Answer:
(b) a hydrogen atom
Explanation:
Wavelength is inversely proportional to energy. So we have to find the system for which the second photon has smaller energy than the first one.
For the harmonic oscillator, the energy level spacing remains the same as the quantum number increases.
For the hydrogen atom, the energy level spacing decreases as the quantum number increases.
For the particle in a box, the energy level spacing increases as the quantum number increases.
We have a quantum-mechanical system where the energy levels are not evenly spaced and the energy between consecutive levels decreases as the level increases, the correct option is B: the hydrogen atom.
So we have a quantum-mechanical system that increases its energy level when it absorbs a photon, there are two increases, from the ground state to first existed and from first exited to second exited.
Such that the second absorbed photon has less energy than the first one (because it has larger wavelength).
From this, we conclude that the energy levels are not evenly spaced.
We know that the energy levels of the harmonic oscillator are:
[tex]E_n = \omega*\hbar*(n + 1/2)[/tex]
So the levels are evenly spaced, we can discard this option.
The hydrogen atom levels are proportional to 1/n^2
So, as n increases, the energy difference between consecutive levels decreases, so to increase the level we need less energy, thus the hydrogen atom can be the described quantum-mechanical system.
Finally, for the particle in a box, we have the energy levels proportional to n^2
So as we increase the value of n, the difference between consecutive energy levels increases.
This means that the photon needed to go from the first to the second excited level must be more energetic than the first photon, so we can also discard this option.
Finally, we conclude that the correct option is B, the hydrogen atom.
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