Page 114 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Problems 105

(a) Calculate hAi.
(b) Express the normalized wave function ö of the system in terms of the
^
eigenfunctions of A.
3.17 The wave function Ø(x) for a particle in a one-dimensional box of length a is
ðx
Ø(x) ˆ C sin 7 ; 0 < x < a
a
where C is a constant. What are the possible observed values for the energy and
their respective probabilities?
^
3.18 If jøi is an eigenfunction of H with eigenvalue E, show that for any operator A ^
^
^
the expectation value of [H, A] vanishes, i.e.,
^
^
høj[H, A]jøiˆ 0
3.19 Derive both of the Ehrenfest theorems using equation (3.72).
3.20 Show that
"
ÄHÄx > h^ p x i
2m