Review problem. An electron moves in a three-dimensional box...

Review problem. An electron moves in a three-dimensional box of edge length $L$ and volume $L^{3} .$ The wave function of the particle is $\psi=A \sin \left(k_{x} x\right) \sin \left(k_{y} y\right) \sin \left(k_{z} z\right) .$ Show that its energy is given by Equation $43.20,$ $$ E=\frac{\hbar^{2} \pi^{2}}{2 m_{e} L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right) $$ where the quantum numbers $\left(n_{x}, n_{y}, n_{z}\right)$ are integers $\geq 1 .$ Suggestions: The Schrödinger equation in three dimensions may be written $$ \frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\frac{\partial^{2} \psi}{\partial z^{2}}\right)=(U-E) \psi $$ To confine the electron inside the box, take $U=0$ inside and $U=\infty$ outside.

Review problem. An electron moves in a three-dimensional box of edge length $L$ and volume $L^{3} .$ The wave function of the particle is $\psi=A \sin \left(k_{x} x\right) \sin \left(k_{y} y\right) \sin \left(k_{z} z\right) .$ Show that its energy is given by Equation $43.20,$
$$
E=\frac{\hbar^{2} \pi^{2}}{2 m_{e} L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)
$$
where the quantum numbers $\left(n_{x}, n_{y}, n_{z}\right)$ are integers $\geq 1 .$ Suggestions: The Schrödinger equation in three dimensions may be written
$$
\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\frac{\partial^{2} \psi}{\partial z^{2}}\right)=(U-E) \psi
$$
To confine the electron inside the box, take $U=0$ inside and $U=\infty$ outside.

Key Concepts

Energy Quantization

Energy quantization is a key result of the wave-like nature of particles and the imposition of boundary conditions in confined systems. Only specific energy values are allowed, corresponding to the eigenvalues of the Schrödinger equation, and in a three-dimensional box, these energies depend on the sums of the squares of the quantum numbers associated with each spatial dimension.

Quantum Numbers

Quantum numbers are integers that arise from quantizing the solutions of the Schrödinger equation under specific boundary conditions. In the case of the three-dimensional box, the quantum numbers (n_x, n_y, n_z) indicate the number of half-wavelengths that fit along each dimension of the box, and they determine the energy levels of the electron within the box.

Separation of Variables

Separation of variables is a mathematical method used to solve partial differential equations by assuming that the solution can be written as a product of functions, each depending on a single coordinate. This method simplifies the three-dimensional Schrödinger equation into separate one-dimensional ordinary differential equations, each of which is easier to solve.

Schrödinger Equation

The Schrödinger equation is the foundational equation of non-relativistic quantum mechanics, which describes how the quantum state of a physical system changes with time. In the context of stationary states, it provides an eigenvalue problem where the energy levels of the system are found by solving the equation subject to specific potentials and boundary conditions.

Infinite Potential Well

The infinite potential well is a model that confines a particle in a limited region of space by setting the potential energy to zero within a finite region and to infinity outside. This model emphasizes the idea of complete confinement, forcing the particle's wave function to vanish at the boundaries, and leading to discrete, quantized energy levels.

Boundary Conditions

Boundary conditions are constraints that the wave function must satisfy at the edges of the defined region, particularly in problems with infinite potential barriers. They are crucial for ensuring physically acceptable solutions, which in turn leads to the quantization of energy. For a three-dimensional box, these conditions require the wave function to be zero at the walls of the box.

Transcript

Please give Ace some feedback