An electron is confined in a three-dimensional cubic space of \(L^{3}\) with infinite potentials. a) Write down the normalized solution of the wave function in the ground state. b) How many energy states are available up to the second excited state from the ground state? (Take the electron spin into account.)

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. In the textbook exercise, we explore a simplified version known as the time-independent Schrödinger equation, which is used to calculate the energy levels of a system that does not change with time.

The equation takes the form:
\[ (-\frac{\hbar^2}{2m}abla^2 + V(r))\psi(r) = E\psi(r) \]
where:\begin{itemize}\item \( \hbar \) represents the reduced Planck's constant, indicating the scale of quantum effects. \item \( m \) is the mass of the particle, such as an electron. \item \( abla^2 \) is the Laplacian operator, describing how the wave function curves in space. \item \( V(r) \) is the potential energy as a function of position. \item \( \psi(r) \) is the quantum wave function. \item \( E \) is the energy of the quantum state.\end{itemize}
In this specific exercise, the equation simplifies due to the infinite potential outside of a 'cubic box,' leaving us with \( -\frac{\hbar^2}{2m}abla^2 \psi(r) = E\psi(r) \). Here, we are dealing with a particle trapped in a three-dimensional box, a classic case in quantum mechanics that illustrates discretized energy levels and particle confinement.

The quantum wave function, denoted as \( \psi \), is a mathematical function that describes the quantum state of a particle or system. In the context of our problem, the wave function represents an electron confined in a cubic box.

For the ground state of a particle in a three-dimensional box, the normalized wave function is:
\[ \psi_{111}(x,y,z) = \sqrt{\frac{8}{L^3}}\sin(\frac{\pi x}{L})\sin(\frac{\pi y}{L})\sin(\frac{\pi z}{L}) \]
The quantum wave function has several key characteristics:

• It contains all the information about the system.
• Its square magnitude \(|\psi|^2\) gives the probability density of finding the particle at a particular location.
• It must be normalized so that the total probability of finding the particle within the entire space is 1.

The exercise guides us through the process of calculating and normalizing the wave function for an electron in a 'quantum box.' This concept cements the probabilistic nature of quantum mechanics, differing from classical mechanics, where objects have definite positions and velocities.

Quantum energy states reflect the discrete energy levels available to a particle confined in a quantum system, such as an electron in a box. These states are a direct consequence of the wave-like nature of particles in quantum mechanics, leading to quantization of physical properties.

The energy of a quantum state in our cubic box problem is given by:
\[ E_{n_x n_y n_z} = \frac{\hbar^2 \pi^2}{2mL^2}(n_x^2 + n_y^2 + n_z^2) \]
This expression shows that the energy levels depend on the quantum numbers (\( n_x, n_y, n_z \)), which represent the states of motion along the respective axis of the box. The ground state has the lowest energy, and higher energy states correspond to higher quantum numbers.

When considering electron spin, each energy level can host two electrons due to the Pauli exclusion principle—one with spin-up and one with spin-down. Therefore, for each energy level derived from the quantum numbers, we can have two distinct states due to spin. In the example, the increment from the ground state to the first and second excited states is demonstrated, revealing the count of available energy states for an electron in a three-dimensional box up to the second excited state, which totals to 13 including spin considerations.