   Chapter 10, Problem 10.92E

Chapter
Section
Textbook Problem

Construct plots of the probabilities of the first three wavefunctions for a particle in a one-dimensional box having length a . Identify where the nodes are.

Interpretation Introduction

Interpretation:

The plots of the probabilities of the first three wavefunctions for a particle in a one-dimensional box having length a are to be constructed. The position of the nodes for three wavefunctions is to be predicted.

Concept introduction:

In quantum mechanics, the wavefunction is given by Ψ. The wavefunction contains all the information about the state of the system. The wavefunction is the function of the coordinates of the particles and time. The square of the probability function, |Ψ|2, relates to the probability density.

Explanation

The wavefunction for a particle in a one-dimensional box having length a is given by,

Ψn(x)=2asin(nπxa)

Where,

x is the position of a particle.

a is the length of a 1-D box.

Similarly, the equation for n=1, n=2 and n=3 wavefunctions for a particle in a one-dimensional box is given by,

Ψ1(x)=2asin(πxa)Ψ2(x)=2asin(2πxa)Ψ3(x)=2asin(3πxa)

The number of nodes is calculated by the formula,

Numberofnodes=n1

Where,

n is the principal quantum number

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