Using the concept of standing waves, de Broglie was able to...

Using the concept of standing waves, de Broglie was able to derive Bohr's stationary orbit postulate. He assumed a confined electron could exist only in states where its de Broglie waves form standing-wave patterns, as in Figure $28.6$. Consider a particle confined in a box of length $L$ to be equivalent to a string of length $L$ and fixed at both ends. Apply de Broglie's concept to show that (a) the linear momentum of this particle is quantized with $p=m v=n h / 2 L$ and $(\mathrm{b})$ the allowed states correspond to particle energies of $E_{n}=n^{2} E_{0}$, where $E_{0}=h^{2} /\left(8 m L^{2}\right)$.

Key Concepts

Standing Waves

Standing waves occur when two waves of identical frequency and amplitude travel in opposite directions, interfering in such a way that certain points (nodes) remain fixed. In confined systems, standing wave patterns emerge due to boundary conditions, which is central to understanding how allowed wave modes, and thereby quantized states, arise.

de Broglie Hypothesis

The de Broglie hypothesis posits that particles exhibit wave-like behavior, with a wavelength given by ? = h/p, where h is Planck’s constant and p is the particle’s momentum. This fundamental idea links particle properties to wave phenomena and enables the derivation of quantization conditions in confined systems.

Quantization in Confined Systems

Quantization in confined systems occurs because the boundary conditions force the allowed wavelengths to be discrete multiples of a fundamental unit. This restriction results in a discrete set of allowed momenta and energy levels, reflecting the wave nature of particles and the influence of spatial constraints.

Particle in a Box

The particle in a box model is a foundational quantum mechanics problem where a particle is confined within an impenetrable region, leading to standing wave solutions. The model illustrates that the particle can only occupy specific energy levels and momentum states, as determined by the condition that its de Broglie wavelength forms a standing wave within the box.

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