Question

A) Calculate the mean values <x^2> and <p^2x> for stationery state n.

B) Calculate the rms deviations sigma(x) and sigma(px) for stationery state n. Check that the Heisenberg uncertainty principle is satisfied.

1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 saSa otherwise. V (z) = For simplicity, we may take the 'universe' here to be the region of 0 sas a, which is where the wave function is nontrivial. Consequently, we may express stationary state n as where E, is the associated mechanical energy. It can be shown that (z) = a/2 and (ps) stationary state n. 0 for (a) Calculate the mean values (x2) and (p2) for stationary state n. (b) Calculate the rms deviations Oz, and σΡ, for stationary state n. Check that the H Heisenberg uncertainty principle is satisfied. Which state comes closest to the uncertainty limit?

Answer 1

Psi_n (x, t) = squareroot 40 [sin [n pi x/a]] e^- I Ent/n Now < x^2 > = < Psi_n Vertical-bar x^2 Vertical-bar Psi_n > = 2/a integral^a_0 x^2 sin^2 (n pi x/a) dx = 1/a integral^a_0 x^2 [1 - cos [2n pi x/a] dx] = 1/a [a^3/3 - integral^a_0 x^2 cos (2 pi n x/a) dx] now integral^a_0 x^2 cos (2n pi x/a) dx = x^2 (a/2n pi) sin (2n pi/x) Vertical-bar ^a_0 - 2a/2n pi integral^a_0 x sin (2n pi x/a) = -2a/2n pi [- x a/2n pi cos (2n pi x/a) + a/2n pi integral^a_0 cos (2n pix/a)] = (a/2npi)^2