Expectation values of main observables

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

$\Psi(x,0) ={\begin{matrix} A(a^2-x^2) & if -a \leq x\leq a\\ 0 & otherwise \end{matrix}}$

a) Determine the normalization constant $A$

b) What is the expectation value of $x$ (at time t=0)?

c) What is the expectation value of $p$ (at time t=0)? (Note: you cannot get it from p=(m*d<x>)/d t)

d) Find the expectation value of $x^2$.

e) Find the expectation value of $p^2$.

f) Find the uncertainty in $x$ (i.e. $\sigma_x$).

g) Find the uncertainty in $p$ (i.e. $\sigma_p$)

h) Verify Heisenberg’s uncertainty principle for this wave function.

a) Normalization constant

$1 =\int_{-a}^a |\Psi|^2dx=A^2*\int_{-a}^a (a^2-x^2)^2dx=…=(16/15)*A^2a^5$

$A = \sqrt {\frac {15}{16a^5}}$

b) Expectation value of $x$

$<x> =\int_{-a}^a \Psi*x*\Psi^*d x =A^2\int_{-a}^a x(a^2-x^2)^2dx=A^2*\frac {-(a^2-x^2)^3}{6} |_{-a}^a =$

$=A^2*0 =0$

This was to be expected since the wave function is “centered” on $x=0$.

c) Expectation value of $p$

 $\hat p =-i\hbar*(d/d x)$

$<p> =\int_{-a}^a \Psi(-i\hbar\frac{d}{d x})\Psi^*d x=-i\hbar*A^2\int_{-a}^a -2x(a^2-x^2)d x =$


d) Expectation value of $x^2$

$<x^2> =\int_{-a}^a \Psi x^2\psi^*d x =A^2\int_{-a}^a x^2(a^2-x^2)^2=…=(a^2/7)$

e) Expectation value of $p^2$

One needs to know that

$\hat p^2 =-\hbar^2*(d^2/d x^2)$

$<p^2> =-\hbar^2\int_{-a}^a \Psi\frac{d^2}{d x^2}\Psi^*d x=2\hbar^2 A^2\int_{-a}^a (a^2-x^2)d x=…=\frac{5\hbar^2}{2a^2}$

f) Uncertainty in $x$ is $\sigma x$

$(\sigma x)^2=<x^2> -<x>^2 =(a^2/7)-0 =(a^2/7)$

g) Uncertainty in $p$ is $\sigma p$

$(\sigma p)^2=<p^2> -<p>^2 =\frac{5\hbar^2}{2a^2}$

h) Heisenberg principle:

$(\sigma x)^2*(\sigma p)^2=\frac{a^2}{7}*\frac{5\hbar^2}{2a^2} =\frac{5}{14}\hbar^2 =0.357*\hbar^2> (\hbar^2/4)$