Tfy-44.192 MANY-PARTICLE PHENOMENA
4. Exercise 9.10.2000; return solutions before 9.10. 11.00 to the box in the lobby.
Salomaa/Virtanen
1. Verify in detail that (Fetter & Walecka (8.29))
i.e. that the contraction of a field operator and its adjoint is the single-particle Green’s function of the
noninteracting system.
2. Consider one-dimensional quantum anharmonic oscillator, i.e. a particle in a potential
#% ),+
"!$#
'&(!*)
Express its Hamilton operator in terms of the standard operators - and - of the harmonic oscillator,
/ 10 /
and calculate the difference .
, where . denotes normal ordering.
3. (Fetter & Walecka, Problem 3.9.)
Make the canonical transformation to particles and holes for fermions
23*4 657 ! 0 !$8 -,3*4 % 75 *! 8 0 ! 9:; 3*4 +
By applying Wick’s theorem, prove the relation
2 < 2 # 2 ) 2= . 2 < 2 # 2 ) 2= % 75 ! 8 0 $! # > ? @# ) . 2 <
% 57 !*8 0 ! < > ? < = . 2 #
% 57 !*8 0 ! < 5E !$8 0 !
2= A0B?
2 ) A0B?
:> ? < ?
# =
< DC
#=. 2 2)
< 2 2 = DC
) . <#
0F? ? = CD
#@)
) #
where the normal-ordered products on the right side now refer to the new particle and hole operators,
HGAJIE
and the subscripts indicate the quantum numbers
.
4. (Fetter & Walecka, Problem 3.10.)
Verify the cancellation of disconnected diagrams [Eqs. (8.11), (9.3), and (9.4)] explicitly to second
order in the interaction potential.
<
5. (Fetter & Walecka, Problem 3.11.)
Consider a = system of noninteracting spin- # fermions in an external static potential with a hamiltonian
/LKDM ONQP HR HR HR
.
a) Use Wick’s theorem to find the Feynman rules for the single-particle Green’s function in the presence of the external potential.
b) Show that Dyson’s equation becomes
; < W =YX
DK M @O H S0B %UVT
P 4 HS0 X Z4 4Y[ ]\$ DK^4 M [ X @ +
c) Express the ground-state energy in a form analogous to Eqs. (7.23) and (7.31). What happens if the
particles also interact?