2. (QM) A normalised solution of the time-independent Schrödinger equation for a particle of mass m in the potential V(x) = kr² is ^ (2)² where A²=2/(√) and a²=h/√mk. Show by calculation that is an eigenfunction of the Hamiltonion operator, II. Hence, derive the energy of this state in terms of w = √√k/m, i.e. show that x= Exp. Show that this wave function predicts uncertainties on the observed position (2) and momentum (p) that are consistent with the Heisenberg uncertainty principle, i.e. Araph/2. Note that the standard definite integrals required are all in the Maths Handbook. Hint: there are a few "short cuts" to this answer if write down in terms of and calculate Ar before Ap.

2. (QM) A normalised solution of the time-independent Schrödinger equation for a particle of mass m in the potential V(x) = kr² is ^ (2)² where A²=2/(√) and a²=h/√mk. Show by calculation that is an eigenfunction of the Hamiltonion operator, II. Hence, derive the energy of this state in terms of w = √√k/m, i.e. show that x= Exp. Show that this wave function predicts uncertainties on the observed position (2) and momentum (p) that are consistent with the Heisenberg uncertainty principle, i.e. Araph/2. Note that the standard definite integrals required are all in the Maths Handbook. Hint: there are a few "short cuts" to this answer if write down in terms of and calculate Ar before Ap.