Find the wavelengths emitted in all allowed transitions between the first three rotational states in the \(n=1\) level to any states in the \(n=0\) level in \(\mathrm{H}_{2},\) whose rotational inertia and classical vibration frequency are \(4.60 \times 10^{-48} \mathrm{kg} \cdot \mathrm{m}^{2}\) and \(3.69 \times 10^{14} \mathrm{Hz}\), re- spectively.

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It introduces the concept that energy is quantized and cannot be any arbitrary value. In quantum mechanics, particles are described not as single points in space but as waves and they can exist in multiple states simultaneously. This dual nature of matter and energy is one of the key features that sets quantum mechanics apart from classical mechanics.

Rotational motion, which is relevant for our textbook problem, is a clear example of quantum mechanics at work. In such systems, rotational energy is also quantized, leading to discrete energy levels that an object, such as a molecule, can occupy.

In quantum mechanics, rotational energy levels refer to the discrete energy states that a rotating system, such as a molecule, can occupy. Each of these energy levels corresponds to a different quantum state, characterized by a quantum number. For linear molecules, the energy of a rotational state can be represented by the formula \(E = \frac{h^2n(n+1)}{8\pi^2I}\), where \(n\) is the rotational quantum number (which can take values starting from 0), \(I\) is the moment of inertia, and \(h\) is Planck's constant.

The quantization of rotational energy means that the molecule can only rotate with specific angular velocities, leading to unique spectral lines when the molecule transitions between these energy levels.

Planck's constant (represented by the symbol \(h\)) is a fundamental constant in quantum mechanics that has the value of approximately \(6.62607015 \times 10^{-34} \text{J}\cdot\text{s}\). It relates the energy of a photon to its frequency, with the formula \(E = hv\), where \(E\) is the energy and \(v\) is the frequency of the photon. Planck's constant essentially sets the scale for the quantization in quantum mechanics, serving as the proportionality constant between the energy and frequency of photons. It is this constant that determines the energy levels for all quantum systems, including the rotational states of molecules.

Photon emission occurs when an electron or molecule transitions from a higher to a lower energy state, releasing a photon in the process. The energy of the emitted photon corresponds to the difference in energy between the two states. In our textbook problem, the molecule hydrogen (\(\mathrm{H}_{2}\)) undergoes transitions between rotational states, each change corresponding to the emission or absorption of a photon. The frequency, and thus the energy of the photon, is directly related to the change in rotational energy as described by Planck's formula \(E = hv\). Photon emission is a cornerstone of spectroscopy, as it is responsible for the creation of the spectral lines we observe.

Spectral lines are unique identifiers of the chemical composition and physical conditions of a light-emitting source. They are calculated by finding the wavelengths of light that correspond to the energy transitions between quantized energy levels of molecules or atoms. Following the steps outlined in our textbook solution, we begin by calculating the energy levels of the system, then finding the energy difference between these levels, and finally converting this energy to a wavelength using the formula \(\lambda = \frac{c}{v}\), where \(c\) is the speed of light and \(v\) is the frequency of the light. By applying this process to all the allowed transitions, we can create a complete picture of the rotational spectrum of \(\mathrm{H}_{2}\), helpful for understanding molecular structure and dynamics.