91Ó°ÊÓ

Suppose we have a 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: $\mathbf{0}.\mathbf{3}\text{ }\mathrm{e}\mathbf{V}$(infrared), $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{\text{ }}\mathbf{eV}$(visible), and $\mathbf{2}\mathbf{.}\mathbf{3}\mathbf{\text{ }}\mathbf{eV}$(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. The energy$\mathbf{K}+\mathbf{U}$ of the ground state is $\mathbf{-}\mathbf{4}\mathbf{eV}$. Label the energies of each level ( $\mathbf{K}+\mathbf{U}$, which is negative).

(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 ($\mathbf{K}+\mathbf{U}$, which is negative).

(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).

Step by step solution

01

Identification of the given data

The given data can be listed below as:

• The first emitted energy is ${\mathrm{E}}_1=0.3\text{ e³Õ}$.
• The second emitted energy is ${\mathrm{E}}_2=2.0\text{ e³Õ}$.
• The third emitted energy is ${\mathrm{E}}_3=2.3\text{ e³Õ}$.
• The ground state energy is ${\mathrm{E}}_0=−4\text{ e³Õ}$.

02

Significance of the energy level diagram

The energy level diagram is described as the diagram of the different discrete level of energy. This diagram also has both vibrational and electronic states.

03

(a) Determination of the energy level diagram

The energy level diagram has been drawn below:

In the diagram, it has been identified that the transition occurring from the first or second excited state to the ground state. The energies are $−4\text{ e³Õ}$, $−2\text{ e³Õ}$and$−1.7\text{ e³Õ}$ respectively.

THUS,

04

(b) Determination of the photon energies

As the quantum object has no emission energy at the ground state, then if the laser is being pointed into the system and gradually the energy is increased, then it can be identified that the system will jump from the ground state to the first and the second excited state which are$2\text{ e³Õ}$ and$2.3\text{ e³Õ}$ respectively.

Thus, the photon energies observed by the light beam are$2\text{ e³Õ}$ and $2.3\text{ e³Õ}$respectively.

05

(c) Determination of the transition and labelling the energy

The diagram has been drawn below:

In the above diagram, the difference is from the above diagram to the diagram in part (a) is the excited level’s position. The transition is that the energy falls from the higher excited states to the lower excited states.

THUS,

06

(d) Determination of the photon energies

As the quantum object has no emission energy at the ground state, then if the laser is being pointed into the system and gradually the energy is increased, then it can be identified that the system will jump from the ground state to the first and the second excited state which are$0.3\text{ e³Õ}$ and$2.3\text{ e³Õ}$ respectively.

Thus, the photon energies observed by the light beam are$0.3\text{ e³Õ}$ and $2.3\text{ e³Õ}$respectively.

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