**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{-x^{2}/2b^{2}}exp{ip_{0}x/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^{-31}kg,
ħ = 1.05*10^{-34 }J s, c = 3*10^{8 }m/s).

Hint: ∫_{-∞}^{+∞}dx
exp(-(ax^{2} + bx + c)) = (π/a)^{1/2} exp((b^{2} -
4ac)/(4a)).

To obtain, for example, ∫_{-∞}^{+∞}dx
x exp(-ax^{2}), differentiate
with respect to b and then set b = c = 0.

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 > E_{0}, where E_{0} 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 L^{2} and its z component L_{z} have
eigenvectors |Elm> that are simultaneous eigenvectors of H,
i.e.

H|Elm> = E|Elm>, L^{2}|Elm> = ħ^{2}l(l + 1)|Elm>, L_{z}|Elm> = ħm|Elm>,

and these eigenvectors form a complete set of states, show that [H,L^{2 }] = [H,L_{z}] = 0.