Skip to main content
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 32c
Chapter 40, Problem 32c

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.

Verified step by step guidance
1
Understand the problem: The particle is confined in a rigid one-dimensional box, which means its wave functions are solutions to the Schrödinger equation for a particle in a box. The energy levels are quantized, and the wave functions correspond to standing waves inside the box.
Recall the general form of the wave function for a particle in a box: \( \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right) \), where \( L \) is the length of the box, \( n \) is the quantum number, and \( x \) is the position within the box.
For the \( n+1 \) energy level, the wave function will have \( n+1 \) nodes (points where the wave function crosses zero) within the box. This is a key characteristic of the wave function for higher energy levels.
Sketch the wave function: To sketch \( \psi_{n+1}(x) \), plot a sine wave with \( n+1 \) complete oscillations over the length of the box (10 fm). Ensure the wave starts and ends at zero, as the boundary conditions require the wave function to vanish at the edges of the box.
Label the sketch: Clearly indicate the box length (10 fm), the quantum number \( n+1 \), and the nodes of the wave function. The amplitude of the wave function is proportional to \( \sqrt{\frac{2}{L}} \), but the exact amplitude is not required for the sketch.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?

Key Concepts

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

Quantum Mechanics and Particle in a Box

In quantum mechanics, a particle confined in a rigid one-dimensional box is a fundamental model that illustrates how particles behave in confined spaces. The particle can only occupy specific energy levels, which are quantized. The wave function describes the probability amplitude of finding the particle in a given position within the box, and the allowed energy levels depend on the length of the box and the quantum number n.
Recommended video:
Guided course
04:27
Power of Pushing a Box

Wave Function and Energy Levels

The wave function, denoted as Ψ, represents the state of a quantum system and contains all the information about the particle's position and momentum. For a particle in a box, the wave functions are sinusoidal functions that correspond to different energy levels. The energy levels are given by 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.
Recommended video:
Guided course
08:30
Intro to Wave Functions

Sketching Wave Functions

Sketching the wave function for a specific energy level involves plotting the function Ψ against position within the box. For the n+1 level, the wave function will have one additional node compared to the n level, indicating a change in the probability distribution of the particle's position. The amplitude of the wave function at any point reflects the likelihood of finding the particle there, with nodes representing points of zero probability.
Recommended video:
Guided course
08:30
Intro to Wave Functions
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

In most metals, the atomic ions form a regular arrangement called a crystal lattice. The conduction electrons in the sea of electrons move through this lattice. FIGURE P40.34 is a one-dimensional model of a crystal lattice. The ions have mass m, charge e, and an equilibrium separation b. Suppose this crystal consists of aluminum ions with an equilibrium spacing of 0.30 nm. What are the energies of the four lowest vibrational states of these ions?

1
views
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.

1
views
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?

1
views
Textbook Question

CALC Consider a particle in a rigid box of length L. For each of the states n = 1,n = 2, and n = 3: Where, in terms of L, are the positions at which the particle is most likely to be found?

1
views
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.

1
views