A quantum well device is designed to trap electrons in a 1 nm wide region.Assuming that such a device acts as a one dimensional box,what wavelength of light would these electrons absorb in making a transition from the first quantum state to the third quantum state?
A)350 nm
B)399 nm
C)550 nm
D)633 nm

Question

Quizplus · Accepted Answer

The energy difference between quantum states in a one-dimensional box is given by the formula $\Delta E = \frac{h^2}{8mL^2}(n_f^2 - n_i^2)$, where $h$ is Planck's constant, $m$ is the mass of an electron, $L$ is the width of the box, $n_f$ is the final quantum number, and $n_i$ is the initial quantum number. For a transition from the first to the third quantum state ($n_i = 1$, $n_f = 3$), and given $L = 1$ nm, we can calculate the energy difference. The energy of a photon is given by $E = \frac{hc}{\lambda}$, where $c$ is the speed of light and $\lambda$ is the wavelength of the light. By equating the energy difference to the photon energy and solving for $\lambda$, we find that the wavelength of light absorbed corresponds to the energy difference between these states. Plugging in the values for $h$, $m$, and $c$, and solving for $\lambda$, we find that the wavelength that matches the energy difference for an electron making a transition from the first to the third quantum state in a 1 nm wide quantum well is closest to the value given in option B, 399 nm.

A Quantum Well Device Is Designed to Trap Electrons in a 1