Question

 

Problem 1. At time t = 0 the state of a particle in one dimension (1D) is given by ψ(x, 0) = A x 2 + a 2 Here a and A are some positive constants.

i) Find A

ii) Sketch the graph of the probability density of ψ

iii) Find the probability that the particle is within −a < x < a and − √ 2a < x < √ 2a

iv) Find the expectation value of the momentum operator hpi

Problem 2. Consider a state of a 1D particle at time t = 0 given by ψ(x, 0) = Ae −x 2 2s2 Here s is some positive constant, and A is the normalization factor.

i) Find A

ii) Find hpi, hp 2 i

iii) Find σx and σp and verify that they satisfy the uncertainty relation. iv Find hpi and hp 2 i for ψ(x, 0) = A exp − x 2 2s 2 + ikx where k is some constant.

Problem 3. A 1D particle of mass m is in a state given by ψ(x, t) = ( A(1 + cos( x L ))e i~ 2mL2 t for |x| < πL 0 otherwise Here L is some constant length, and A is the normalisation constant.

i) Find A

ii) Find the potential V (x) for |x| < πL such that ψ(x, t) satisfies the Schr¨odinger equation.

iii) Find hpi and hp 2 i Extra Credit: Find hxi, hx 2 i and show that σx and σp are consistent with the uncertainty relation. Look up any integrals you need or use Wolfram Alpha to find the expression for hx 2 i.

Answer 1