Question

infinite Demonstrate that for a particle trapped in an wel 12 2 where n is the energy level of the particle. Comment on whether or not such a particle is in accordance with Heisenberg's uncertainty principle.

Answer 1

For particle in a box psi_n (x) = squareroot 2/a sin n pi x/a Delta x = squareroot < x^2 > - < x > ^2 Delta p = squareroot < p^2 > - < p > ^2 Now < x > = integral^a_0 psi_n^* x psi_n dx = integral^a_0 squareroot 2/a sin n pi x/a middot x squareroot 2/a sin n pi x/a dx = 2/a integral^a_0 sin^2 n pi x/a middot x dx = 2/a integral^a_0 (1 - cos 2n pi x/a)x/2 dx = 1/a [integral^a_0 x dx - integral^a_0 cos 2n pi x/a middot x dx] = 1/a middot [x^2/2]^a_0 = 1/a middot a^2/2 = a/2 < x > = a/2 < x^2 > = integral^a_0 psi_n^* x^2 psi_n dx = integral^a_0 squareroot 2/a sin n pi x/a middot x^2 squareroot 2/a sin n pi x/a dx = 2/a integral^a_0 x^2 sin^2 n pi x/a dx 2/a integral^a_0 x^2 (1 - cos 2n pi x/a)/2 dx = 1/a [integral^a_0 x^2 dx - integral^a_0 x^2 cos 2n pi x/a dx] = 1/a [x^3/3]^a_0 - {[sin 2n pi x/a \r\n < null >
i have solved it clearly if any doubt plz comment !!!!!