Exploring Quantum Physics - Week 1 Question 5

Question:

Suppose we have a particle in 1-dimension, with wavefunction $Ae^{-\frac{|x|}{2d}}$. What is the probability to find the particle in the interval $[0, d]$?

Solution:

The born's interpretation tell us the probability is:

$\begin{eqnarray*} P &=& \int\limits_{0}^{d}{\psi(x, t)^2dx} \\ &=& \int\limits_{0}^{d}{(Ae^{-\frac{|x|}{2d}})^2dx} \\ &=& A^2\int\limits_{0}^{d}{(e^{-\frac{x}{d}})dx} \\ &=& A^2(-de^{-\frac{x}{d}})|_0^d \\ &=& A^2(-de^{-1}-(-d)) \\ &=& A^2d(1-e^{-1}) \\ \end{eqnarray*}$

If we wish, we can substitute the $A = \frac{1}{\sqrt{2d}}$ in it.