∙(a) An electron moves with a speed of 4.70 ×10^6 m / s . What is its de Broglie wavelength? (b)

step 1 The DeBrobley relation says that lambda, the wavelength of a particle is equal to Plank's consta

∙(a) An electron moves with a speed of 4.70 ×10^6 m / s ....

$\bullet$ (a) An electron moves with a speed of $4.70 \times 10^{6} \mathrm{m} / \mathrm{s}$ . What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

Key Concepts

de Broglie Wavelength

This is a fundamental concept in quantum mechanics that describes the wave-like behavior of matter. It is given by the equation ? = h/p, where ? is the wavelength, h is Planck's constant, and p is the momentum of the particle. The de Broglie wavelength demonstrates that as particles like electrons and protons move, they can be associated with a wavelength inversely proportional to their momentum, emphasizing the dual nature of matter.

Momentum

Momentum, defined as the product of mass and velocity (p = mv), is a key parameter in determining the de Broglie wavelength. Since the wavelength is inversely proportional to momentum, particles with a higher mass moving at the same velocity will have smaller de Broglie wavelengths compared to lighter particles. This relationship underpins how differences in mass affect the quantum behavior of particles.

Planck's Constant

Planck's constant (h) is a fundamental physical constant that plays a critical role in quantum mechanics. It appears in the de Broglie wavelength formula as the proportionality constant linking a particle's momentum to its wavelength, emphasizing the quantum nature of matter and the quantization of energy.

Wave-Particle Duality

Wave-particle duality is a core principle of quantum physics, stating that particles such as electrons and protons exhibit both classical particle properties and wave-like behavior. This duality is essential for understanding phenomena at the quantum scale and is exemplified by the de Broglie wavelength, which provides a link between a particle's motion and its associated wave-like properties.

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