Question

A particle moves in an infinite potential well described by The eigenfunctions are of the form (r) = A For n = 3. e3(r) = (v/2/n) cos(3mr/n) for lrl cos (knr), or er (r) = Dn sin (k, r), depending E0 for o/2 and t's(r)- (a) What are the expectation values of r and 2 in the n 3 state. (b) What are the expectation values of p and p2 in the n 3 state. To calculate the expectation value for momentum, we need to use the mo- mentum oDerator -ihlo/ox)

Answer 1

A particle move in infinite potential well described by v(x) = {0 Vertical-bar x Vertical-bar > a/2 infinity Vertical-bar x Vertical-bar lessthanorequalto a/2 Psi_n = A_n cos k_n x for odd value of n or Psi_n = B_n sin k_n x for even value of n. For n = 3 Psi_3 = squareroot 2/a cos(3 pi x/a) for Vertical-bar x Vertical-bar lessthanorequalto a/2 psi_3 = 0 for Vertical-bar x Vertical-bar > a/2 (a) expectation value of position vector < Psi_3 Vertical-bar x Vertical-bar Psi_3 > = integral Psi_3 x Psi_3 dz = integral 2/a x cos^2(3 pi x/a)dx = 2/a integral x cos^2(3 pi x/a)dx \r\n < Psi_3 Vertical-bar x Vertical-bar Psi_3 > = 2/a integral^a/2_-a/2 x(1 + cos(6 pi x/a/2)dx = 2/2a integral^a/2_-a/2 (x + x cos 6 pi x/a)dx = 1/a[x^2/2]^a/1_-a/2 = 1/2a [a^2/4 - (-a/2)^2] Vertical-bar Psi_3 Vertical-bar x Vertical-bar Psi_3 > = 0 < Psi_3 Vertical-bar x^2 Vertical-bar Psi_3 > = integral^a/2_-a/2 2/a x^2 cos^2 3 pi x/a dx = integral^a/2_-a/2 2x^2/a (1 + cos(6 pi x/a)/2dx = 1/a [x^3/a]^a/2_-a/2 \r\n < Psi_3 Vertical-bar x^2 Vertical-bar Psi_z > = 1/3a[a^3/8 - (-a^3/8)] = 1/3a [2a^3/8] < Psi_3 Vertical-bar x^3 Vertical-bar Psi_z > = a^2/12 (b) expectation value of momentum vector