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 $E_{0}=h^{2} /\left(8 m L^{2}\right)$.

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

Quantization via Boundary Conditions

Quantization via boundary conditions refers to the restriction of a system’s allowed states due to spatial constraints. When a wave, such as a de Broglie wave, is confined within a region with fixed boundaries, only specific wavelengths that satisfy the standing wave conditions are allowed, leading to discrete (quantized) values of momentum and energy.

Energy Quantization in Confined Systems

Energy quantization in confined systems arises because only certain discrete wave modes can exist when a particle is confined within a finite region. Each allowed mode corresponds to a specific energy level, and this discretization is a direct consequence of imposing boundary conditions on the wavefunction in quantum systems.

Standing Waves

Standing waves are wave patterns that remain stationary in space, characterized by fixed nodes and antinodes. They occur when two waves of the same frequency and amplitude travel in opposite directions and superpose, satisfying specific boundary conditions that lead to discrete wavelengths allowed in the system.

de Broglie Hypothesis

The de Broglie hypothesis posits that particles exhibit wave-like behavior, associating a wavelength with every moving particle. This concept is fundamental in quantum mechanics as it bridges the gap between classical particles and quantum waves, providing a basis for understanding phenomena such as quantization in confined systems.

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