Question

The electrons in a rigid box emit photons of wavelength 1483.0nm during the 3 to 2 transition. How long is the box in which the electrons are confined? Submit Answer Tries 0/10

Answer 1

The electron is in a rigid box. This means that upto a certain length "L" , the potential is zero and this is where the electron is and then beyond that, on either sides of the box, the potential is assumed to be infinite( since the box is rigid).

The energy of a particle confined in such a box is given by:
$E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}=\frac{n^2h^2}{8mL^2}$

The transition that is being seen is a 3 to 2 transition and the wavelength that is being emitted is 1483.0 nm.

Thus,

$E_3-E_2=\Delta E=\frac{hc}{\lambda}=\frac{h^2}{8mL^2}(3^2-2^2)\newline\newline \implies \frac{c}{\lambda}=\frac{5h}{8mL^2}\implies L=\sqrt{\frac{5h\lambda}{8mc}}$

using c = 3 x 10^8 m/s, h = 6.626 x x10^-34 J.s, lambda = 1483 x x10^-9 m, m=9.11 x x10^- 31 kg

we get:

L = 1.499 x 10^-9 m = 1.5 nm

Which is the required length.