Electron confined to a rectangle with walls

[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.

 

[Meir Achuz]

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

 

[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.

 

[Meir Achuz]

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

 

[quantum_prince]

Hi,

Could you elaborate a bit more.I don't follow.

Regards.
QP

 

[Meir Achuz]

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

 

[quantum_prince]

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

Regards.

QP.

 

[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.

 

[Meir Achuz]

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

Regards.

QP.

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

 

[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