Exercises for T2, Summer term 2017, Sheet 2
1) De Broglie wavelength
Calculate the de Broglie wavelength of a proton with a kinetic energy of 1 eV, 100 eV,
100 keV (mp ' 938 MeV/c2 ). What is the de Broglie-wavelength of a human with a mass
of 70 kg, who is moving at 1 m/s? Compare the obtained results with the size of a proton
and the size of a human. What is your conclusion concerning the wave character of a
human being, compared to that of a proton?
2) De Broglie wavelength of nonrelativistic particles
Write down the general (relativistic) relation between the de Broglie wavelength and the
kinetic energy T = E − mc2 of a massive particle (m 6= 0). Find an approximation for
the nonrelativistic case (T mc2 ).
3) Scattering of neutrons
Q uQ
Q
Q
Q
~eout *
Q
Q
ein
QQ ~
QQ
sQ
Q
Q
Q
Q ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp Qpp pp pp pp pp pp pp pp pp pp
p p p pp pp
p p p p
pp pp Q
pp pp ppp ppp ppp ppp ppp ppp ppp ppp ppp
pp pp pp pp pp pp pp pp pp pp pp
D u
crystal
The atoms of a crystal lattice are located at the points ~x~n = a~n, ~n ∈ Z3 , ni = −N, −N + 1, . . . , N
(i = 1, 2, 3). Furthermore there is a source Q (RQ N a) at the point ~xQ = −RQ ~ein
(|~ein | = 1) which emits neutrons with momentum p. In addition a neutron-detector
(RD N a) is located at ~xD = RD ~eout (|~eout | = 1). The amplitude hD out|Q ini, that the
detector D detects a neutron which originated from Q is of the form
hD out|Q ini ∼
X eip|~xQ −~x~n |/h̄
~
n
|~xQ − ~x~n |
W~n
eip|~xD −~x~n |/h̄
.
|~xD − ~x~n |
(multiple scattering is omitted). Calculate this expression assuming
that W~n is the same
q
for all atoms. Use a suitable approximation for |~xQ − ~x~n | = (~xQ − ~x~n )2 which should
refect that |~xQ | = RQ |~x~n |. (analogous for |~xD − ~x~n |.)
By following that strategy you should get an expression like
hD out|Q ini ∼
X
~ n/h̄
ipa∆·~
e
=
3
N
Y
X
eipa∆i ni /h̄ ,
~ = ~ein − ~eout
∆
i=1 ni =−N
~
n
This means it is necessary to calculate a geometric series of the type
s(α) =
N
X
eiαn
n=−N
Now show that this equals
s(α) =
sin α(N + 12 )
sin α2
Discuss the behavior of this function. For which values of α are there distinct extrema?
Show that in the case of neutron scattering
pa ~
∆ = ~ν ∈ Z3
2πh̄
leads to distinct maxima of the interference pattern. This is Laue’s condition for interference for the simple cubic lattice. It is also possible to write it in the form
a ~
kin − ~kout = ~ν ∈ Z3
2π