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Ch 40: One-Dimensional Quantum Mechanics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 40: One-Dimensional Quantum MechanicsProblem 32a
Chapter 40, Problem 32a

A particle confined in a rigid one-dimensional box of length 10 fm has an energy level En = 32.9 MeV and an adjacent energy level En+1 = 51.4 MeV. Determine the values of n and n+1.

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Step 1: Recall the formula for the energy levels of a particle in a one-dimensional box: Eₙ = (n²h²)/(8mL²), where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.
Step 2: Write the formula for the difference between adjacent energy levels: ΔE = Eₙ+₁ - Eₙ. Substitute the given values for Eₙ = 32.9 \, \(\text{MeV}\) and Eₙ+₁ = 51.4 \, \(\text{MeV}\) to find ΔE = 18.5 \, \(\text{MeV}\).
Step 3: Express the difference in energy levels using the formula: ΔE = \(\frac{h²}{8mL²}\)((n+1)² - n²). Expand the term (n+1)² to get n² + 2n + 1, and simplify the expression to ΔE = \(\frac{h²}{8mL²}\)(2n + 1).
Step 4: Rearrange the formula to solve for n: n = \(\frac{ΔE \cdot 8mL²}{2h²}\) - \(\frac{1}{2}\). Substitute the known values for ΔE = 18.5 \, \(\text{MeV}\), L = 10 \, \(\text{fm}\), and the constants h and m (Planck's constant and the mass of the particle, typically the mass of an electron or proton depending on the context).
Step 5: Once n is determined, calculate n+1 by simply adding 1 to the value of n. Verify that the calculated energy levels match the given values of Eₙ and Eₙ+₁.

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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 concepts such as quantization of energy levels, where particles can only occupy specific energy states. This framework is essential for understanding phenomena like the energy levels of a particle in a potential well, such as a one-dimensional box.
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Particle in a Box Model

The particle in a box model is a fundamental quantum mechanics problem that describes a particle confined to a one-dimensional region with infinitely high potential walls. The energy levels of the particle are quantized and can be calculated using the formula Eₙ = n²h²/(8mL²), where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box. This model helps in determining the allowed energy states of the particle.
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Energy Level Calculation

In quantum mechanics, energy levels are calculated based on the quantum number associated with the state of the particle. For a particle in a box, the energy levels are proportional to the square of the quantum number (n²). By knowing the energy of adjacent levels, one can derive the quantum numbers n and n+1, which correspond to the specific energy states of the particle in the box.
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Related Practice
Textbook Question

INT Model an atom as an electron in a rigid box of length 0.100 nm, roughly twice the Bohr radius. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label λn→m to indicate the transition.

Textbook Question

A particle confined in a rigid one-dimensional box of length 10 fm has an energy level En = 32.9 MeV and an adjacent energy level En+1 = 51.4 MeV. Draw an energy-level diagram showing all energy levels from 1 through n+1. Label each level and write the energy beside it.

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Textbook Question

A particle confined in a rigid one-dimensional box of length 10 fm has an energy level En = 32.9 MeV and an adjacent energy level En+1 = 51.4 MeV. What is the mass of the particle? Can you identify it?

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Textbook Question

A particle confined in a rigid one-dimensional box of length 10 fm has an energy level En = 32.9 MeV and an adjacent energy level En+1 = 51.4 MeV. Sketch the n+1 wave function on the n+1 energy level.

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Textbook Question

INT Model an atom as an electron in a rigid box of length 0.100 nm, roughly twice the Bohr radius. What are the four lowest energy levels of the electron?

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Textbook Question

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.