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PHY–396 K. Problem set #3. Due September 23, 2003.
- The first exercise is about first-quantized v. second-quantized forms of one-body and two-body
operators acting on identical bosons. In class, we wrote the wave function of an N –particle
state |α 1
,... , α N
{n β
as
φ α 1 ,...,αN
(x 1
,... , x N
β
n β
N!
distinct permutations
(˜α 1 ,...,α˜N ) of (α 1 ,...,αN )
φ α˜ 1
(x 1
) · · · φ α˜N
(x N
and we defined the annihilation operators ˆa α
according to
ˆa α
{n β
nα
{n
′
β
= n β
− δ αβ
(a) Consider an N –particle state |N, Ψ〉 with a completely generic totally-symmetric wave
function Ψ(x 1 ,... , x N
). Show that the (N − 1)–particle state |(N − 1), Ψ
′
〉 = ˆaγ |N, Ψ〉
has wave function
′
(x 1
,... , x N − 1
N
d
3
x N
φ
∗
γ
(x N
) Ψ(x 1
,... , x N − 1
, x N
Hint: First verify this formula for Ψ of the form (1), and then generalize to arbitrary (but
totally-symmetric) Ψ by linearity.
Now consider a one-body operator
R
1
. In the first-quantized formalism
R
tot
acts on N –particle
states according to
R
(1)
tot
N ∑
i=
R
1
(i
th
particle) (4)
while in the second-quantized formalism it becomes
R
(2)
tot
α,β
〈α|
R 1 |β〉 ˆa
†
α
ˆa β
(b) Use eq. (3) to verify that for any two N –particle states 〈N, Ψ 1
| and |N, Ψ 2
〈N, Ψ
1
R
(1)
tot
|N, Ψ
2
〉 = 〈N, Ψ
1
R
(2)
tot
|N, Ψ
2
Hint: Use
R
1
α,β
|α〉 〈α|
R
1
|β〉 〈β|.
Next, consider a two-body operator
S
2
which acts in the first-quantized formalism according
to
S
(1)
tot
1
2
i 6 =j
S 2 (i
th
and j
th
particles) (7)
and in the second-quantized formalism according to
S
(2)
tot
α,β,γ,δ
〈α| ⊗ 〈β|
S
2
|γ〉 ⊗ |δ〉 ˆa
†
α
aˆ
†
β
ˆa γ
ˆa δ
(c) Again, show that for any two N –particle states 〈N, Ψ 1
| and |N, Ψ 2
〈N, Ψ 1 |
S
(1)
tot
|N, Ψ 2 〉 = 〈N, Ψ 1 |
S
(2)
tot
|N, Ψ 2 〉. (9)
(d) Finally, let
A
1
be a one-body operator, let
B
2
and
C
2
be two-body operators, and let
A,
B and
C be the corresponding second-quantized operators defined similar to eqs. (5)
and (8).
Show that if
C
2
[(
A
1
st
) +
A
1
nd
)
B
2
]
then
C = [
A,
B]
Hint: First, calculate the commutator [ˆa
†
αˆa
†
β
ˆaγ ˆa δ
, aˆ
†
μˆaν ].
- The second problem is about coherent states of harmonic oscillators and free quantum fields.
Let us start with a harmonic oscillator
H = ¯hωˆa
† ˆa.
(a) For any complex number ξ we define a coherent state |ξ〉
def
= exp
ξˆa
† − ξ
∗ ˆa
| 0 〉. Show
that
|ξ〉 = e
−|ξ|
2 / 2
e
ξˆa
†
| 0 〉 and ˆa |ξ〉 = ξ |ξ〉. (10)
(b) Calculate the uncertainties ∆q and ∆p for a coherent state |ξ〉 and verify their minimality:
∆q∆p =
1
2
¯h. Also, verify δn =
n¯ where ¯n
def
= 〈nˆ〉 = |ξ|
2
.
Hint: use ˆa |ξ〉 = ξ |ξ〉 and 〈ξ| ˆa
† = ξ
∗ 〈ξ|.
(c) Show that for ξ(t) = ξ 0
e
−eωt the coherent state |ξ(t)〉 satisfies the time-dependent
Schr¨odinger equation i¯h
d
dt
|ξ(t)〉 =
H |ξ(t)〉.
