Download Series 5 - Problems

Physics IV
Prof. Dr. Simon Lilly
SS 2006
Series 5 - Problems
Tips: 18/22 May 2006 / Hand in: 1 June/29 May 2006 / Return: 8/12 June 2006
1
1D wave packet
At time t = 0, we consider a 1D wave packet ψ(x, 0), with average position x0 and average
momentum p0 , defined by:
x2
ψ(x, 0) = eip0 x/~ f (x − x0 ) with f (x) = C exp − 2
(1)
2σ
The Fourier transform of f is:
1
f˜(p) = √
2π~
Z
+∞
e
−∞
−ipx/~
2 2
p σ
σ
f (x)dx = C √ exp − 2
2~
~
(2)
a) Give the expression for ψ̃(p, 0) and draw the shape of |ψ(x, 0)|2 and |ψ̃(p, 0)|2 .
b) The wave packets evolves freely. We note Ĥ =
the expression for ψ̃(p, t).
p2
2m
is the system hamiltonian. Determine
c) Do the 1st order approximation in p of H:
H(p) ≃ H(p0 ) + (p − p0 )
∂H
∂p
(3)
p=p0
Deduce an expression for ψ(x, t). Give the shape of |ψ(x, t)|2
d) Same question as previously but developing H up to the 2nd order in p.
e) Apply those results to an electron (m ≃ 10−30 kg) and to a dust grain (m ≃ 10−15 kg).
We’ll take σ = 10−6 m.
2
Detection of a weak magnetic field
There is a strange effect in quantum mechanics that shows that a magnetic field B can influence the
motion of electrons even if it exists in regions where the probability of finding them is arbitrarily
weak. The influence shows up through interference phenomena.
a) Consider the Schroedinger equation for electrons with a vector potential A(r) and a potential
U (r) that are time independant:
1
(P − eA)2 ψ + U ψ = Eψ
2m
a.1) In the region where the magnetic field B is zero, show that:
" Z
#
ie M
(0)
ψ(r) = ψ (r) exp
A(r) · dl
~ M0
(4)
(5)
is a solution of equation (4). ψ (0) being the solution for the same equation but with A = 0.
The curvilinear integral is calculated along a path (γ) that goes from the fixed point M0 ,
located in r0 to the point M located in r
1
Source of r
electrons 0
111111
000000
000000
111111
000000
111111
000000
111111
000000
111111
000000
111111
Magnetic
flux
r
Detector
Figure 1: Experimental setup
a.2) Show that the integral doesn’t depend on the path as long as this one is in a region where
B = 0.
b) In the experimental setup described on Fig. 1, an electron beam starting from r0 is split in
two and recombines in the interference region r. This is similar to the Young holes in optics
except that the grey region, between the beams, contains a magnetic field B perpendicular
to the surface of the figure while it is 0 on the path of the beams.
Let ψ1 (r) and ψ2 (r) denote the amplitude of the wave function of each beam at location r.
(0)
(0)
We suppose that, when the magnetic flux is zero, we have ψ1 (r) = ψ2 (r).
What is the expression
R R of the presence probability I of the electron in r as a function of the
B · n dS?
magnetic flux φ =
S
Compare the quantum mechanical and classical case.
c) This interference phenomenon is observed in structures of semi-conductors. It allows to
detect very weak variations in the magnetic field. Suppose that the surface enclosed by the
paths is 20 µm × 20 µm. Compute the magnetic field variation ∆B that can be detected
thanks to this interferometric device?
Compare to the terrestrial magnetic field of 40 µT.
3
Particle in a box II
As a simple (but instructive) example of time evolution, let’s consider the first physical scenario
we learned for time-independent quantum mechanics - the particle in a box. Take V (x) = 0 for
0 < x < L and V (x) = ∞ everwhere else.
a) What are the energy eigenstates, the energy eigenvalues (in terms of the quantum number
n) and the spacing between energy levels for this physical situation?
b) Let’s make it move. What is now ψn (x, t)? Do the wavefunction and the phase change?
c) Calculate |ψn (x, t)|2 . Is it time-dependent? Plot this probability distribution for n=1,2,3.
d) Are the energy eigenstates stationary states? Is this result true, in general?
e) Now consider a linear combination of energy eigenstates, φ = c1 ψn1 + c2 ψn2 . In general, is
φ an energy eigenstate? If yes, find its eigenvalues. If no, are there any situations in which
it could be an eigenstate?
2
f) Find φ(x, t). Write your answer in terms of c1 (t) and c2 (t). What does this tell you (physically) about (x, t)? What does the relative contribution of each ”basis” state look like over
time? Does φ(x, t) ever equal φ(x, 0) for t 6= 0? Compute |φ(t = 0)|2 then |φ(t)|2 . Take
n1 = 1, n2 = 2 and c1 = c2 = √12 . Sketch the probability distribution for several timesteps.
Does |φ(t)|2 ever equal |φ(0)|2 ?
g) What is the expectation value for the time-evolution operator for this state? i.e. calcuR +∞
it
late −∞ φ∗ e− ~ H φdx. Try the general (n1 , n2 ) case first, then derive the specific case of
(1,2). This expectation value is the probability amplitude for the state φ(t) beeing equal
toR its initial state at t = 0. We call this the survival probability amplitude. Calculate
it
+∞
| −∞ φ∗ e− ~ H φdx|2 for general and specific values of n.
R +∞
R +∞
h) Calculate −∞ ψn∗ (t) x ψn (t)dx and | −∞ ψn∗ (t) x ψn (t)dx|2 . (Hint: No integral needed,
remember that the ψn ’s are stationary states.)
R +∞
R +∞
Calculate −∞ φ∗ (t) x φ(t)dx and | −∞ φ∗ (t) x φ(t)dx|2 .
3