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PHY–396 K. Problem set #3. Due October 2, 2008.
- An operator acting on identical bosons can be described in terms of N –particle wave
functions (the first-quantized formalism) or in terms of creation and annihilation operators
in the Fock space (the second-quantized formalism). This exercise is about converting the
operators from one formalism to another.
The key to this conversion are the single-particle wave functions φ α
(x) of states |α〉 and
the symmetrized N -particle states wave functions
φ αβ···ω
(x 1
, x 2
... , x N
D
distinct permutations
of (α,β,...,ω)
∑
(˜α,
˜ β,...,ω˜)
φ α˜
(x 1
) × φ ˜ β
(x 2
) × · · · × φ ω˜
(x N
T
D
all permutations
of (α,β,...,ω) ∑
(˜α,
˜ β,...,ω˜)
φ α˜
(x 1 ) × φ ˜ β
(x 2 ) × · · · × φ ω˜
(x N
of N -boson states |α, β,... , ω〉. In eqs. (1), D is the number of distinct (i.e., non-trivial)
permutations of single-particle states (α, β,... , ω) and T is the number of trivial permu-
tations. In terms of the occupation numbers n γ
T =
γ
n γ
! , D =
N!
T
(a) Consider a generic N -particle quantum state |N ; ψ)〉 with a totally symmetric wave-
function Ψ(x 1
,... , x N
). Show that the (N +1)–particle state |N + 1, ψ
′ 〉 = ˆa
†
α
|N ; ψ〉
has wave function
ψ
′
(x 1
,... , x N
N + 1
N + ∑
i=
φ α
(x i
) × ψ(x 1
,... , 6 x i
,... , x N +
Hint: First prove this for wave-functions of the form (1). Then use the fact that
states |α 1
,... , α N
〉 form a complete basis of the N -boson Hilbert space.
(b) Show that the (N − 1)–particle state |N − 1 , ψ
′′ 〉 = ˆa α
|N ; ψ〉 has wave-function
ψ
′′ (x 1
,... , x N − 1
N
d
3 x N
φ
∗
α
(x N
) × ψ(x 1
,... , x N − 1
, x N
Hint: for any |N − 1 ,
ψ〉, 〈N − 1 ,
ψ| ˆa α
|N, ψ〉 = 〈N, ψ| ˆa
†
α
|N − 1 ,
ψ〉
∗
.
Now consider one-body operators, i.e. additive operators acting on one particle at a time.
In the first-quantized formalism they act on N –particle states according to
A
(1)
net
N ∑
i=
A
1
(i
th
particle) (5)
where
A
1
is some kind of a one-particle operator (such as momentum ˆp, or kinetic energy
1
2 m
pˆ
2 , or potential V (ˆx), etc., etc.). In the second-quantized formalism such operators
become
A
(2)
net
α,β
〈α|
A 1 |β〉 ˆa
†
α
ˆa β
(c) Verify that the two operators have the same matrix elements between any two N -
boson states |N, ψ〉 and |N,
ψ〉, 〈N,
ψ|
A
(1)
net
|N, ψ〉 = 〈N,
ψ|
A
(2)
net
|N, ψ〉.
Hint: use
A
1
α,β
|α〉 〈α|
A
1
|β〉 〈β|.
Finally, consider two-body operators, i.e. additive operators acting on two particles at a
time. Given a two-particle operator
B
2
— such as V (ˆx 1
− xˆ 2
) — the net B operator acts
in the first-quantized formalism according to
B
(1)
net
1
2
i 6 =j
B
2
(i
th
and j
th
particles), (7)
and in the second-quantized formalism according to
B
(2)
net
1
2
α,β,γ,δ
(〈α| ⊗ 〈β|)
B 2 (|γ〉 ⊗ |δ〉) ˆa
†
α
aˆ
†
β
ˆa γ
ˆa δ
(d) Again, show these two operators have the same matrix elements between any two
N -boson states, 〈N,
ψ|
A
(1)
net
|N, ψ〉 = 〈N,
ψ|
A
(2)
net
|N, ψ〉 for any 〈N,
ψ| and |N, ψ〉.
(e) Consider time-dependent coherent states |ξ(t)〉. Show that for ξ(t) = ξ 0
e
−iωt , the
state |ξ(t)〉 satisfies the time-dependent Schr¨odinger equation i¯h
d
dt
|ξ(t)〉 =
H |ξ(t)〉.
(f) The coherent states are not quite orthogonal to each other.
Calculate their overlap 〈η|ξ〉.
Now consider coherent states of multi-oscillator systems and hence quantum fields. In
particular, let us focus on the creation and annihilation fields
†
(x) and
Ψ(x) for non-
relativistic spinless bosons.
(g) Generalize (a) and construct coherent states |Φ〉 which satisfy
Ψ(x) |Φ〉 = Φ(x) |Φ〉 (11)
for any given classical complex field Φ(x).
(h) Show that for any such coherent state, ∆N =
N where
N
def
N |Φ〉 =
dx |Φ(x)|
2
. (12)
(i) Let
H =
dx
¯h
2
2 M
†
(x) · ∇
Ψ(x) + V (x) ×
†
(x)
Ψ(x)
and show that for any classical field configuration Φ(x, t) that satisfies the classical
field equation
i¯h
∂t
Φ(x, t) =
¯h
2
2 M
2
Φ(x, t),
the time-dependent coherent state |Φ〉 satisfies the true Schr¨odinger equation
i¯h
∂t
H |Φ〉. (13)
(j) Finally, show that the quantum overlap | 〈Φ 1
2
2 between two different coherent
states is exponentially small for any macroscopic difference δΦ(x) = Φ 1
(x) − Φ 2
(x)
between the two field configurations.
