Question about an electron confined in an infinite well

[lilfinger]

Homework Statement

An electron confined in an infinite well (1 dimensional) can absorb a photon with a maximum wavelength of 1520 nm, what is the length of the well?

Homework Equations

λ=2L/n
E = hf (photon)
E = n^2*h^2/8*m*L^2

The Attempt at a Solution

I honestly don't know what to start with, I solved for the energy of the 1520 nm photon, but I'm not sure what that tells me. Also the fact that this is the maximum wavelength that the electron can absorb is supposed to tell me something which I'm not entirely sure what. I looked around and read my textbook many times but I'm still not sure. Any hint would be great, thanks.

EDIT: I should add, this is a first year physics course intro to quantum mechanics chapters

 

[fzero]

Suppose the electron is in an initial state with energy $E_n$. For the electron to absorb a photon, there must be another state with energy $E_m>E_n$, such that $E_m-En$ is equal to the energy of the photon. This is conservation of energy combined with the quantization of the energy levels of the electron in the well. Use the bound on the wavelength of the photon to construct a bound on the energy of the photon. Is this an upper or lower bound? Does the type of bound suggest anything about which energy eigenstates must be involved in the transition?

 

[lilfinger]

Right, that makes sense. I would think this is an upper bound because it says "maximum" wavelength. In terms of which energy states are involved, can I assume I'm starting at the ground energy state?

 

[Dick]

Right, that makes sense. I would think this is an upper bound because it says "maximum" wavelength. In terms of which energy states are involved, can I assume I'm starting at the ground energy state?

Well, "maximum" wavelength would correspond to minimum energy of the photon. So yes, you want to find two energy states with a minimum of separation.

 

[lilfinger]

Oh, I see, yes that makes perfect sense, since the photon emitted in the transition from n=2 to n=1 has the highest energy, right?
I thought of this, but rejected this idea because I thought why does it need to go between two consecutive energy states; so what in the wording of the question, or information give, suggests that the energy states involved are consecutive? For example, couldn't the electron absorb a photon and go from n=1 to n=3 (or some other arbitrary levels)?

 

[Dick]

Oh, I see, yes that makes perfect sense, since the photon emitted in the transition from n=2 to n=1 has the highest energy, right?
I thought of this, but rejected this idea because I thought why does it need to go between two consecutive energy states; so what in the wording of the question, or information give, suggests that the energy states involved are consecutive? For example, couldn't the electron absorb a photon and go from n=1 to n=3 (or some other arbitrary levels)?

Why do you think the transition from n=2 to n=1 would have the highest energy? This is an infinite square well potential, it's not hydrogen. Look at your equation for the energy.

 

[lilfinger]

Oh.. that's right. Sorry I'm still a bit rough on quantum. I'm not really sure then what to do here.

 

[Dick]

Oh.. that's right. Sorry I'm still a bit rough on quantum. I'm not really sure then what to do here.

That's ok. So think about it. The largest wavelength you can absorb will correspond to the SMALLEST energy gap. What's that?

 

[lilfinger]

From what I see in my textbook, the smallest energy gap is in between the first and second energy states (n=1 to n=2) and this is for quantized energies for a particle in a box.

 

[Dick]

From what I see in my textbook, the smallest energy gap is in between the first and second energy states (n=1 to n=2) and this is for quantized energies for a particle in a box.

Yes, exactly. The energy levels go like n^2=1,4,9,16,25... The smallest gap is between the first two.

 

[lilfinger]

I see, then to solve for the length of the well I can set the energy of the photon equal to the energy at n=2 minus the energy at n=1 and solve for L. I really appreciate your help, thank you!