Question

2. [16 points] What is the solution of the time-dependent Schrödinger Equation Ψ(x, t) for the solution of the time-independent Schrödinger Equation Ψ(x) = ,in (m) in the particle in the box model? Write ω =-explicitly in terms of the parameters of the problem. Explicily show that W,(Cx.t) solves the time-dependent Schrödinger Equation 2

Answer 1

Given that time independent solution of particle in box is psi_4 (x) = squareroot 2/a sin (4 pi x/a) (X) psi_4 (x, t) = psi_4 (x) e^-i/planckconstantovertwopi E_4 t Since most generally psi_4 (x, t) = psi_n (x) e^-i/planckconstantovertwopi E_n t Here E_4 = 4^2 pi^2 planckconstantovertwopi^2/2m a^2 = 16 pi^2 planckconstantovertwopi^2/2ma^2 psi_4 (x, t) = psi_4 (x) e^-i(16 pi^2 2 planckconstantovertwopi/2ma^2) t Now we show that this solution of time dependence schrodinger equation + i planckconstantovertwopi partial differential psi_n(x, t)/partial differential t = -planckconstantovertwopi^2/2m d^2/dx^2 psi_n (x, t) (1) {2 v = 0 inside L.H.S n = 4 i planckconstantovertwopi partial differential/partial differential t {psi_4 (x, t)){ = i planckconstantovertwopi partial differential/partial differential t { psi_4 (x) e^-i(16 pi^2 planckconstantovertwopi/2 m a^2)t } = i planckconstantovertwopi psi (x) {-i pi^2 planckconstantovertwopi/2 m a^2 }e^-i(16 pi^2 planckconstantovertwopi/2 m a^2) t
please thumb up