Download Homework 2

Prof. Dr. S. Dietrich
Dr. M. Gross ([email protected])
MSc. G. Zarubin
Advanced Quantum Theory
winter semester 2016/17
Problem sheet no. 2
(http://www.is.mpg.de/dietrich/teaching/AQT)
31 October 2016
4. Compton effect
An incident photon with wave vector k0 is scattered from a particle with rest mass m0
and initial momentum p0 = 0. The scattered photon has the wave vector k.
Based on the conservation of the relativistic energy and momentum, determine the shift
λ − λ0 of the wave length of the scattered photon as a function of the scattering angle θ
between k and k0 .
(Remark: Special relativity yields the relation E 2 = c2 p2 + m2 c4 between the energy E
and the momentum p of an object of rest mass m.)
5. Bohr model
Consider the motion according to classical mechanics of an electron of charge −e in the
electrostatic field of a proton of charge +e.
(a) Determine the effective potential for the motion of the electron around the nucleus as
a function of the radial distance.
(b) According to the Bohr model, electrons move on circular orbits and the angular momentum L can assume the values L = n~, n ∈ {1, 2, . . .}. Determine the possible
energies En , orbital radii rn , and velocity ratios vn /c, where c is the speed of light.
(c) Bohr assumed that only radiation of frequency
ω(n1 , n2 ) =
En1 − En2
,
~
n1 > n2
(1)
can be emitted. Compare ω(n + 1, n) for n → ∞ with the orbital frequency which
results from classical mechanics using the results of part (b).
(d) For velocities v c, the power radiated by an accelerated charge is given by the
classical expression
e2
P =
v̇2
(2)
6π0 c3
in SI units. For the orbit corresponding to energy E1 , estimate the time during which
the electron (considered to be a classical particle) would crash into the nucleus.
please turn over
1
6. Fourier transform
Given a function f : R → C, its Fourier transform f¯, provided it exists, is defined by
Z ∞
dx
¯
√ f (x) exp(−iqx).
f (q) :=
(3)
2π
−∞
The following relations hold:
Z ∞
dq
√ f¯(q) exp(iqx)
f (x) =
2π
−∞
Z ∞
Z ∞
2
dx |f (x)| =
dq |f¯(q)|2
−∞
Consider the Gaussian
(inverse Fourier transform)
(4)
(Parseval’s theorem).
(5)
−∞
x2
f (x) = √
exp − 2 .
2σ
2πσ 2
Calculate the Fourier transform f¯, plot f (x) and f¯(q), and check Eqns. (4) and (5).
1
2
(6)