Question

5) A particle of mass m is in the ground state of the infinite square well 0 < x < a At t-0 the right hand wall suddenly moves to x = 2a, doubling the size of the well. Assume that this expansion happens on a time scale so fast that the initial wave function (at t0+) is the same as just before the expansion (at t-0-) (This is called the "sudden" approximation.) a) What is the probability that a measurement of the system immediately after the expansion (at t-0+) would find the particle to be in the ground state of the new expanded well? The first excited state of the well? b) Find the mean value of the energy immediately after the expansion

Answer 1

The wave function for infinite square well 0 < x < a is psi_n (x) = squareroot 2/a sin (n pi x/a) for ground state implies n = 1 psi_1 (x) = squareroot 2/a sin (pi x/a) after expansion, doubling the size of well therefore psi_n (x) = squareroot 2/2 a sin (n pi x/2 a) psi_a (x) = squareroot 1/a sin (n pi x/2 a) for ground state implies n = 1 squareroot 1/a sin (pi x/2 a) for first excited state implies n = 2 implies psi_2 (x) = squareroot 1/a sin (pi x/a) probability ground state p_1 = integrtal^2 a_0 Vertical-bar psi_1 (x)^2} dx = integral^2 a_0 1/a sin^2 (pi x/2 a) dx = 1/a integral^2 a_0 1 - cos pi x/a/2 dx = 1/2 a [integral^2 a_0 (1 - cos pi x/a) dx] = 1/2 a [(x)^2 a_0 - a/pi (sin pi x/a)^2 a_0] = 1/2 a [2 a - a/pi (0 - 0)]