An electron has a de Broglie wavelength equal to the...

Name: Problem 10, Chapter 5: Modern Physics
Uploaded: 2020-09-07T08:01:47-08:00
Duration: 4 min
Channel: Manish Kumar
Description: An electron has a de Broglie wavelength equal to the diameter of the hydrogen atom. What is the kinetic energy of the electron? How does this energy compare with the ground-state energy of the hydrogen atom?

An electron has a de Broglie wavelength equal to the diameter of the hydrogen atom. What is the kinetic energy of the electron? How does this energy compare with the ground-state energy of the hydrogen atom?

Key Concepts

Kinetic Energy Calculation

Once the momentum of a particle is determined from the de Broglie relation, its kinetic energy can be calculated using the classical expression KE = p²/(2m), where m is the mass of the particle. This calculation bridges the quantum description of the particle’s wavelength with its macroscopic kinetic energy, even as quantum effects play a significant role in the behavior of subatomic particles.

Hydrogen Atom Ground State Energy

The hydrogen atom's ground state energy, typically characterized as -13.6 eV in the Bohr model, represents the lowest possible energy state of an electron bound to a proton. Understanding and comparing kinetic energies to this value helps in analyzing how free electron energies relate to those in a bound atomic system, thus providing insights into atomic structure and stability.

De Broglie Wavelength

This concept stems from the idea of wave-particle duality, stating that every particle exhibits wave-like properties. The de Broglie wavelength is inversely proportional to the momentum of the particle, which links its classical and quantum behaviors. Understanding this concept is crucial when analyzing scenarios where particle dimensions and their wave characteristics are comparable.

Momentum-Wavelength Relationship

In quantum mechanics, the core relation is given by p = h/?, where p is the momentum of the particle, h is Planck’s constant, and ? is the wavelength. This relationship is key to linking the observable wave nature of particles to their physical motion, and it forms the basis of many quantum mechanical calculations, such as determining energy levels or tunneling effects.

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