Oct. 30, 2002
Quantum Physics 1
Exercise No. 3: Second Quantization
1. Calculate the following commutators for fermions:
(a) [Ntot , (a†i )n ]
(b) [Ntot , ani ]
(c) [Ntot , A†ij ]
(d) [A†ij , A†kl ]
(e) [Aij , A†kl ]
where A†ij = a†i a†j and Ntot =
P
i
Ni =
P
i
a†i ai .
2. Show that for fermions the particle number operator N =
mutes with the Hamiltonian
H=
X
hi|T |jia†i aj +
ij
P
i
a†i ai com-
1X
hij|V |klia†i a†j al ak .
2 ijkl
3. Consider the Hamiltonian
H = a† a +
X
k a†k ak + a† a
k
X
Mk (ak + a†k ),
k
where is the fermion energy of a particular state, k is the energy
of a phonon (which is a boson) with a wave vector k and Mk are kdependent coupling constants. Define the following transformation for
any operator A:
Ā = eS Ae−S = A + [S, A] +
1
[S, [S, A]] + . . .
2!
where S is an operator. In particular we take S = a† a
†
Mk
k k (ak
P
− ak ).
(a) Explain the possible physical meaning of every term in the Hamiltonian.
(b) Show that ā = aX, ā† = a† X † , āk = ak − Mkk a† a, ā†k = a†k − Mkk a† a
h
where X = exp −
†
Mk
k k (ak
P
i
− ak ) (note that X † = X −1 ).
M2
(c) Show that H̄ = a† a( − ∆) + k k a†k ak where ∆ ≡ k kk . Verify
that this form agrees with your initial interpretation of H.
P
1
P