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.