An electron is confined in a one-dimensional potential well of width 2.5 Å with infinitely high walls. Find its energy in the ground state and in the first excited state, in joule and eV. (\(h=6.63\times10^{-34}\) J s, \(m_e=9.1\times10^{-31}\) kg)

Answer: \(E_1=9.66\times10^{-19}\) J \(=6.04\) eV; \(E_2=3.86\times10^{-18}\) J \(=24.15\) eV.

For a particle in an infinitely deep well of width \(L\): \[E_n=\frac{n^2h^2}{8mL^2},\quad n=1,2,3,\dots\]

Given \(L=2.5\ \text{Å}=2.5\times10^{-10}\) m.

Ground state (\(n=1\)):

\[E_1=\frac{(6.63\times10^{-34})^2}{8\times9.1\times10^{-31}\times(2.5\times10^{-10})^2}=\frac{4.396\times10^{-67}}{4.55\times10^{-49}}=9.66\times10^{-19}\ \text{J}\]

In eV: \(E_1=\dfrac{9.66\times10^{-19}}{1.6\times10^{-19}}=6.04\) eV.

First excited state (\(n=2\)): \(E_2=4E_1=3.86\times10^{-18}\) J \(=24.15\) eV.