Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by , , where n_x and n_y are integers.At most the number of electrons that can have energy 8(h²/8mL²)is: A)1 B)2 C)3 D)4 E)any number

Question

Quizplus · Accepted Answer

The energy level of an electron in a two-dimensional square potential well is given by:
E =$\frac{h^2}{8m}(n_x^2+n_y^2)$
where n_x and n_y are the quantum numbers in the x and y directions, respectively.

At most, the number of electrons that can have energy 8$(\frac{h^2}{8mL^2})$ is two because the energy level is given by:
E =$\frac{h^2}{8mL^2}(n_x^2+n_y^2)$

Setting this equal to 8$(\frac{h^2}{8mL^2})$, we get:
$n_x^2+n_y^2 = 8$
The possible combinations of n_x and n_y that satisfy this equation are:
(2, 2) and (±2, ±2)
Therefore, at most, two electrons can have energy 8$(\frac{h^2}{8mL^2})$.

Electrons Are in a Two-Dimensional Square Potential Energy Well with Sides