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. The number of single-particle states with energy 5(h²/8mL²) is: A) 1 B) 2 C) 3 D) 4 E) 5

Question

Quizplus · Accepted Answer

The allowed energy levels for a two-dimensional square potential well are given by:

E(n1,n2) = (n1² + n2²)h² / 8mL²

where n1 and n2 are positive integers.

The question gives a specific energy level:

E = 11h² / 8mL²

We need to find the number of single-particle states with energy equal to this level.

Plugging in the given expression for E into the above equation, we get:

11h² / 8mL² = (n1² + n2²)h² / 8mL²

Simplifying, we get:

n1² + n2² = 11

The possible values of n1 and n2 that satisfy this equation are:

n1 = 1, n2 = 3
n1 = 3, n2 = 1
n1 = -1, n2 = -3
n1 = -3, n2 = -1

Note that negative values are allowed because the energy levels are degenerate (i.e. multiple states can have the same energy).

Therefore, there are two single-particle states with energy 11h² / 8mL², corresponding to the two sets of (n1, n2) values above.

The correct answer is B) 2.

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