Ch 40: One-Dimensional Quantum Mechanics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 40: One-Dimensional Quantum Mechanics

Problem 27a

Chapter 40, Problem 27a

A 2.0-μm-diameter water droplet is moving with a speed of 1.0 μm/s in a 20-μm-long box. Estimate the particle’s quantum number.

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Step 1: Understand the problem. The quantum number of a particle is related to its energy levels and the quantization of its motion. For this problem, we will use the concept of the de Broglie wavelength and the particle-in-a-box model to estimate the quantum number.

Step 2: Calculate the de Broglie wavelength of the water droplet. The de Broglie wavelength is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum of the particle. The momentum \( p \) can be calculated using \( p = mv \), where \( m \) is the mass of the droplet and \( v \) is its velocity.

Step 3: Determine the mass of the water droplet. The mass \( m \) can be estimated using the formula \( m = \rho \cdot V \), where \( \rho \) is the density of water (approximately \( 1000 \; \text{kg/m}^3 \)) and \( V \) is the volume of the droplet. The volume of a spherical droplet is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the droplet.

Step 4: Relate the de Broglie wavelength to the quantum number. In the particle-in-a-box model, the quantum number \( n \) is related to the length of the box \( L \) and the de Broglie wavelength \( \lambda \) by the equation \( n = \frac{2L}{\lambda} \). Substitute the values of \( L \) (20 μm) and \( \lambda \) into this equation to estimate \( n \).

Step 5: Verify the assumptions and approximations. Ensure that the calculated quantum number is reasonable given the size and speed of the droplet. Discuss how the classical and quantum mechanical descriptions converge for macroscopic objects like water droplets.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quantum Number

In quantum mechanics, a quantum number is a value that quantizes certain properties of particles, such as energy levels, angular momentum, and position. For a particle in a confined space, like the water droplet in the box, the quantum number can help determine its allowed energy states and behavior. The quantum number is often denoted by 'n' and is integral to understanding the particle's wave function and its spatial distribution.

De Broglie Wavelength

The de Broglie wavelength is a fundamental concept that relates a particle's momentum to its wave-like behavior. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. For small particles like the water droplet, this wavelength becomes significant in determining the quantum effects that may be observed, especially in confined spaces.

Particle in a Box Model

The particle in a box model is a simplified quantum mechanical model that describes a particle confined to a finite region of space, where it cannot exist outside the boundaries. This model helps in calculating the energy levels and quantum states of the particle. The quantization of energy levels in this model leads to discrete values for the quantum number, which is essential for estimating the behavior of the water droplet in the given scenario.

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