Question

An electron trapped in a one-dimensional box of length L = 1 Å [a typical atomic diameter] is 

described by the normalized wave function

𝛹 = √2

𝐿

sin

𝑛𝜋

𝑥𝐿

 In the ground state, what is the probability that the electron lies between (a) 0.09 

Å and 0.11 Å (b) 0.0 Å 

 and 0.25 Å?

Answer 1

a)

$P=\int_{0.09}^{0.11}\psi ^2dx=\int_{0.09}^{0.11}\frac 2L\sin^2\frac {n \pi x}{L} dx=0.00383,$

b)

$P=\int_{0.09}^{0.11}\psi ^2dx=\int_{0}^{0.25}\frac 2L\sin^2\frac {n \pi x}{L} dx=0.09085.$