Question

(a) The Schrodinger wave equation for hydrogen atom is
${\Psi }_{2s}=\frac{1}{4(2\pi {)}^{1/2}}{\left(\frac{1}{a_0}\right)}^{3/2}\left(2-\frac{r}{a_0}\right)e^{-r/2a_0}$
where, $a_0$ is Bohr's radius. Let the radial node in $2s$ be at $r_0$. Then, find $r$ in terms of $a_0$.
(b) A base ball having mass $100g$ moves with velocity $100m/s$. Find out the value of wavelength of base ball.

Answer 1

(a) At radial node, ${\psi }^2$ must vanishes, i.e.
${\psi }_{2s}^2=0={\left[\frac{1}{4\sqrt{2}\pi }\right]}^2{\left(2-\frac{r_0}{a_0}\right)}^2{e}^{-\frac{r_0}{a_0}}$
$\Rightarrow 2-\frac{r_0}{a_0}=0\Rightarrow r=2a_0$
(b)$\lambda =\frac{h}{mv}=\frac{6.625\times 10^{-34}}{100\times 10^{-3}\times 100}=6.625\times 10^{-35}m$
$=6.625\times 10^{-25}\mathring{A}$ (negligibly small)