Chapter 3
Quantum statistics
“Each photon then interferes only with itself. Interference between two di
ff
erent photons never
occurs.”
– Paul Dirac, The Principles of Quantum Mechanics, Fourth Edition, Chapter 1
Note 1:
Quantum interference is interference between di
ff
erent possibilities (no matter whether they
follows the classical physical laws) of a single object, being one photon, two photons, or else, no be
tween di
ff
erent objects. Ignore those hairsplitters.
Note 2:
Possibilities which can interference must be associated with probability
amplitudes
(complex
numbers), instead of probabilities (positive real numbers) as in classical statistics.
Note 3:
Probabilities! Thus PHYS 5510.
3.1
Quantum measurement
In quantum mechanics, the state of an object is a vector

ψ
⟩
in the Hilbert space. A physical
quantity is an Hermitian operator
O
. The eigen states
{
ψ
n
⟩}
of the operator satisfy
O

ψ
n
⟩
=
O
n

ψ
n
⟩
,
(3.1)
where the eigen value
O
n
is a real number. The eigen states form a complete orthogonal basis,
in which the operator can be written as
O
=
∑
n

n
⟩
O
n
⟨
n

.
(3.2)
In general, the quantum object can be in a superposition of the eigen states,

ψ
⟩
=
∑
n
C
n

ψ
n
⟩
,
where
C
n
’s are complex numbers satisfying the normalization condition
∑
n

C
n

2
=
1.
The
state vector is defined up to a phase factor, i.e.,
e
i
ϕ

ψ
⟩
and

ψ
⟩
describe the same physical state,
so a physical state is indeed represented by a vector ray in the Hilbert space. A particularly
interesting observable is the Hamiltonian
H
which determines the evolution of a quantum state
by the Schr¨odinger equation
i
~
∂
t

ψ
⟩
=
H

ψ
⟩
.
(3.3)
29
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CHAPTER 3. QUANTUM STATISTICS
The eigen values
{
E
n
}
of Hamiltonian are energy of the eigen states. A superposition of di
ff
erent
energy eigen states evolves as

ψ
(
t
)
⟩
=
∑
n
e
−
iE
n
t
/
~
C
n

n
⟩
.
Randomness is intrinsic in quantum mechanics. This is due to the following quantum mea
surement postulate:
For a quantum object in the state

ψ
⟩
, the measurement of the physical observable
O
will yield
randomly
an eigen value
O
n
with
probability p
n
=
⟨
n

ψ
⟩
2
=
⟨
n

ψ
⟩⟨
ψ

n
⟩
.
Thus in general a physical quantity cannot be determined with 100% precision no matter how
well the measurement could be done. For systems at the same quantum state, the measurement
results would not be the same but there would be random results with distribution
p
n
.
From the quantum measurement postulate, the average or expectation value of an observable
is
⟨
O
⟩
=
∑
n
p
n
O
n
=
∑
n
⟨
n

ψ
⟩⟨
ψ

O

n
⟩
=
∑
n
⟨
ψ

O

n
⟩⟨
n

ψ
⟩
=
⟨
ψ

O

ψ
⟩
.
(3.4)
Note that unlike in classical ensembles, the expectation value may never be expected in quantum
mechanics. For example, for a spin1
/
2, the output of
s
z
is either
+
1
/
2 or
−
1
/
2. The quantum
object in the state (

+
1
/
2
⟩
+
 −
1
/
2
⟩
)
/
√
2 would yield expectation value of
s
z
to be zero, but
none of the outputs would be expected to be zero. The variance of the observable is
∆
O
2
=
⟨
O
2
⟩ − ⟨
O
⟩
2
=
⟨
ψ

O
2

ψ
⟩ − ⟨
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 Spring '10
 DrKwong
 mechanics, Photon, Statistical Mechanics, Quantum entanglement, density matrix, Quantum state

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