Calculate the de Broglie wavelength for each of the...

Calculate the de Broglie wavelength for each of the following.
a. an electron with a velocity $10 . \%$ of the speed of light
b. a tennis ball $(55 \mathrm{~g})$ served at $35 \mathrm{~m} / \mathrm{s}(\sim 80 \mathrm{mi} / \mathrm{h})$

Key Concepts

Quantum versus Classical Behavior

This concept differentiates how quantum effects, such as observable wavelengths from the de Broglie relationship, are significant for microscopic particles like electrons, but become negligible in macroscopic objects such as a tennis ball, reinforcing the boundary between quantum and classical regimes.

Wave-Particle Duality

Wave-particle duality is a core concept in quantum mechanics indicating that particles, like electrons, exhibit both particle-like and wave-like behavior. This duality explains why microscopic particles have measurable wavelengths, while macroscopic objects, with extremely short wavelengths, do not exhibit noticeable quantum effects.

de Broglie Wavelength

This concept states that every particle can be associated with a wavelength, known as the de Broglie wavelength, which is inversely proportional to its momentum. It is described by the formula ? = h/(mv), where h is Planck’s constant, m the mass, and v the velocity of the particle, linking the wave nature with the particle nature.

Momentum

Momentum is a measure of the motion of an object and is given by the product of its mass and velocity. In the context of de Broglie’s hypothesis, a particle’s momentum determines its associated wavelength, implying that lighter and faster particles have longer wavelengths compared to heavier and slower ones.

Planck’s Constant

Planck’s constant is a fundamental constant in quantum mechanics that sets the scale for the quantization of energy and action in nature. It plays a crucial role in linking the particle properties with wave phenomena as seen in the de Broglie wavelength formula.

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