Answer
See the explanation
Work Step by Step
The nodes of a wave function are the points where the wave function goes to zero. For the given wave function \( \psi_{300} \), the position of the nodes can be found by setting the wave function equal to zero and solving for \( \sigma \).
Setting \( \psi_{300} \) equal to zero gives:
\[ \frac{1}{81 \sqrt{3 \pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2}\left(27-18 \sigma+2 \sigma^{2}\right) e^{-\sigma / 3} = 0 \]
Since the exponential term is never zero, we can focus on the polynomial part:
\[ \left(27-18 \sigma+2 \sigma^{2}\right) = 0 \]
Solving this quadratic equation gives the values of \( \sigma \) at which the wave function goes to zero. The solutions are:
\[ \sigma = \frac{18\pm\sqrt{324-216}}{4}=\frac{9\pm3\sqrt 3}{2} \]
$\sigma_1\approx 1.9$
$\sigma_2\approx 7.1$
We can find the positions of the nodes in terms of the Bohr radius:
$r_1 = \sigma_1a_{0}=1.9\times 5.29\times 10^{-11}\approx 1.005\times 10^{-10}\text{ m}$
$r_2 = \sigma_2a_{0}=7.1\times 5.29\times 10^{-11}\approx 3.756\times 10^{-10}\text{ m}$
