Problem 1:

An electron is moving freely in the x-direction.  At t = 0 the electron is described by the wave function (neglect spin)
Ψ(x,0) = Aexp{-x²/2b²}exp{ip₀x/ħ}.
(a)  Compute the constant A such that ∫_-∞^+∞|ψ(x,0)|² dx = 1.
(b)  Compute ∆x at t = 0.
(c)  Compute ∆p at t = 0 and show that  for the electron ∆x∆p = ħ/2.
(d)  Assume that the electron has a position uncertainty of ∆x = 10^-10 m.  Compute its velocity uncertainty compared to the speed of light.  (m_e = 9.1*10^-31kg,  ħ = 1.05*10^-34 J s,  c = 3*10⁸ m/s).
Hint:  ∫_-∞^+∞dx exp(-(ax² + bx + c)) = (π/a)^1/2 exp((b² - 4ac)/(4a)).
To obtain, for example, ∫_-∞^+∞dx x exp(-ax²), differentiate with respect to b and then set b = c = 0.

Problem 2:

The wave function of a particle at t = 0 is 
Ψ(x) = 1/L^1/2,  |x| < L/2,  Ψ(x) = 0 otherwise. 
At t = 0, what possible values of the momentum of the particle can be found, and with what probability?

Problem 3:

A particle of mass m is confined to an infinite one-dimensional potential well of width L, i.e. V(x) = 0,  0 < x < L,  V(x) = ∞ everywhere else.  At t = 0 the particle is equally likely to be found in the ground state or the firsts excited state.

(a)  What is the expectation value of the energy of the system?
(b)  Write down a properly normalized wave function to describe the system at subsequent times.
(c)  Find <p_x> for times  t > 0.

Problem 4:

Consider a quantum system for which the exact Hamiltonian is H.  Assume the quantum system is of bounded spatial extend, so that it is known rigorously that the eigenstates of H, {|Ψ_n>}, are complete. 
(a) Show that if |Ψ_n> and |Ψ_m> are two eigenstates of H with eigenvalues E_n and E_m
with E_n ≠ E_m, then <Ψ_n|Ψ_m> = 0.
(b) Suppose E_n = E_m, with n ≠ m.  Can we still have <Ψ_n|Ψ_m> = 0 ?
(c) The problem H|Ψ> = E|Ψ> is very complicated but it is suggested that we use a trial function |Ψ_trial> for |Ψ> and approximate E by E = <Ψ_trial|H|Ψ_trial>/<Ψ_trial|Ψ_trial>.
Show that E > E₀, where E₀ is the lowest eigenvalue of H.

Problem 5:

Assume that the Hamiltonian H for a quantum system is Hermitian. 
(a) Show that its eigenvalues E are real.
(b) Show that the eigenvectors |E> and |E’> corresponding to different eigenvalues E ≠ E’ are orthogonal.
(c) If the square of the angular momentum operator L² and its z component L_z have eigenvectors |Elm> that are simultaneous eigenvectors of H, i.e.
H|Elm> = E|Elm>,  L²|Elm> = ħ²l(l + 1)|Elm>,   L_z|Elm> = ħm|Elm>,
and these eigenvectors form a complete set of states, show that [H,L² ] = [H,L_z] = 0.