Question

(a) The Schrodinger wave equation for hydrogen atom is
$\Psi_{2 s}=\frac{1}{4(2 \pi)^{1 / 2}}\left(\frac{1}{a_{0}}\right)^{3 / 2}\left(2-\frac{r}{a_{0}}\right) e^{-r / 2 a_{0}}$
where, $a_{0}$ is Bohr's radius. Let the radial node in $2 s$ be at $r_{0}$. Then, find $r$ in terms of $a_{0}$.
(b) A base ball having mass $100 \,g$ moves with velocity $100\, m / s$. Find out the value of wavelength of base ball.

Answer 1

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