Problem 19.71

จากสมมติฐานที่ \(1\) จะได้ \(\Sigma \vec{F}_{r}= m\vec{a}\)

\(\frac{k(Ze)(e)}{r^{2}}=\frac{mv^{2}}{r}\)

\(v^{2} =\frac{kZe^{2}}{mr}\cdots \cdots (1)\)

จากสมมติฐานที่ \(2\) จะได้ \(E = K + U\)

\(E=\frac{1}{2}mv^{2}+\frac{k(Ze)(-e)}{r}\)

\(E=\frac{1}{2}m(\frac{kZe^{2}}{mr}) -\frac{kZe^{2}}{r}; (1)\)

\(E = -\frac{kZe^{2}}{2r}\cdots \cdots (2)\)

จากสมมติฐานที่ \(4\) จะได้ \(L = n\hbar \)

\(mvr = n\hbar\)

\(v^{2} =\frac{n^{2}\hbar ^{2}}{m^{2}r^{2}}\cdots \cdots (3)\)

\((1) = (3) ;\frac{kZe^{2}}{mr}=\frac{n^{2}\hbar ^{2}}{m^{2}r^{2}}\)

\(r =\frac{n^{2}\hbar ^{2}}{mkZe^{2}}\cdots \cdots (4)\)

\((4)\) แทนใน \((2) ; E = - \frac{kZe^{2}}{2}(\frac{mkZe^{2}}{n^{2}\hbar ^{2}})\)

\(E_{n}= -\frac{mk^{2}e^{4}Z^{2}}{2\hbar ^{2}n^{2}}\)

\(E_{n}= -\frac{13.6Z^{2}}{n^{2}}eV \;\blacksquare\)