Answers for Assigned Problems

4.2 (a) y(x, y, z) = (2/a)^3/2 sin(n_xpx/a) sin(n_ypy/a) sin(n_zpz/a)
            E = [p² hbar² / (2 m a²)] [n_x² + n_y² + n_z²]

      (b) E₁ = 3 [p² hbar² / (2 m a²)]     d = 1
            E₂ = 6 [p² hbar² / (2 m a²)]     d = 3
            E₃ = 9 [p² hbar² / (2 m a²)]     d = 3
            E₄ = 11 [p² hbar² / (2 m a²)]     d = 3
            E₅ = 12 [p² hbar² / (2 m a²)]     d = 1
            E₆ = 14 [p² hbar² / (2 m a²)]     d = 6

      (c) E₁₄ = 27 [p² hbar² / (2 m a²)]     d = 4
Two different sets of n_i, (3,3,3) and (5,1,1), contribute to this energy, something which is called "accidental degeneracy."

4.3 See table, p. 139 of text.

4.5 Y_k^k = (1/k!)sqrt[(2k+1)!/(4p)] (- e^if sinq / 2)^k

4.7 See table, p. 143 of text.

4.9 The energy is given by - tan ka = k / l , where k = sqrt(2mE)/hbar and l = sqrt(2m(V₀-E))/hbar

Last revised 2005/04/12