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When a photon is emitted, it means it carries energy with it, and the final energy of the atom would be less than the initial energy. An atom gains energy when a photon is absorbed. So the final quantum number would be higher than the initial number.

The relation between the photon energy and its wavelength is expressed as follows:

$E=\frac{hc}{\mathrm{λ}}$

Here, E is the photon energy, h is the Plank’s constant, c is the light speed in vacuum and$\mathrm{λ}$ is the wavelength.

From the above relation, one can observe that the longest possible wavelength of a photon would correspond to its minimum energy. This happens when the electron jumps from the third excited state n = 4 to the next lower state n = 3 . Therefore, the quantum number to emit light with the longest possible wavelength would be n = 3 .

Thus, the value of the quantum number corresponds to emit light with the longest possible wavelength is n = 3 .

The largest possible energy arises when the electron jumps from the third excited state n = 4 to the ground state n = 1 that emits light with the smallest possible wavelength. Therefore, the quantum number to emit light with the smallest possible wavelength would be n = 1 .

Thus, the value of the quantum number corresponds to emit light with the smallest possible wavelength is n = 1 .

The absorbed photon has the longest possible wavelength when its energy is the smallest, i.e., the electron jumps from the third excited state n = 4 to the next higher state n = 5 . Therefore, the quantum number to absorb light with the longest possible wavelength would be n = 5 .

Thus, the value of the quantum number corresponds to absorb light with the longest possible wavelength is n = 5 .