All Problems Wave Properties of particles. Schrodinger Equation.
Problem 6.74
Making use of the uncertainty principle, evaluate the minimum permitted energy of an electron in a hydrogen atom and its corresponding apparent distance from the nucleus.
6.74. Taking into account that \(p \sim \Delta p \sim \hbar / \Delta r\) and \(\Delta r \sim r,\) we get \(E=p^{2} / 2 m-e^{2} / r \approx \hbar^{2} / 2 m r^{2}-e^{2} / r\). From the condition \(d E / d r=0\) we find \(r_{e f f} \approx \hbar^{2} / m e^{2}=53 \mathrm{pm}, E_{m i n} \approx-m e^{4} / 2 \hbar^{2}=\) \(=-13.6 \mathrm{eV}\)