# Electron confined to a rectangle with walls

1. Nov 3, 2007

### quantum_prince

An electron is confined to a rectangle with infinitely high walls.I need to calculate the ground state energy of the electron.

Can I treat a rectangle with infinite high walls to be the same as 2d box as mentioned here.

http://en.wikipedia.org/wiki/Particle_in_a_box

or it should be treated in some other way.

The problem mentions rectangle and not rectangular box as such, but since high walls are mentioned I thought it should be treated as 2d.

Will this apply for rectangle with high walls?.

For the 2-dimensional case the particle is confined to a rectangular surface of length Lx in the x-direction and Ly in the y-direction.

I think infinite high walls is mentioned only to inform us that the potential is zero inside the rectangle and infinite at the walls. If the rectangle had side lengths in nanometers can we still treat it as a 2-dimension box problem.

Regards.
QP.

2. Nov 3, 2007

### Meir Achuz

It is a 2D box if the walls are infinite.

3. Nov 3, 2007

### quantum_prince

Thanks.

How can I figure this out to compare the wavelength of a photon emitted in a transition from the first excited state to the ground state with the spectrum of visible light in such a box.

4. Nov 3, 2007

### Meir Achuz

The evs are just the sum of two 1D evs.
Then use hf=E12-E11.

5. Nov 4, 2007

### quantum_prince

Hi,

Could you elaborate a bit more.I dont follow.

Regards.
QP

6. Nov 4, 2007

### Meir Achuz

The energy levels are
$$E_{mn}= \frac{h^2}{8M}\left[\left(\frac{m}{L_x}\right)^2 +\left(\frac{n}{L_y}\right)^2\right].$$
Then hf=E12-E11.

7. Nov 5, 2007

### quantum_prince

How did you derive this equation?.Can you post me a link of where you got this from.

Regards.

QP.

8. Nov 5, 2007

### fasterthanjoao

Which kind of lecture course are you being asked to tackle this problem? I would say this is a classic introductory quantum mechanics problem - one of the first I remember looking at anyway. This question is the same as the box where the side-wall potentials aren't infinite and in fact it is actually easier. To derive the formula Meir Achuz has given is very instructive - you will no-doubt be asked to find similar results for various types of potentials and as such I would strongly recommend getting your notes/textbook out and attempting to work through it.

9. Nov 5, 2007

### Meir Achuz

You could just look at your Wikipedia and add the two 1D eigenvalues given there.

10. Nov 11, 2007

### quantum_prince

Hi Meir,

I use the following formulae.

$$\lambda = \frac{hc}{E12-E11}$$

In such a case

I have $$\lambda = \frac {8McL_y^2}{3h}$$

Taking Mass of electron as 9*10^31 kg

c as 3*10^8 m/sec Ly as 400nm and h as 6.634*10^-34, I get

$$\lambda = 0.16 m$$ which is huge where as most of the wavelengths we see are in between the range of 400-800nm.How would this be possible?.

As I understand h in your equation is the same as h what I have in what equation I have with me.You have used Planck's constant instead of dirac constant right?.

Regards.
Q.P

Last edited: Nov 11, 2007