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Chapter 7: Problem 7

Can the de Broglie wavelength of a particle be known precisely? Can the position of a particle be known precisely?

Short Answer

Expert verified
No, the de Broglie wavelength of a particle cannot be known precisely due to the uncertainty in momentum. Similarly, the position of a particle cannot be known precisely, according to the Heisenberg Uncertainty Principle.

Step by step solution

01

Understand the de Broglie Hypothesis

The de Broglie hypothesis states that every particle has both wave-like and particle-like properties, which leads to the concept of wave-particle duality. According to this hypothesis, the wavelength \(\lambda\) associated with a particle is given by the equation: \(\lambda = \frac{h}{p}\) where \(h\) is Planck's constant, and \(p\) is the particle's momentum.
02

Understand the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the position \(x\) and momentum \(p\) of a particle with complete certainty. Mathematically, this principle is represented as: \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\) where \(\Delta x\) is the uncertainty in position and \(\Delta p\) is the uncertainty in momentum.
03

Analyzing the de Broglie Wavelength

Since the de Broglie wavelength depends on the particle's momentum \(p\), we can say that uncertainty in momentum also introduces uncertainty in the de Broglie wavelength. Therefore, the de Broglie wavelength cannot be known precisely and will always be subject to some level of uncertainty.
04

Analyzing the Position of a Particle

Similarly, based on the Heisenberg Uncertainty Principle, the position of a particle cannot be known precisely due to the trade-off between the knowledge of position and momentum. There will always be some uncertainty in the position measurement.
05

Summarizing the Answers

1. Can the de Broglie wavelength of a particle be known precisely? No, the de Broglie wavelength cannot be known precisely due to the uncertainty in momentum. 2. Can the position of a particle be known precisely? No, the position of a particle cannot be known precisely, according to the Heisenberg Uncertainty Principle.

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