Suppose a particle of mass m is confined to a one-dimensional box of width L. According to quantum theory, the particle's wave (with λ = h/mv) is a standing wave with nodes at the edges of the box.

(a) Show the possible modes of vibration on a diagram.

(b) Show that the kinetic energy of the particle has quantized energies given by KE = n2h2/8mL2, where n is an integer.

(c) Calculate the ground-state energy (n = 1) for an electron confined to a box of width 0.50 × 10-10 m.

(d) What is the ground-state energy, and speed, of a baseball (m = 140g) in a box 0.65 m wide?

(e) An electron confined to a box has a ground-state energy of 22 eV. What is the width of the box?