In one line
Use \(E_n = n^2h^2/(8mL^2)\) in SI units for the required level, and use \(\Delta E = E_{n_2} - E_{n_1}\) with \(\lambda = hc/\Delta E\) for transitions between levels.
Explained simply
Box problems are mostly "plug and convert" problems. The main skill is keeping units straight (Å or nm to metres, joules to eV) and remembering that energy grows as \(n^2\), so you only calculate \(E_1\) and multiply.
How it works
- Convert \(L\) to metres (1 Å \(= 10^{-10}\) m, 1 nm \(= 10^{-9}\) m).
- Compute \(E_1 = h^2/(8mL^2)\) in joules, then convert to eV.
- Higher levels: \(E_n = n^2E_1\).
- For a transition: \(\Delta E = (n_2^2 - n_1^2)E_1\), and emitted photon \(\lambda = hc/\Delta E\) (shortcut: \(\lambda\) in nm \(= 1240/\Delta E\) in eV).
- For the minimum energy, use \(n = 1\). For the box width that gives a certain energy, rearrange: \(L = nh/\sqrt{8mE}\).
- For momentum: \(p_n = \sqrt{2mE_n} = nh/2L\).
Key formula
\(E_n = \dfrac{n^2h^2}{8mL^2}\); \(\lambda = \dfrac{hc}{\Delta E}\). \(m_e = 9.11\times10^{-31}\) kg, \(h = 6.63\times10^{-34}\) J s, \(c = 3\times10^8\) m/s, 1 eV \(= 1.6\times10^{-19}\) J.
Quick example
(a) Electron in a box of width 1 nm: \(E_1 = (6.63\times10^{-34})^2/(8\times9.11\times10^{-31}\times10^{-18}) \approx 6.0\times10^{-20}\) J \(\approx 0.376\) eV. \(E_2 = 1.50\) eV, \(E_3 = 3.38\) eV.
(b) Photon emitted for \(n = 2 \to 1\): \(\Delta E = 3E_1 = 1.13\) eV, \(\lambda = 1240/1.13 \approx 1100\) nm (infrared).
(c) A 1 g marble in a 10 cm box: \(E_1 = (6.63\times10^{-34})^2/(8\times10^{-3}\times10^{-2}) \approx 5.5\times10^{-63}\) J, utterly negligible, which is why quantisation is invisible for everyday objects.
Common mistakes
- Not squaring \(L\), or using \(L\) in Å directly.
- Calculating \(\Delta E\) as \(E_1\times(n_2 - n_1)^2\) instead of \((n_2^2 - n_1^2)E_1\).
- Leaving energy in joules when the question asks for eV.
How MU asks it
Asked as 3 to 5-mark numericals: "Find the lowest three energy levels of an electron in a box of width ...", "Find the wavelength of radiation emitted in a transition", or "Find the width of the box". Write the formula, convert units first, compute \(E_1\), then scale by \(n^2\), with units at every step.