have learned time-independent schrodinger 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 dimention?
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 Schrodinger 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-dimentional 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).

 

[paranormal]

The Schrodinger equation for three dimensions is indeed an extension of the time-independent equation you have learned. In three dimensions, we consider the position of a particle in three coordinates: x, y, and z. This means that the wave function, ψ, will also have three variables: ψ(x,y,z). The potential energy, U, will also be a function of all three coordinates: U(x,y,z).

Therefore, the Schrodinger equation for three dimensions is:

Eψ(x,y,z) = - hbar^2 / 2m [ψ''(x,y,z) + ψ''(x,y,z) + ψ''(x,y,z)] + U(x,y,z) ψ(x,y,z)

The wave equation, or the solution to the Schrodinger equation, will now be a three-dimensional function. In your example, the wave function for one dimension is ψ(x) = Asinkx + Bcoskx. In three dimensions, it will look something like ψ(x,y,z) = Asinkx + Bcoskx + Csinky + Dcosky + Esinkz + Fcoskz. This is just an example, as the exact form of the wave function will depend on the specific potential energy and boundary conditions of the system you are studying.

I hope this clarifies the Schrodinger equation for three dimensions. Keep in mind that this is just a basic overview and there are many more complexities and applications of this equation in quantum mechanics. It is a fundamental tool in understanding the behavior of particles at the atomic and subatomic level.