Problem 1.16 – Griffith’s Intro to QM

Problem 1.16

A particle is represented (at time $t=0$ ) by the wave function

$\Psi (x,0) = \begin{cases} A(a^2-x^2), -a \le x \le a \\ 0, \text{otherwise} \end{cases}$

(a) Determine the normalization constant A.
(b) What is the expectation value of x?
(c) What is the expectation value of p? (Note that you cannot get it from $\langle p \rangle =md \langle x \rangle /dt$ . Why not?)
(d) Find the expectation value of $x^2$ .
(e) Find the expectation value of $p^2$ .
(f) Find the uncertainty in x $(\sigma_x)$ .
(g) Find the uncertainty in p $(\sigma_p)$ .
(h) Check that your results are consistent with the uncertainty principle.