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Chapter 10: Problem 22

What is the de Broglie wavelength? What happens to the de Broglie wavelength of an electron when its speed is increased?

Short Answer

Expert verified
Answer: When the speed of an electron is increased, its de Broglie wavelength decreases.

Step by step solution

01

Introduction to de Broglie Wavelength

The de Broglie wavelength, named after French physicist Louis de Broglie, is a concept in quantum mechanics that assigns a wavelength to any particle with momentum. It helps to describe the wave-particle duality of particles, stating that particles can exhibit both wave-like and particle-like characteristics. In this exercise, we will be focusing on the de Broglie wavelength of an electron.
02

Formula for de Broglie Wavelength

The de Broglie wavelength (λ) of a particle can be calculated using the following formula: \[λ = \frac{h}{p}\] Here, h is the Planck's constant (approximately \(6.63 \times 10^{-34} Js\)), and p is the momentum of the particle. For an electron, we can express the momentum as the product of its mass (m) and speed (v): \[p = m_e \times v\] where \(m_e\) is the mass of the electron (\(9.11 \times 10^{-31} kg\)). Thus, the equation becomes: \[λ = \frac{h}{m_e \times v}\]
03

Relationship between de Broglie Wavelength and Electron Speed

Now, let's analyze the relationship between the electron's de Broglie wavelength and its speed. From the above formula, we can see that the de Broglie wavelength (λ) is inversely proportional to the electron's speed (v): \[λ ∝ \frac{1}{v}\] This means that if the speed of the electron is increased, the de Broglie wavelength will decrease, and vice versa. In other words, when the electron's speed increases, its wave-like behavior becomes less pronounced, and its particle-like nature becomes more dominant.

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