A particle of mass m moves in a one-dimensional box of length L. (Take the potential energy of the particle in the box to be zero so that its total energy is its kinetic energy p²/2m.) Its energy is quantized by the standing-wave condition n(λ/2) = L, where λ is the de Broglie wavelength of the particle and n is an integer. 

(a) Show that the allowed energies are given by E_n = n²E₁, where E₁ = h²/8mL². 

(b) Evaluate E_n for an electron in a box of size L = 0.1 nm and make an energy-level diagram for the state from n = 1 to n = 5. Use Bohr’s second postulate f = ΔE/h to calculate the wavelength of electromagnetic radiation emitted when the electron makes a transition from 

(c) n = 2 to n = 1,

(d) n = 3 to n = 2, and 

(e) n = 5 to n = 1.