What Are the Energy Levels of a Quantum Object with Three Bound States?

[clutch12]

Homework Statement

Suppose we have reason to suspect that a certain quantum object has only three quantum states. When we excite a collection of such objects we observe that they emit electromagnetic radiation of three different energies: 0.2 eV (infrared), 2.1 eV (visible), and 2.3 eV (visible).
(a) Draw a possible energy-level diagram for one of the quantum objects, which has three bound states. On the diagram, indicate the transitions corresponding to the emitted photons, and check that the possible transitions produce the observed photons and no others. When you are sure that your energy-level diagram is consistent with the observed photon energies, enter the energies of each level (K+U, which is negative). Enter ALL levels before submitting; all of the energies must be correct to be properly scored. The energy K+U of the ground state is -5 eV.

(b) The material is now cooled down to a very low temperature, and the photon detector stops detecting photon emissions. Next a beam of light with a continuous range of energies from infrared through ultraviolet shines on the material, and the photon detector observes the beam of light after it passes through the material. What photon energies in this beam of light are observed to be significantly reduced in intensity ("dark absorption lines")?

Energy of highest-energy dark line: eV
Energy of lowest-energy dark line: eV

(c) There exists another possible set of energy levels for these objects which produces the same photon emission spectrum. On an alternative energy-level diagram, different from the one you drew in part (a), indicate the transitions corresponding to the emitted photons, and check that the possible transitions produce the observed photons and no others. When you are sure that your alternative energy-level diagram is consistent with the observed photon energies, enter the energies of each level (K+U, which is negative). Enter ALL levels before submitting; all of the energies must be correct to be properly scored.

eV = energy of highest level (2nd excited state)
eV = energy of next highest level (1st excited state)
-5 eV = energy of ground state

(d) For your second proposed energy-level scheme, what photon energies would be observed to be significantly reduced in intensity in an absorption experiment ("dark absorption lines")? (Given the differences from part (b), you can see that an absorption measurement can be used to tell which of your two energy-level schemes is correct.)

Energy of highest-energy dark line: eV
Energy of lowest-energy dark line: eV

Homework Equations

I don't think i need equations.

The Attempt at a Solution

for part a, I was able to solve it by 2.3 from 2.1 and then subtracting 5 from 2.7
so my answers for a were -0.2 and -2.3

for b to d, i have no clue what I am doing wrong. I've been subtracting numbers and have been getting the wrong answers. Anyone have an idea why this is happening?

 

[mfb]

for part a, I was able to solve it by 2.3 from 2.1 and then subtracting 5 from 2.7

Why? 2.7 eV isn't even in the problem statement.

An energy level at -0.2 eV would produce a transition going from -0.2 eV to -5 eV, emitting and absorbing photons with 4.8 eV.

A three-level system only has three transitions, we have three transition energies so they must all occur. The highest energy difference (2.3 eV) will be between the ground state at -5 eV and the highest energy level, so this energy level must be at -5 V - (-2.3 eV) = -2.7 eV. The other one must be in between, which leads to two options: 0.2 eV above the ground state (at -4.8 eV) or 0.2 below the highest state (at -2.9 eV). Each position automatically puts it 2.1 eV away from the other state, as needed for the third transition. Just with the given information we can't distinguish between these cases, so one option is used for (a) and (b) and the other is used for (c) and (d).