Question

For the cuhic \(3 \mathrm{D}\) intinite well wave function $$ \forall(x, y, z)=A \sin \frac{n_{x} \pi x}{L} \sin \frac{n_{y} \pi y}{L} \sin \frac{n_{z} \pi z}{L} $$ thow that the correct nornulization constant is \(A=(2 / L)^{1 / 2}\)

Short answer

Following through the normalization integral, the normalization constant \(A\) is found to be \(A = (2 / L)^{1/2}\)

Step-by-step solution

Setup Normalization Integral

The normalization condition is \(\int_{all \, space} |\phi|^2 \, dV = 1\). Hence, \(|\phi|^2 = |A|^2 \sin^2 \left( \frac {n_x \pi x}{L} \right) \sin^2 \left( \frac {n_y \pi y}{L} \right) \sin^2 \left( \frac {n_z \pi z}{L} \right)\). The volume element in Cartesian coordinates is \(dV = dx \, dy \, dz\). Hence, the integral becomes: \(\int_{0}^{L} \int_{0}^{L} \int_{0}^{L} |A|^2 \sin^2 \left( \frac {n_x \pi x}{L} \right) \sin^2 \left( \frac {n_y \pi y}{L} \right) \sin^2 \left( \frac {n_z \pi z}{L} \right) dx \, dy \, dz = 1\).

Solve Independent Integrals

The integral is a product of three identical independent integrals in terms of \(x\), \(y\), and \(z\). So it can be written as \(|A|^2 \left[\int_{0}^{L}\sin^2 \left( \frac {n_x \pi x}{L} \right) dx \right]^3 = 1\). Each integral can be solved individually as \(\int_{0}^{L}\sin^2 \left( \frac {n_x \pi x}{L} \right) dx = \frac {L}{2}\). So the overall integral is \(\frac {|A|^2 L^3}{8} = 1\)

Solve for \(A\)

Rearranging the resulting equation gives \(|A|^2 = \frac {8}{L^3}\). Take square root of both sides to get \(|A| = \sqrt{\frac {8}{L^3}} = 2 / L^{1/2}\). As \(A\) is a constant, the real part should be considered, so the normalization constant \(A = (2 / L)^{1/2}\)

Key concepts

Wave Function

In quantum mechanics, the wave function is a fundamental concept that describes the quantum state of a system. It provides a complete description of the probabilities of a particle's position and momentum. For a 3D infinite potential well, the wave function captures behavior within a confined space between robust barriers where the particle cannot escape.

The wave function is usually denoted by \( \phi(x, y, z) \) and takes the form of mathematical functions such as sine or cosine, which relate to the spatial structure of the system. In our specific example, the wave function is:

• \[\phi(x, y, z) = A \sin \left( \frac{n_x \pi x}{L} \right) \sin \left( \frac{n_y \pi y}{L} \right) \sin \left( \frac{n_z \pi z}{L} \right)\]

This expression represents standing waves confined in each spatial dimension, characterized by quantum numbers \( n_x, n_y, \) and \( n_z \), and a boundary defined by the dimension \( L \). Techniques like separation of variables help in solving these types of equations, creating a product of single-dimensional solutions for such bounded problems.

Understanding the wave function is key because it gives insight into which states are allowed for a system and the likelihood of finding a particle in those states.

Normalization Condition

The normalization condition ensures that the probability of finding a particle in all of space sums to one. This requirement is critical in quantum mechanics, reflecting the probabilistic nature of quantum states.

For a 3D wave function, the normalization integral spreads across the cube from \(0\) to \(L\) in each direction, reflecting the infinite potential well boundaries. The condition is outlined by:

• \[\int_{0}^{L} \int_{0}^{L} \int_{0}^{L} |\phi(x, y, z)|^2 \, dx \, dy \, dz = 1\]

Each of the separate integrals in the \(x\), \(y\), and \(z\) directions can be solved independently due to the product structure of the wave function. The solution of \(\int_{0}^{L} \sin^2 \left( \frac{n \pi x}{L} \right) dx = \frac{L}{2}\) simplifies the evaluation for each dimension.

When calculated collectively, the integration produces the relation \(|A|^2 \left(\frac{L}{2}\right)^3 = 1\), which leads to finding the normalization constant \(A\). This procedure establishes the mathematical rigor needed to analyze physical systems accurately, confined with quantum mechanics rules.

Infinite Potential Well

The infinite potential well is a classic problem in quantum mechanics. It's a model used to describe a particle confined in a space that is impenetrable at its boundaries. This confinement creates a scenario where the only allowed energies are quantized.

In a 3D infinite well, the boundaries are typically set at \(0\) to \(L\) for each dimension \(x, y,\) and \(z\). Within these walls, the potential energy is zero, but it's considered infinite outside these limits, forcing particles to stay within the region and avoiding any probability of being found beyond the well.

The quantized nature means:

• Only specific wave numbers \(\left( \frac{n_x \pi}{L}, \frac{n_y \pi}{L}, \frac{n_z \pi}{L} \right)\) are allowed, reflecting standing wave conditions within the well.
• Each quantized level corresponds to a specific energy, discrete and non-continuous.

This system helps introduce concepts like quantization of energy levels and wave-particle duality, essential for understanding more complex phenomena in quantum mechanics. It's a stepping stone toward realistic quantum models and provides key insights into the microscopic world of particles.