(a) The one-dimensional time-independent Schrodinger equation is
(− ℏ2/2m) (d2ψ(x))/dx2 + U(x)ψ(x) = Eψ(x)
Give the meanings of the symbols in this equation.
(b) A particle of mass m is contained in a one-dimensional box of width a. The potential energy U(x) is infinite at the walls of the box (x = 0 and x = a) and zero in between (0 < x < a).
Solve the Schrodinger equation for this particle and hence show that the normalized solutions have the form ψn(x) = (2/a)1/2 sin(nπx/a), with energy En = h2n2/8ma2, where n is an integer (n > 0).
(c) For the case n = 3, find the probability that the particle will be located in the region a/3 < x < 2a/3.
(d) Sketch the wave-functions and the corresponding probability density distributions for the cases n = 1, 2 and 3.