Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 38: Quantization

Problem 23

Chapter 38, Problem 23

The diameter of the nucleus is about 10 fm. A simple model of the nucleus is that protons and neutrons are confined within a one-dimensional box of length 10 fm. What are the first three energy levels, in MeV, for a proton in such a box?

Verified step by step guidance

Step 1: Understand the problem. The nucleus is modeled as a one-dimensional box of length 10 fm (femtometers). The energy levels of a particle confined in such a box can be calculated using the quantum mechanical formula for a particle in a one-dimensional infinite potential well.

Step 2: Write the formula for the energy levels of a particle in a one-dimensional box: \( E_n = \frac{n^2 h^2}{8 m L^2} \), where \( n \) is the quantum number (1, 2, 3,...), \( h \) is Planck's constant, \( m \) is the mass of the particle (proton in this case), and \( L \) is the length of the box.

Step 3: Convert the given length of the box (10 fm) into meters for consistency in SI units: \( 1 \, \text{fm} = 10^{-15} \, \text{m} \), so \( L = 10 \times 10^{-15} \, \text{m} \).

Step 4: Use the mass of the proton \( m_p \approx 1.67 \times 10^{-27} \, \text{kg} \) and Planck's constant \( h \approx 6.626 \times 10^{-34} \, \text{J·s} \). Substitute these values into the formula for \( E_n \).

Step 5: Calculate the energy levels for \( n = 1, 2, \text{and} 3 \) using the formula \( E_n = \frac{n^2 h^2}{8 m L^2} \). Convert the resulting energy values from joules to MeV using the conversion factor \( 1 \, \text{MeV} = 1.602 \times 10^{-13} \, \text{J} \).

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Key Concepts

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

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It introduces the concept of quantization, where certain properties, such as energy, can only take on discrete values. This is crucial for understanding how particles like protons behave in confined spaces, such as a one-dimensional box.

Particle in a Box Model

The particle in a box model is a fundamental concept in quantum mechanics that simplifies the analysis of a particle confined to a specific region of space. In this model, the particle can only occupy certain energy levels, which are determined by the dimensions of the box. For a one-dimensional box, the energy levels are quantized and can be calculated using specific formulas.

Energy Levels Calculation

The energy levels of a particle in a one-dimensional box can be calculated using the formula E_n = n^2 * (h^2 / (8mL^2)), where E_n is the energy of the nth level, h is Planck's constant, m is the mass of the particle, and L is the length of the box. For a proton in a box of length 10 fm, this formula allows us to determine the first three quantized energy levels, which are essential for understanding the behavior of nucleons within the nucleus.

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