91Ó°ÊÓ

The ground-state wavefunction of a hydrogen atom has the form \(\psi(r)=N \mathrm{e}^{-b r}, b\) being a collection of fundamental constants with the magnitude \(1 / a_{0},\) with \(a_{0}=53 \mathrm{pm}\) Normalize this spherically symmetrical function. Hint. The volume element is \(\mathrm{d} \tau=\sin \theta \mathrm{d} \theta \mathrm{d} \varphi r^{2} \mathrm{d} r,\) with \(0 \leq \theta \leq \pi\) \(0 \leq \varphi \leq 2 \pi,\) and \(0 \leq r<\infty,\) 'Normalize' always means 'normalize to 1 ' in this text.

Construct quantum mechanical operators in the position representation for the following observables: (a) kinetic energy in one and in three dimensions, (b) the inverse separation, \(1 / x,\) (c) electric dipole moment \(\left(\Sigma_{i} Q_{i} r_{i} \text { where } r_{i}\right.\) is the position of a charge \(Q_{i}\) ), (d) \(z\) -component of angular momentum \(\left(x p_{y}-y p_{x}\right),\) (e) the mean square deviations of the position and momentum of a particle from the mean values.

Write the time-independent Schrödinger equations for (a) the hydrogen atom, (b) the helium atom, (c) the hydrogen molecule, (d) a free particle, (e) a particle subjected to a constant, uniform force. Hint. Identify the appropriate potential energy terms and express them as operators in the position representation.