A particle is confined to a finite box of length L. In the nth state, the wave function has n − 1 nodes. The wave function must make a smooth transition from sinusoidal inside the box to a decaying exponential outside-there can't be a kink at the wall.
(a) Make some sketches to show that the wavelength λn inside the box must fall in the range 2L/n < λn < 2L/(n − 1).
(b) Show that the energy levels E n in the finite box satisfy (n − 1)2 E1 < En < n2 E1, where E1 = h2 /(8 mL2) is the ground-state energy for an infinite box of length L.