Should the Infinite Potential Well Proposal Be Approved?

[45ias]

Infinite potential well "proposal"

Homework Statement

An experimental physicist submits a proposal to a granting agency requesting support to construct an infinite potential well analogous to the one shown in Figure 3.5. Specifically, the proposal is to build a well with L = 1mm, inject some electrons into it, and then measure the wavelengths of photons emitted during low-n transitions via optical spectroscopy. As an expert on quantum mechanics, you are asked to evaluate the proposal. What is your recommendation? (Reject or approve)

Homework Equations

I'm sorry my equations aren't pretty I'll get better at the latex thing I promise!

(1.) Quantized Energy state solution to Schrodinger Equation for an infinite potential well: E=(π²hbar²n²)/(2mL²)

(2.) also E=hc/λ

The Attempt at a Solution

My general thought was that I could use the above solution to the Schrodinger Equation for an infinite potential well of the given dimension, and the other equation to figure out what wavelength of light would be emitted during transitions. If the wavelength was too low or high to be useful in an experiment then the proposal should be rejected. I used the following steps:

1. Used a transition from n=2 to n=1 as an example "low-n transition."

2. Calculated the energies for n=1 and 2 for an electron using equation (1.) simply plugging in the given value of L and the other physical constants.

3. Calculated the difference between the two energies figuring that would be equal to the energy of the photon given off in a transition from n=2 to n=1.

4. Used this energy value in equation (2.) solving for lambda (wavelength of the emitted photon).

I found lambda was extremely large, thus the spectroscope would not detect the photons given off in the transitions and the proposal should be rejected.

I'm really not sure if I'm right... it's more of a guess than anything else, as there were no real criteria or reasons given for why a proposal should be rejected or approved.
Thanks

 

[Simon Bridge]

Welcome to PF;
This is an under-specified assignment, making it rather open ended.
You are taking the correct approach by specifying criteria for an acceptable experiment.

You have chosen that the experiment may be funded if optical spectroscopy may, as claimed by the researcher, be used to detect the transitions of interest.

What is the wavelength range for optical spectroscopy to be valid?

You chose to work in wavelengths rather than energy ranges - either is fine but you'd find energy easier.

The approach seems like what you'd be expected to produce at this level.
You have asserted "wow that's a really long wavelength!" that the lowest transition lies outside the range of optical spectroscopy (but you need to say what that range is) ... but maybe there are sufficient other low-n transitions that it is still worthwhile?

Can you work out how big (in terms of changing n) the smallest detectable transition will be - and how big the largest detectable transitions would be?

Your report would then state this range of transitions which could be usefully investigated by this approach, and comment on what that means.

There are other ways to approach this assignment of course.
You could look at safety concerns or the rate at which to expect transitions (maybe the optical lines are too faint, maybe some initial large-n transitions produce dangerous radiation?) But I'd stick with refining what you are doing to start with.