Chapter 7, Problem 7.9P

Interpretation Introduction

Interpretation:

The location for particle present in box of length L at which its probability is found 50 percent of its maximum probability when n is equal to 1 has to be identified.

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.