Chapter 7, Problem 7.5P

Interpretation Introduction

Interpretation:

The sketch has to be given for given wave function. The highest probability of finding the particle has to be identified. The value at which the probability of finding the particle has been reduced to 50 percent from maximum has to be calculated.

Concept Introduction:

For a particle in $1D$, the wave function is continuous and it is zero outside of the box hence at x values 0 and L its value should be 0. At $n=0$ wavelengths are positive hence a negative value for n does not exist.

The wave function expression is given as follows,

$\begin{array}{c}{\text{ψ}}_{\text{n}}\text{=}\text{N}\text{sin}\left(\frac{\text{nπx}}{\text{L}}\right)\\ \text{n = }\text{1,2,}\mathrm{...}\\ {\text{here,ψ}}_{\text{n}}=wavefunction\\ \text{N}=Normalization\text{constant}\\ N={\left(\frac{2}{L}\right)}^{1/2}\\ \text{L}=lengthofthebox\\ \end{array}$

The wave function is considered to be normalized if the probability of finding the particle in considered domain is equal to 1 and therefore, the integral function of given wave function in considered domain is equal to 1.