The particle in a box model is often used to make rough estimates of energy level spacings. For a metal wire \(10 \mathrm{cm}\) long, treat a conduction electron as a particle confined to a one-dimensional box of length \(10 \mathrm{cm} .\) (a) Sketch the wave function \(\psi\) as a function of position for the electron in this box for the ground state and each of the first three excited states. (b) Estimate the spacing between energy levels of the conduction electrons by finding the energy spacing between the ground state and the first excited state.

For a one-dimensional box from \(0\) to \(L\), the wave function for the electron can be expressed as: \(\psi _n(x) = \sqrt{\frac{2}{L}}\sin{\frac{n \pi x}{L}}\) where \(n\) is the principal quantum number (\(n = 1, 2, 3, ...\)) and \(L\) is the length of the box. For the ground state, \(n = 1\): \(\psi _1(x) = \sqrt{\frac{2}{10}}\sin{\frac{1 \pi x}{10}}\) For the first excited state, \(n = 2\): \(\psi _2(x) = \sqrt{\frac{2}{10}}\sin{\frac{2 \pi x}{10}}\) For the second excited state, \(n = 3\): \(\psi _3(x) = \sqrt{\frac{2}{10}}\sin{\frac{3 \pi x}{10}}\) For the third excited state, \(n = 4\): \(\psi _4(x) = \sqrt{\frac{2}{10}}\sin{\frac{4 \pi x}{10}}\) Based on these expressions, the student can sketch the wave function \(\psi\) as a function of position for the ground state and the first three excited states.

To estimate the energy level spacing, we need to find the energy levels. The energy of a particle in a one-dimensional box can be expressed as: \(E_n = \frac{n^2 \pi ^2 \hbar ^2}{2m_e L^2}\) where \(n\) is the principal quantum number, \(\hbar\) is the reduced Planck constant, \(m_e\) is the mass of the electron, and \(L\) is the length of the box. The spacing between the ground state (\(n=1\)) and the first excited state (\(n=2\)) can be calculated as: \(\Delta E_{1 \to 2} = E_2 - E_1 = \frac{2^2 \pi ^2 \hbar ^2}{2m_e L^2} - \frac{1^2 \pi ^2 \hbar ^2}{2m_e L^2}\) This simplifies to: \(\Delta E_{1 \to 2} = \frac{3 \pi ^2 \hbar ^2}{2m_e L^2}\) Now, we can substitute the values of the constants and the length of the box: \(\Delta E_{1 \to 2} = \frac{3 \pi ^2 (6.582 \times 10^{-16} \mathrm{eV \cdot s})^2}{2(9.109 \times 10^{-31} \mathrm{kg})(10 \times 10^{-2} \mathrm{m})^2}\) \(\Delta E_{1 \to 2} = 1.964 \times 10^{-9}\, \mathrm{eV}\) So, the spacing between the energy levels of the conduction electrons is approximately \(1.964 \times 10^{-9}\, \mathrm{eV}\).