A particle of mass m is moving in a one-dimensional potential. (x) are the orthonormal energy eigenstates

(their explicit form is not needed) and the energy eigenvalues are En nE with n= 1, 2, 3, ... and E some real

constant. In this problem no integrations are needed; use orthonormality of the Un and/or any results derived

in class.

At t=0 the system is prepared in the state:

(a) Find a by requiring that (x, 0) is correctly normalised. You may assume that a is a real and positive

number.

[2]

(b) At t = 0 a measurement of energy is performed and the most likely energy eigenvalue is found. Which

eigenvalue is this? Find the corresponding probability and the wave function after the measurement at t > 0. [3]

[3]

(c) Calculate the expectation values (E) = (Î) and (E²) = (Ĥ²), and the energy uncertainty AE.