The wave nature of particles results in the quantum-...

The wave nature of particles results in the quantum- mechanical situation that a particle confined in a box can assume only wavelengths that result in standing waves in the box, with nodes at the box walls, (a) Show that an electron confined in a one- dimensional box of length $L$ will have energy levels given by $$E_{n}=\frac{n^{2} h^{2}}{8 m L^{2}}$$ (Hint: Recall that the relationship between the de Broglic wave- length and the speed of a nonrelativistic particle is $m v=h / \lambda$ . The energy of the particle is $\frac{1}{2} m v^{2}, )(b)$ If a hydrogen atom is modeled as a one-dimensional box with length equal to the Bohr radius, what is the energy (in electron volts) of the lowest energy level of the electron?

The wave nature of particles results in the quantum- mechanical situation that a particle confined in a box can assume only wavelengths that result in standing waves in the box, with nodes at the box walls, (a) Show that an electron confined in a one- dimensional box of length $L$ will have energy levels given by
$$E_{n}=\frac{n^{2} h^{2}}{8 m L^{2}}$$
(Hint: Recall that the relationship between the de Broglic wave- length and the speed of a nonrelativistic particle is $m v=h / \lambda$ . The energy of the particle is $\frac{1}{2} m v^{2}, )(b)$ If a hydrogen atom is modeled as a one-dimensional box with length equal to the Bohr radius, what is the energy (in electron volts) of the lowest energy level of the electron?

Key Concepts

Energy Quantization

Due to the wave nature of particles and the constraints imposed by the boundary conditions in a confined space, only certain wavelengths (and hence momenta) are allowed. Since the energy of a particle is related to its momentum, this results in discrete, quantized energy states. This quantization is a direct outcome of the requirement that the particle’s wavefunction forms a standing wave within the confinement, leading to a relationship between the energy levels and the size of the confinement.

de Broglie Hypothesis

The de Broglie hypothesis posits that particles exhibit wave-like properties, with a wavelength inversely proportional to their momentum. This wave-particle duality is a cornerstone of quantum mechanics and is instrumental in linking the energy, momentum, and confinement of particles. In the context of a particle in a box, it provides the foundation for relating the wavelength of the standing wave inside the box to the quantized energy levels.

Particle in a Box

This concept describes a quantum mechanical model where a particle is confined within an impenetrable region, typically represented by infinitely high potential walls. In this model, the particle may only exist within a finite space, and its wavefunction is required to vanish at the boundaries. This restriction leads to quantized solutions for the particle’s energy and momentum, reflecting the fundamental differences between classical and quantum systems.

Standing Waves and Boundary Conditions

In a confined system, the wavefunction of a particle must satisfy specific boundary conditions, typically that it vanishes at the walls of the box. This requirement leads to the formation of standing waves, where only certain wavelengths are allowed. The nodes (points of zero amplitude) appear at the boundaries, and only waves that fit an integer number of half-wavelengths into the box are permitted, directly resulting in a set of discrete energy states.

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