(a) Show explicitly that [x, p?_x] = i?, i.e. that the momentum operator p?_x = (? / i ? / ?x) does not commute with position. (Hint: apply the commutator [x, p?_x] = xp?_x - p?_xx to an arbitrary function f(x)).
(b) A quantum mechanical problem for a Hamiltonian ? has time-independent (normalized) solutions ?_i(x) with corresponding energies E_i (i.e. ??_i(x) = E_i?_i(x)). An arbitrary wavefunction ?(x, t) in this system can be obtained as a linear combination of the ?_i(x)’s like:
?(x, t) = ?_{i=0}^? c_i ?_i(x) e^{-iE_it/?}, (1)
where the c_i’s are coefficients (complex in general). If ?(x, t) is known at t = 0 (?(x, 0) ? f(x)), show how you obtain the coefficients c_i. (Hint: use the orthogonality of the ?_i(x)’s).
(c) For the wavefunction in Eq. 1, show that energy (i.e. the expectation value of ?) is constant over time (conservation of energy in quantum mechanics). Show also that, regardless of the particular ?(x, t) being used, the condition:
?_{i=0}^? |c_i|^2 = 1 (2)
always holds (if it holds at t = 0). Describe (very briefly) the physical meaning of this result.
(d) Given the commutation relations for the different Cartesian components of the angular momentum:
[L_x, L_y] = i? L_z; [L_y, L_z] = i? L_x; [L_z, L_x] = i? L_y; (3)
show that one of the components, say L_z, commutes with L^2 = L_x^2 + L_y^2 + L_z^2,
i.e.:
[L^2, L_z] = 0. (4)
Comment (very briefly) on the physical significance of this result.
