Show that the allowed energies of a mass M confined in a...

Show that the allowed energies of a mass $M$ confined in a three-dimensional rectangular rigid box with sides $a, b$, and $c$ are $$ E=\frac{\hbar^{2} \pi^{2}}{2 M}\left(\frac{n_{x}^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}+\frac{n_{z}^{2}}{c^{2}}\right) $$ where the three quantum numbers $n_{x}, n_{y}, n_{z}$ are any three positive integers $(1,2,3, \ldots) .[$ Hint: Use separation of variables, and seek a solution of the form $\psi=X(x) Y(y) Z(z)$. Note that by setting $a=b=c$ one obtains the cubical box of Example 8.2.]

Show that the allowed energies of a mass $M$ confined in a three-dimensional rectangular rigid box with sides $a, b$, and $c$ are
$$
E=\frac{\hbar^{2} \pi^{2}}{2 M}\left(\frac{n_{x}^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}+\frac{n_{z}^{2}}{c^{2}}\right)
$$
where the three quantum numbers $n_{x}, n_{y}, n_{z}$ are any three positive integers $(1,2,3, \ldots) .[$ Hint: Use separation of variables, and seek a solution of the form $\psi=X(x) Y(y) Z(z)$. Note that by setting $a=b=c$ one obtains the cubical box of Example 8.2.]

Key Concepts

Schrodinger Equation

The time-independent Schrödinger equation is the foundation of non-relativistic quantum mechanics and describes how the quantum state of a physical system changes spatially. In this scenario, it is applied to a particle confined in a potential-free region inside a box, leading to eigenvalue problems whose solutions provide both the eigenfunctions and the corresponding energy eigenvalues.

Separation of Variables

Separation of variables is a mathematical technique used to simplify partial differential equations by assuming that the solution can be written as a product of functions, each depending on a single coordinate. This method is particularly effective in systems with rectangular symmetry, such as the rigid box, which enables the reduction of the three-dimensional Schrödinger equation into three independent one-dimensional equations.

Boundary Conditions

Boundary conditions are essential in quantum mechanics for ensuring that the wavefunctions satisfy physical constraints imposed by the system. For a particle in a rigid box, the wavefunction must vanish at the boundaries of the box, which leads to the quantization of the allowed wave numbers and thus determines the discrete energy levels.

Quantization of Energy

The quantization of energy arises from the imposition of the boundary conditions on the solutions of the Schrödinger equation. Because the confined particle’s wavefunction must have nodes at the walls of the box, only specific wavelengths and corresponding wave numbers are permitted. This restriction leads directly to discrete, or quantized, energy levels that depend on integer quantum numbers associated with each spatial dimension.

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