CEM 993 Spring 2014 Assignment No. 2 (20 pts.) Due Date

CEM 993
Spring 2014
Assignment No. 2 (20 pts.)
Due Date: February 14, 2014
1. (2 pts.) Use Wick’s theorem to show that
Xp Xq Xr† Xs† = Xr† Xs† Xp Xq + δpr Xs† Xq − δps Xr† Xq − δqr Xs† Xp + δqs Xr† Xp + δps δqr − δpr δqs .
(1)
2. (5 pts.) Use Wick’s theorem and rules of calculating the expectation values of normal
products n[. . .] with or without contractions relative to the true vacuum to evaluate
the following matrix element:
h{p1 p2 }|Z|{q1 q2 }i,
where
Z=
X
r,s
(2)
hr|z|siXr† Xs
(3)
is a one-body operator and
|{p1 p2 }i = Xp†1 Xp†2 |0i
(4)
|{q1 q2 }i = Xq†1 Xq†2 |0i
(5)
and
are the 2-fermion Slater determinants (as usual, |0i designates the true vacuum state).
Do not use the general Slater rules for matrix elements involving one-body operators
derived in class, but, rather, calculate the matrix element given by Eq. (2) directly
by applying to it the rules of determining the expectation values of the products of
the creation or annihilation operators in the true vacuum state, resulting from Wick’s
theorem. Once you are done, demonstrate that the resulting expression for the matrix
element given by Eq. (2) recovers the Slater rules for the two cases that yield the
nonzero results, i.e., h{p1 p2 }|Z|{p1 p2 }i and h{p1 p2 }|Z|{q1 p2 }i.
3. (3 pts.) Calculate the following N -products:
(a) N [Xa Xb Xi† Xj† ],
(b) N [Xi Xj Xa† Xb† ],
(c) N [Xi Xa Xj† Xb† ].
4. (4 pts.) Let χ(p) and π(p) be the “step” functions defined as follows:
χ(p) =
Show that
(
1, for p = i
,
0, for p = a
π(p) =
(
0, for p = i
.
1, for p = a
(6)
(a) Yp Xq = χ(q)δpq ,
(b) Yp Xq† = π(q)δpq ,
(c) Yp† Xq = 0,
(d) Yp† Xq† = 0.
5. (6 pts.) Let ZN be the one-body operator in the normal-product form,
ZN =
X
p,q
hp|z|qiN [Xp† Xq ],
(7)
and let T1 be the one-body particle-hole excitation operator,
T1 =
X
i,a
ha|t1 |iiN [Xa† Xi ].
(8)
Use the hole-particle variant of Wick’s theorem to prove that
[ZN , T1 ] |Φi =
X
a,i


X
b
ha|z|bihb|t1 |ii −
X
j


X
ha|t1 |jihj|z|ii |Φa i+ ha|t1 |iihi|z|ai |Φi,

i
a,i
(9)
where [ , ] is the commutator, |Φi is the Fermi vacuum state, and
are the singly
excited determinants relative to |Φi. You will have a chance to prove Eq. (9) using
diagrams later. This equation is a demonstration of the fact (used, for example, in
coupled-cluster theory) that commutators of many-body operators produce connected
quantities (diagrams) only.
|Φai i

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