A particle of mass m is in a one-dimensional infinite square well that extends from x = –a to x = a.
a) Find the energy eigenfunctions ψn (x) and corresponding eigenvalues En of this particle. (Hint: you may use the results of the book for an infinite square well between x=0 and x=a, appropriately modified!)
b) The parity operator Π is defined as: Π ψ(x) = ψ (–x) for any function ψ(x). Does Π commute with the Hamiltonian H of this particle?
c) Are the energy eigenfunctions ψn (x) also eigenfunctions of Π and, if yes, with what eigenvalue each?
The wavefunction of the particle at some initial time is ψ = C sin |πx/a| , with C a real positive constant. ( ψ = 0 for |x| > a )
d) Normailize the wavefunction by calculating the appropriate value of C.
e) Calculate the expectation value of the energy of this particle.
f) Is the above wavefunction an eigenfunction of Π and, if yes, with what eigenvalue?
g) What is the probability that a measurement of the energy of this particle will yield the value E2 ? (Hint: the result of (c) and (f) may help you.)
Can anyone help me with this? Thanks.