Übungsblatt Nr.1

Institut für Theoretische Physik
Institut für Experimentelle und Angewandte Physik
Prof. Dr. J. Repp, Prof. Dr. Klaus Richter
SS 2014
Übungen zum Integrierten Kurs: Quantenmechanik
Blatt 1
Assistants: Gerhard Münnich, Juan-Diego Urbina
Aufgabe 1: Operator Analysis in Dirac notation
a. Let  and B̂ be hermitian operators, i.e. † =  and B̂ † = B̂, where the superscript †
stands for transpose and complex conjugate, with (normalized) eigenstates |aj i and |bj i,
j = 1, 2, 3, . . . and eigenvalues aj and bj , respectively:
Â|aj i = aj |aj i
B̂|bj i = bj |bj i
Show that the following properties hold:
i) aj , bj ∈ R ∀j.
ii) haj |ak i = hbj |bk i = δjk
iii) det  = eTr{log Â} .
∞
P
dj f (x) j
x
,
f
 |ai i = f (ai ).
iv) For any function f (x) =
dxj x=0
j=0
v) Tr{ÂB̂} = Tr{B̂ Â}.
b. Show that for an unitary operator Û , i.e. Û † = Û −1 , det Û = eiα with α ∈ R.
c. Let |aj i and |bj i two orthonormal basis sets. Show that the operator Û with entries Ujk =
haj |bk i is unitary, i.e. Û † = Û −1 .
Aufgabe 2: Kommutatoren
Der Kommutator zweier Operatoren Â, B̂ wird definiert durch [Â, B̂] = ÂB̂ − B̂ Â, und
[Â, B̂](1) = [Â, B̂], [Â, B̂](n+1) = [Â, [Â, B̂](n) ].
a. Zeigen Sie mit vollständiger Induktion:
[Â, B̂]
(n)
n
X
n n−l
=
(−1)
 B̂ Âl
l
l=0
l
b. Zeigen Sie die Baker-Hausdorff-Relation
Â
e B̂e
−Â
∞
X
1
= B̂ +
[Â, B̂](n)
n!
n=1
1
Aufgabe 3: Uncertainty principle
R∞
1
˜
√
f (x)e−ikx dx its
Let f (x) be a L -integrable function, f (x) also L -integrable and f (k) =
2π −∞
Fourier transform. Moreover let the integrals
0
2
2
Z∞
x̄ =
−∞
Z∞
(∆x)2 =
−∞
Z∞
k̄ =
−∞
Z∞
(∆k)2 =
|f (x)|2 xdx
|f (x)|2 (x − x̄)2 dx
|f˜(k)|2 kdk
|f˜(k)|2 k − k̄
2
dk
−∞
exist and be finite. Show that
R∞
R∞
a.
|f˜(k)|2 dk =
|f (x)|2 dx (assume in the following, that f (x) is normalized such that these
−∞
−∞
integrals are euqal to 1) and
2
−x /4
b. ∆x∆k = 21 for f (x) = e√
.
4
2π
c. optionally: ∆x∆k ≥ 12
R∞
Hint: Use the Cauchy-Schwarz inequality
−∞
2
|f (x)| dx
R∞
−∞
∞
2
R ∗
|g(x)| dx ≥ f (x)g(x)dx ,
2
−∞
where the ∗ denotes complex conjugation.
Aufgabe 4: Gaußsche Wellenpakete von freien Teilchen
Betrachten Sie ein Wellenpaket ψ(x, t), das die freie Bewegung eines Teilchens mit Masse m im
eindimensionalen Raum beschreibt und zum Anfangszeitpunkt t = 0 durch
1
x2
ψ(x, 0) = p √ exp − 2 + ik0 x
2a
a π
gegeben ist. Die Zeitentwicklung dieses Wellenpakets wird durch die eindimensionale SchrödingerGleichung im freien Raum beschrieben.
a) Berechnen Sie die Zeitentwicklung des Wellenpakets ψ(x, t) für t > 0. Machen Sie dafür eine
Fourier-Transformation der Wellenfunktion gemäß
Z ∞
1
ψ(x, t) = √
ψ̂(k, t) eikx dk.
2π −∞
Zur Bestimmung der Zeitentwicklung der Fourier-Komponenten ψ̂(k, t), führen Sie eine FourierTransformation der Schrödinger-Gleichung durch und verwenden Sie deren Lösung, welche im
Ortsraum die oben angegebene Anfangsbedingung erfüllt.
2
Ergebnis:
(x − v0 t)2
exp − 2
+ i (k0 x − ω0 t)
ψ(x, t) = p √
2a (1 + it/τ )
a π(1 + it/τ )
mit
1
~k02
ω0 =
,
2m
~k0
v0 =
,
m
ma2
τ=
.
~
b) Zeigen Sie, dass für den zeitabhängigen Ortserwartungswert hxit , definiert durch
Z ∞
x|ψ(x, t)|2 dx ,
hxit =
−∞
gilt: hxit = v0 t.
p
c) Zeigen Sie, dass für die Standardabweichung ∆xt = h(x − hxit )2 it des Ortserwartungswerts,
definiert durch
Z ∞
2
(x − hxit )2 |ψ(x, t)|2 dx ,
h(x − hxit ) it =
−∞
a p
gilt: ∆xt = √
1 + (t/τ )2 .
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d) Betrachen Sie jetzt das Wellenpaket eines freien Elektrons, dessen Breite ∆xt=0 anfangs dem
Bohrschen Radius entspricht. Welche Breite ∆xt hätte es nach einer Sekunde freier Propagation erreicht? Bei welcher Anfangsbreite ∆xt=0 ergäbe sich formal, dass das Wellenpaket mit
Lichtgeschwindigkeit auseinanderfließt?
Aufgabe 5: Photoelectric effect
a. A beam of photons with maximum wave length λ0 = 248 nm is needed so that electrons are
emitted from Germanium (Ge) surface. Calculate the work function (the minimum energy
required to remove an electron from atomic binding) of Ge.
b. The maximum wave length of photons needed to emit electrons from Sodium (Na) surface
is λ0 = 451 nm. Calculate the kinetic energy of emitted electrons in case that the incident
photons have the wavelength λ = 400 nm.
Aufgabe 6: Black body radiation
The spectral power density emitted by a black body through a small hole is
u(ν, T ) =
hν 3
8π
,
c3 ehν/kT − 1
where ν is the photon frequency, c is the light velocity, h is the Planck’s constant, and T is the
temperature of the black body.
a. The power of photons in the frequency range [ν, ν + dν] emitted by the black body through a
small hole with the area ∆A into a solid angle ∆Ω 4π during a time ∆t can be calculated
as
∆Edν = L(ν, T )dν∆A∆Ω∆t.
Show the relation between the function L(ν, T ) and the spectral power density u(ν, T ).
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b. Calculate the frequency of photons with maximum power, νmax .
c. The total power radiated per unit area per unit time j is followed the Stefan-Boltzmann law
Z
c ∞
u(ν, T )dν = σT 4 .
j=
4 0
Proof the Stefan-Boltzmann law and show the expression of the coefficient σ. Motivate the
prefactor 4c in front of the integral.
Aufgabe 7: Sun as a black body
Consider the Sun as an ideal black body in otherwise an empty space. The Sun is at a temperature
TS = 6000 K. The radius of the Earth is RE = 6.4 × 106 m, and the radius of the Sun is RS =
7 × 108 m, and the Earth-Sun distance is d = 1.5 × 1011 m.
a. Calculate the total radiation flux arriving at the Earth.
b. Suppose the temperature of the Earth is uniform and consider the balance of incoming and
outcoming radiation flux. Calculate the Earth temperature.
Aufgabe 8: Compton scattering
As discussed in the lecture, Compton observed the scattering of X-rays from electrons. Derive the
equation for Compton scattering ∆λ = λC (1 − cos Θ), where ∆λ is the change in wavelength of
the X-ray upon scattering, λC = h/me c is the Compton wavelength, h is Planck’s constant, me the
electron mass, and Θ is the angle between forward direction and scattered X-ray (cf. lecture notes).
You need to consider the relativistic energy and momentum conservation.
Aufgabe 9: Wien’s displacement law
Wien’s displacement law states that the peak wavelength of the emission spectrum of black body
radiation is at λmax = b/T , where b = 2.89 mm K and T is the absolute temperature. Derive this
equation from the power density as a function of λ. Note that the equation is not the same as if
you would convert the result νmax from exercise 2 into a wavelength. Explain why.
Aufgabe 10: Boltzmann statistics
Consider you have a certain total energy E = N that is an integer multiple N of a certain energy
quantum available for your system. Consider now the possibilities to distribute this total energy
amongst individual particles. Each particle can only take up the energy in portions of the same
energy quantum , so its energy is also quantized. Assume that every of these possibilities has
the same probability. However, as the particles should be classical, they are distinguishable from
each other. So you have to consider the combinatorial possibilities of distributing the energies
to the individual distinguishable particles. For every possibility you have a certain occupation of
every state of an energy n with n = 1, 2, 3, . . . . Sum up all these occupation numbers for every
state multiplied by the the combinatorial possibilities. Show that the resulting numbers resemble
a Boltzmann distribution. Below, an example is given for the total energy E = 5.
a. Make an example with a higher number for N . Note that, the higher you choose N , the
better it fits to a Boltzmann distribution, but the higher is also the probability to miss a
configuration.
b. Estimate the value of kB T for the given and your own example.
Please give the solutions of exercises 2, 4, 5 and 6 in written form to Prof. Repp on Mo. 28.04.2012.
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occupation
Abbildung 1: Example for determining the occupation for a total energy of 5 × 10
1
1
2
state
3
4
5
Abbildung 2: Semilogarithmic plot of the summed occupations versus state number. The points are
almost on a straight line. State 5 should not be considered because of rounding problems.
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