Question

(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 ψ_n(x) = (2/a)^1/2 sin(nπx/a), with energy E_n = 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.

Answer 1

(a) The term (−ℏ²d²/2mdx²) is the kinetic energy operator, U(x) is the potential energy operator, ψ(x) is the eigen function and E is the eigen value. 

(b) Put U(x) = 0 in the region 0 < x < a in the Schrodinger equation to obtain

This is an unnormalized solution. The constant A is determined from normalization condition.

(d) ψ(n) and probability density P(x) distributions for n = 1, 2 and 3 are sketched in Fig 3.6