The three lowest energy states of an infinitely deep square well (of width L, between x=0 and x=L) are:

Y₁(x,t) = N sin(px/L) e^iw₁t
Y₂(x,t) = N sin(2px/L) e^iw₂t
Y₃(x,t) = N sin(3px/L) e^iw₃t

• N = sqrt(2/L) is the normalization, to make the total probability = 1.
• Each wave function oscillates with a different frequency, w_i = 2pf_i = 2pE_i/h. This relation between energy and frequency is the same as for photons.

a) Suppose a particle in the well is described by the wave function Y₂. If you measure its energy, what result will you obtain, in terms of the ground-state energy E₁?

E = E₁

b) Suppose the particle's wave function is

Y = Y₁ + Y₂ + Y₃

P(E₁) =    
P(E₂) =    
P(E₃) =

c) Suppose you measure the energies of a large number of particles that are each described by the wave function in part b. What will be the average value of your energy measurements?

<E> = E₁