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

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Quizplus · Accepted Answer

The energy levels of the system are given by En=(h^2π^2/2mL^2)(n^2x+n^2y) where n_x and n_y are integers. We can write n^2x+n^2y as (n^2x+n^2y-2n^2x/2) + (n^2x+n^2y-2n^2y/2), and this equals (n^2-2n^2x/2) + (n^2-2n^2y/2). The quantity in the parentheses is the square of an integer, so En=(h^2π^2/2mL^2)(nx-1/2)^2+(ny-1/2)^2). Therefore, the minimum single-particle energy is (h^2π^2/2mL^2)(1/2)^2+(1/2)^2=2(h^2π^2/2mL^2)/2= (h^2π^2/8mL^2). Thus, the number of single-particle states with energy 5(h^2π^2/8mL^2) is 2, since there are only two possible excited states with energies closest to 5(h^2π^2/8mL^2), which are (nx,ny)=(7,1) and (nx,ny)=(5,5).

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