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 of the electrons is given by the formula: 
E(nSubx, nSuby) = (h^2/8mL^2) * (nSubx^2 + nSuby^2) 
The maximum energy in the question is given as: 
8(h/8mL^2) 
Substituting this value for E(nSubx, nSuby), we get: 
(nSubx^2 + nSuby^2) <= 8 
The number of combinations of (nSubx, nSuby) that satisfy this condition and have energy less than or equal to 8(h/8mL^2) is 2, namely (0, 0) and (1, 1). 
Since each energy level can accommodate 2 electrons (one spin up and one spin down), the maximum number of electrons that can have energy 8(h/8mL^2) is 4. 
Therefore, the answer is (B) 4.

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