Question

8. A particle in a box (0x<L) has wave functions and energies of En 8m2 a) Normalize the wave functions to determine A b) At t-0, ψ(x)-vsv, + ψ2 . 2. c) The particle will oscillate back and forth. Derive an expression for the oscilla- tion frequency in terms of h, m, and L Derive expressions for Ψ(x, t) and |Ψ(x, t)

Answer 1

psi_n = A_n sin[n pi/L x], E_n = h^2/8mL^2 n^2 (a) Normalizing psi_n integral^infinity_-infinity psi_n psi_n^*dx = 1 integral^infinity_-infinity A_n sin[n pi/L x] middot A_n^* sin[n pi/L x]dx = 1 A_n A_n^* integral^infinity_-infinity sin^2 [n pi/L x] dx = 1 A_n^2 integral^L_0 sin^2 [n pi/L x] dx = 1 (because x elementof (0, L)) A_n^2 integral^L_0 1/2 (1 - cos 2n pi/L x) dx = 1 1/2 A_n^2 integral^L_0 (1 - cos 2n pi/L x) dx = 1 1/2 A_n^2 [x - (sin 2n pi/L x)(1/2n pi)]^L_0 = 1 1/2 A_n^2 [L - sin 2n pi (L/2n pi) - 0 + 0) = 1 1/2 A_n^2 [L - 0 - 0 + 0] = 1 1/2 A_n^2 middot L = 1 or A_n^2 = 2/L implies A_n = plusminus squareroot 2/L \r\n (b) psi(x) = 2/squareroot 5 psi_1 + 1/squareroot 5 psi_2 = 2/squareroot 5 middot squareroot 2/L sin(pi/L x) + 1/squareroot 5 middot squareroot 2/L sin(2 pi/L x) psi(x) = squareroot 2/L [2/squareroot 5