The quantum states available for gas atoms of energy \(\epsilon\) in a cubical box of length \(L\) correspond to integer values for cach \(n_{x}, n_{y}\), and \(n_{z}\), according to Eq. (12.1). In a three-dimensional Euclidean space with coordinates \(n_{x}, n_{y}\), and \(n_{z+}\) each unit volume will contain one quantum state. The total number of quantum states \(\boldsymbol{g}^{\prime}\) with energy less than \(\epsilon^{\prime}\) is equal to the volume of the positive octant of a sphere of radius \(r=L\left(8 m \epsilon_{i}\right)^{1 / 2} / h\) (a) Show that $$ g^{\prime}=\frac{4 \pi V\left(2 m \epsilon^{\prime}\right)^{3 / 2}}{3 h^{3}} $$ (b) In a volume of \(1 \mathrm{~cm}^{3}\) of helium gas at \(300 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure, \(\epsilon^{\prime}\) is about \(10^{-5}\) J. Calculate \(g^{\prime}\). (c) Calculate the number \(N\) of helium atoms. (d) Show that \(g^{\prime} \gg N\).

Quantum mechanics is the branch of physics that deals with the behavior of particles at atomic and subatomic levels. Unlike classical mechanics, it introduces the concept that particles have probabilistic rather than deterministic characteristics. This means particles like electrons, photons, and even atoms can exist in multiple states or locations simultaneously. These states are defined by a wave function, denoted by \(\text{Ψ}\), which provides the probabilities of finding a particle in a particular location or state.
In quantum mechanics, the energy of a particle is quantized, meaning it can only take on specific values. These discrete energy levels align with the solutions to the Schrödinger equation, a fundamental equation in quantum mechanics. When considering gas atoms in a confined space, like a cubical box, quantum mechanics provides a detailed framework for understanding how these particles behave as they occupy distinct quantum states.

The cubical box model is a simple, yet powerful tool in quantum mechanics. It helps describe the behavior of particles confined in a three-dimensional space. Imagine placing a particle, such as a gas atom, inside a cube. The walls of the box impose boundary conditions that restrict the particle's motion.
In this model, the wave functions for the particle are determined by the integer quantum numbers \(n_x, n_y, \text{and} n_z\). Each of these integers represents the number of half-wavelengths of the particle's wave function along each axis of the cube. As a result, these quantum numbers dictate the allowed energy levels of the particle.
This model is particularly useful because it simplifies the complex quantum behavior of particles into something more manageable for calculations. For example, it enables us to determine the number of quantum states available for the gas atoms based on their energy.

In the cubical box model, the energy levels of particles, such as gas atoms, are quantized. This means that the particles can only occupy specific energy states determined by their quantum numbers \(n_x, n_y, \text{and} n_z\). Each set of these numbers corresponds to a unique quantum state with a distinct energy.
The energy \( \epsilon \) of a particle in a cubical box is given by the equation:
\( \epsilon = \frac{h^2 (n_x^2 + n_y^2 + n_z^2)}{8mL^2} \).
Here, \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( L \) is the length of the cubical box. Because the quantum numbers are integers, the energy levels are discrete and not continuous.
Understanding these energy levels is crucial for calculating the total number of quantum states, \( g' \), with energy less than a given value, \( \epsilon' \. \)

To find the total number of quantum states \( g' \) with energy less than a specific value, \( \epsilon' \, \), we use a geometrical approach involving the positive octant of a sphere. In a three-dimensional space defined by the quantum numbers \( n_x, n_y, \text{and} n_z \, \), each unit volume contains one quantum state.
The radius \( r \) of the sphere is related to the energy and given by:
\( r = \frac{L (8 m \epsilon_i )^{1/2} }{h} \).
The volume of the positive octant of this sphere represents the total number of quantum states with energy less than \( \epsilon' \). Calculating the volume of this octant and then integrating gives us:
\( g' = \frac{4 \pi V (2 m \epsilon')^{3/2} }{3 h^3} \).
This formula allows us to quantify the number of quantum states available for gas atoms in the system.
By applying this equation and substituting specific values, such as the volume of the cubical box, mass of the particles, and energy value, we can calculate \( g' \). This approach helps us understand the quantum behavior of the particles confined within the box.