Download Quantum Theory 1 - Home Exercise 3

Quantum Theory 1 - Home Exercise 3
1. Consider a system with a real Hamiltonian that occupies a stationary state having a real wave
function both at time t = 0 and t = t1 . Meaning, the wave function satisfies:
ψ ∗ (x, 0) = ψ(x, 0) ,
ψ ∗ (x, t1 ) = ψ(x, t1 ).
(a) Show that such a system must be periodic, i.e. that there exists a period T such that
ψ(x, t) = ψ(x, t + T )
,
for all t.
(b) Calculate the period T .
(c) Show that for such a system, the energy eigenvalues must be integer multiples of 2π~/T .
Hint : Assume the state ψ(x, t) has some defined energy E, then in article (c) show it must
obey the condition given.
2. Consider a normalized wave function ψ(x). Assume that the system is in a state described by
the wave function
Ψ(x) = C1 ψ(x) + C2 ψ ∗ (x),
where C1 and C2 are two known complex numbers.
(a) Write down the condition for the normalization of Ψ(x) in terms of the complex integral,
R∞
−∞ ψ(x)ψ(x)dx = D, assumed to be known.
(b) Obtain an expression for the probability current J (x) for the state Ψ(x). Use a polar
presentation of complex numbers, i.e. ψ(x) = f (x)eiθ(x) .
(c) Calculate the expectation value hpi of the momentum and show that,
Z ∞
hΨ, p̂Ψi = m
J (x)dx
−∞
Show that both probability current and momentum vanish if |C1 | = |C2 |.
3. Consider a free particle moving in one dimension. At time t = 0 its wave-function is
α
2
α
2
ψ(x, 0) = N e− 2 (x+x0 ) + N e− 2 (x−x0 ) ,
where α and x0 are known real parameters.
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(a) Compute the normalization factor |N | and the momentum wave function ψ̃(x).
(b) Find the evolved wave function ψ(x, t) at any time t > 0. Hint: Use the notation
z = 1 + i ~tα
m
(c) Write down the position probability density and discuss the physical interpretation of
each term.
(d) Obtain an expression for the probability current J (x, t).
4. Consider a free particle that is initially (at t = 0) extremely well localized at the origin x = 0
and has a Gaussian wave function,
ψ(x, 0) =
α 1/4
π
e−
αx2
2
eik0 x ,
where α is very large (α → ∞). k0 is also real.
(a) Write down position probability density ρ(x, 0) and show that
lim ρ(x, 0) = δ(x).
α→∞
(b) Keep α finite and calculate ψ(x, t) at times t > 0.
(c) Write down the evolved probability density ρ(x, t). Take the limit α → ∞ and observe
that even for infinitesimal values of time (t ∼ m/~α), the probability density becomes
space independent. Give a physical argument for the contrast of this behavior with the
initial distribution.
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