A particle is confined to a finite box of length L. In the n...

A particle is confined to a finite box of length $L$. In the $n$ th state, the wave function has $n-1$ nodes. The wave function must make a smooth transition from sinusoidal inside the box to a decaying exponential outside- there can't be a kink at the wall. (a) Make some sketches to show that the wavelength $\lambda_{n}$ inside the box must fall in the range $2 L / n<\lambda_{n}<2 L /(n-1) .$ (b) Show that the energy levels $E_{n}$ in the finite box satisfy $(n-1)^{2} E_{1}<E_{n}<n^{2} E_{1}$, where $E_{1}=h^{2} /\left(8 m L^{2}\right)$ is the ground-state energy for an infinite box of length $L$

Key Concepts

Quantum Confinement

This concept refers to the restriction of a particle’s motion to a limited region of space, which leads to discrete (quantized) energy levels. In confined systems such as particles in a box, the allowed energies are determined by the spatial constraints that the particle’s wave function must satisfy.

Finite Potential Well

In a finite potential well, the potential energy outside the well is finite rather than infinite. This results in the wave function decaying exponentially outside the well instead of being exactly zero. The gradual transition enforces additional continuity conditions compared to the infinite well, affecting the energy levels and the shape of the wave function.

Boundary Conditions for the Schrödinger Equation

For physically acceptable solutions of the Schrödinger equation, both the wave function and its derivative must be continuous. These boundary conditions prevent abrupt changes or kinks at the interfaces between different potential regions, ensuring a smooth transition of the wave function from one region (e.g., sinusoidal inside) to another (e.g., decaying exponential outside).

Wave Function Nodes

Nodes are points within the region of confinement where the wave function passes through zero. The number of nodes increases with the energy state of the particle, and each additional node corresponds to a higher quantum number. This nodal structure is directly related to the allowed wavelengths and consequently the energy levels of the system.

Energy Quantization and Wavelength Constraints

The quantization of energy in confined systems arises because only certain wavelengths (and hence energies) can satisfy the spatial boundary conditions. The relationship between the wavelength inside the box and the box length imposes limits (inequalities) that ensure the sinusoidal wave fits within the well without discontinuities, which in turn leads to bounds on the energy levels relative to a reference system like the infinite potential well.

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