Question

Problem 3: A free particle of mass m in one dimension is in the state Hbr Ψ(z, t = 0) = Ae-ar with A, a and b real positive constants. a) Calculate A by normalizing v. b) Calculate the expectation values of position and momentum of the particle at t 0 c) Calculate the uncertainties ΔΧ and Δ1) for the position and momentum at t 0, Do they satisfy the Heisenberg relation? d) Find the wavefunction Ψ(x, t) at a later time t. e) Calculate the expectation values of the position and the momentum of the particle at and Δ1for the position and momentum at a later time t. Do they still satisfy the Heisenberg relation? Do they stay the same, increase or decrease? f) Calculate the uncertainties

Answer 1

a) given psi = A e^- a x^2 + i b x psi^* = A e^- a x^2 - i b x A = ? < pai Vertical-bar psi^* > = 1 Normalization condition 1 = A^2 integral^+ infinity_- infinity e^- a x^2 + i b x e^- a x^2 - i b x d x 1 = A^2 integral^+ infinity_-infinity e^- 2 a x^2 d x 1 = A^2 integral^+ infinity_-infinity e^- 2 a x^2 d x Apply gamma function integral^infinity_0 x^n e^-2 x^2 = 1/ 2 times squareroot n + 1/2 /(infinity)^n + 1/2 1 = A^2 [2 times 1/ 2 squareroot 0 + 1/ 2 /(2 Q)^1/ 2] 1 = A 62 squareroot pi/(2 Q)^1/ 2