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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 29a
Chapter 40, Problem 29a

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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Understand the problem: The electron is modeled as a particle in a one-dimensional rigid box (also called an infinite potential well). The energy levels for such a system are quantized and given by the formula: En = (n2h2)/(8mL2), where n is the quantum number (1, 2, 3,...), h is Planck's constant, m is the mass of the electron, and L is the length of the box.
Substitute the given values into the formula: The length of the box is L = 0.100 \, \(\text{nm}\) = 0.100 \(\times\) 10^{-9} \, \(\text{m}\). The mass of the electron is m = 9.11 \(\times\) 10^{-31} \, \(\text{kg}\), and Planck's constant is h = 6.626 \(\times\) 10^{-34} \, \(\text{J·s}\).
Calculate the energy for the first four quantum states: Use the formula En = (n2h2)/(8mL2) for n = 1, 2, 3, 4. For each value of n, substitute it into the formula and compute the corresponding energy level.
Express the energy levels in electron volts (eV): Since the energy will initially be calculated in joules, convert it to electron volts using the conversion factor 1 \, \(\text{eV}\) = 1.602 \(\times\) 10^{-19} \, \(\text{J}\). Divide each energy value by this factor to express the results in eV.
Summarize the results: The four lowest energy levels correspond to n = 1, 2, 3, 4. List the calculated energy values for these quantum states in eV, ensuring they are clearly labeled as E1, E2, E3, E4.

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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 wave-particle duality, where particles like electrons exhibit both wave-like and particle-like properties. This framework is essential for understanding how electrons occupy discrete energy levels in an atom.
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Particle in a Box Model

The particle in a box model is a fundamental quantum mechanics problem that simplifies the analysis of a particle confined to a rigid, one-dimensional space. In this model, the particle can only occupy certain quantized energy levels, which are determined by the size of the box and the mass of the particle. This model helps in calculating the energy levels of an electron in an atom.
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Energy Quantization

Energy quantization refers to the concept that certain physical systems, such as electrons in an atom, can only exist in specific energy states. In the context of the particle in a box model, the energy levels are determined by the equation E_n = n^2 * h^2 / (8mL^2), where n is a positive integer, h is Planck's constant, m is the mass of the electron, and L is the length of the box. This principle is crucial for determining the allowed energy levels of the electron in the given scenario.
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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

An electron approaches a 1.0-nm-wide potential-energy barrier of height 5.0 eV. What energy electron has a tunneling probability of (a) 10%, (b) 1.0%, and (c) 0.10%?

Textbook Question

CALC Suppose that ψ1(x) and ψ2(x) are both solutions to the Schrödinger equation for the same potential energy U(x). Prove that the superposition ψ(x)=Aψ1(x) + Bψ2(x) is also a solution to the Schrödinger equation.

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.

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. Determine the values of n and n+1.

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