Problem 1.4 – Griffith’s Intro to QM

Problem 1.4

At time $t=0$ a particle is represented by the wave function

$\Psi (x,0) = \begin{cases} A(x/a), 0 \le x \le a \\ A(b-x)/(b-a), a \le x \le b \\ 0, \text{otherwise} \end{cases}$

where $A$ , $a$ , and $b$ are (positive) constants.
(a) Normalize $\Psi$ (that is, find $A$ , in terms of $a$ and $b$ ).
(b) Sketch $\Psi (x,0)$ , as a function of $x$ .
(c) Where is the particle most likely to be found, at $t=0$ ?
(d) What is the probability of finding the particle to the left of $a$ ? Check your result in the limiting cases $b=a$ and $b=2a$ .
(e) What is the expectation value of $x$ ?