Exercises for T2, Summer term 2016, 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−1 ? Compare the obtained results with the size of a
proton and the size of a human. What is your conclusion?
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
D u
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
pp pp pp ppp ppp
pp pp pp pp
pp pp Q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
ppppppppppp
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