A particle is confined to & 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 λ_n inside the box must fall in the range 2L/n < λ_n < 2L/(n − 1). 

(b) Show that the energy levels E_n in the finite box satisfy

 where E₁ = h²/(8mL²) is the ground-state energy for a box of length L with infinite potential energy outside the box.

Transcribed Image Text:

(n-1)² E₁