Hydrogen \(\mathrm{H}_{2}\) molecules are kept at \(300.0 \mathrm{K}\) in a cubical container with a side length of \(20.0 \mathrm{cm} .\) Assume that you can treat the molecules as though they were moving in a one-dimensional box. (a) Find the ground state energy of the hydrogen molecule in the container. (b) Assume that the molecule has a thermal energy given by \(k_{\mathrm{B}} T / 2\) and find the corresponding quantum number \(n\) of the quantum state that would correspond to this thermal energy.

In quantum mechanics, the ground state energy refers to the lowest energy level that a quantum system can occupy. This concept is fundamental in understanding the behavior of particles at the microscopic scale.

Take, for example, a hydrogen molecule (\text{H}_2) in a container. At absolute zero temperature, the molecule would be in its ground state, with quantum number n=1. This state is important because it acts as a benchmark for calculating other energy levels and quantum phenomena.

To calculate the ground state energy, one can use the formula: \[E_1 = \frac{n^2 h^2}{8 m L^2}\] where 'n' is the quantum number, 'h' is Planck's constant, 'm' is the mass of the particle, and 'L' is the length of the container. Using Planck's constant \(h = 6.62607004 \times 10^{-34} m^2 kg / s\), a length of \(L = 0.2 m\), and the mass of a hydrogen molecule \(m = 2.825 \times 10^{-27} kg\), we find the ground state energy to be \(E_1 = 37.0 \times 10^{-25} J\).

This energy level is significant because it provides a baseline for understanding the state of a molecule before any external energy, such as thermal energy, is applied.

The quantum number is an integer that determines the energy level of an electron or any other particle within a quantum system. It's a cornerstone in the quantization of energy levels, which is essential to quantum mechanics.

Each energy state of a particle in a quantum system is associated with a specific quantum number, often denoted by 'n'. For example, the ground state of a system corresponds to the quantum number n=1. As the quantum number increases, so does the energy level of the particle.

In the given problem, the quantum number for the hydrogen molecule in the container can be found using the thermal energy at \(300.0 K\) and the respective formula: \[n = \sqrt{\frac{8 k_B T m L^2}{h^2}}\] Substituting in the known values, we calculate that the quantum number n closely approximates 119 when the molecule has a thermal energy given by \(\frac{1}{2} k_B T\). It's crucial to understand that these quantum numbers are discrete and represent the distinct energy levels that a particle can inhabit.

Thermal energy in quantum mechanics is the energy associated with the random motion of particles at a given temperature. According to the principles of quantum mechanics, energy is quantized, and thermal energy contributes to the changes in the quantum states of particles.

The energy due to temperature is directly proportional to the absolute temperature (T) and is represented using the Boltzmann constant (\text{k}_B). We use the relationship \(E = \frac{1}{2} k_B T\) when referring to the average energy per degree of freedom that a particle has due to thermal fluctuations.

In context with the exercise, at a temperature of \(300.0 K\), a hydrogen molecule would have a certain amount of thermal energy, which can be equated to the quantum mechanical energy of the molecule. Understanding this allows us to find the quantum number n that corresponds to this thermal energy, linking the concepts of temperature, quantum mechanics, and the state of a particle.