Download Review single-particle states Consequences for two

Review single-particle states
• Notation
|...�
• … list of quantum numbers associated with a CSCO
• Normalization �α|β� = δα,β
• Continuous quantum numbers
�
�
�
�r,
m
|r
,
m
�
=
δ(r
−
r
)δms ,m�s
– Example
s
� s
|α� �α| = 1
• Completeness
α
Consequences for two-particle states
• CVS for two particles: product space
|α1 α2 ) = |α1 � |α2 �
� �
(α
α
|α
1 2 1 α2 ) = δα1 ,α�1 δα2 ,α�2
• Orthogonality �
|α1 α2 )(α1 α2 | = 1
• Completeness
• Notation
α1 α2
Identical
Particles
Exchange degeneracy
• Consider
• Then
• All states
α1 �= α2
|α2 α1 ) �= |α1 α2 )
|α1 α2 )
|α2 α1 )
c1 |α1 α2 ) + c2 |α2 α1 )
yield α1 for one particle and α2 for the other upon measurement
• Yet, unclear which state describes this system and therefore
inconsistent with quantum postulates
• Consider permutation operator
with
P12 |α1 α2 ) = |α2 α1 )
2
P12 = P21 and P12
=1
• Hamiltonian for two particles is symmetric for 1 ⇔ 2
Identical
Particles
Development
p21
p22
+
+ V (|r1 − r2 |)
• Typical Hamiltonian H =
2m 2m
• Consider operator acting on particle 1 and corresponding
A1 |α1 α2 ) = a1 |α1 α2 )
eigenvalue
• Similarly, the same operator acting on particle 2 yields
A2 |α1 α2 ) = a2 |α1 α2 )
• Note
• and
P12 A1 |α1 α2 ) = a1 P12 |α1 α2 ) = a1 |α2 α1 ) = A2 |α2 α1 )
−1
−1
P12 A1 |α1 α2 ) = P12 A1 P12
P12 |α1 α2 ) = P12 A1 P12
|α2 α1 )
• Holds for any state; therefore
• It follows that
−1
P12 HP12
= H
−1
P12 A1 P12
= A2
or
[P12 , H] = 0
Identical
Particles
Symmetric and antisymmetric two-particle states
• So
[P12 , H] = 0
• Common eigenkets either
or
�
1 �
|α1 α2 �+ = √ |α1 α2 ) + |α2 α1 )
2
�
1 �
|α1 α2 �− = √ |α1 α2 ) − |α2 α1 )
2
• Eigenstates of the Hamiltonian either symmetric ⇒ bosons
or antisymmetric ⇒fermions
Identical
Particles
Fermions
• Antisymmetry:
|α2 α1 � = − |α1 α2 �
• Both kets represent the same physical state: count only once in
completeness relation “order” quantum numbers
• Ordered:
�
i<j
|1� , |2� , |3� , ...
|ij� �ij| = 1
1 �
|ij� �ij| = 1
• Not ordered:
2! ij
Identical
Particles
N-particle states (fermions)
• Product states
• Normalization
|α1 α2 ...αN ) = |α1 � |α2 � ... |αN �
�
(α1 α2 ...αN |α1� α2� ...αN
)
• Completeness
�
α1 α2 ...αN
=
=
�
�α1 |α1� ��α2 |α2� �...�αN |αN
�
δα1 ,α�1 δα2 ,α�2 ...δαN ,α�N
|α1 α2 ...αN )(α1 α2 ...αN | = 1
• Identical particles: symmetric or antisymmetric states
1 �
• Fermions: use antisymmetrizer A =
(−1)p P
N! p
• Permutation operator product of two-particle permutations
• # of two-particle permutations odd/even ⇒ sign
Identical
Particles
Example for 3 particles
• Check odd/even permutation
1 �
|α1 α2 α3 � = √ |α1 α2 α3 ) − |α2 α1 α3 ) + |α2 α3 α1 )
6
�
− |α3 α2 α1 ) + |α3 α1 α2 ) − |α1 α3 α2 ) .
• Note normalization (6 states)
• Also note antisymmetry |α1 α2 α3 � = − |α2 α1 α3 �
• No two fermions can occupy the same state!!
Identical
Particles
N fermions
• Completeness with ordered single-particle (sp) quantum numbers
ordered
�
α1 α2 ...αN
• Not ordered
|α1 α2 ...αN � �α1 α2 ...αN | = 1
�
1
|α1 α2 ...αN � �α1 α2 ...αN | = 1
N ! α α ...α
1
2
N
• Normalization with ordered single-particle (sp) quantum numbers
�
�α1 α2 ...αN |α1� α2� ...αN
�
=
=
• Not ordered ⇒ determinant
�
�α1 |α1� ��α2 |α2� �...�αN |αN
�
δα1 ,α�1 δα2 ,α�2 ...δαN ,α�N
�
�
� �α1 |α1� � �α1 |α2� � . . . �α1 |αN
�
�
�
� �α2 |α1� � �α2 |α2� � . . . �α2 |αN
�
�
�
�α1 α2 ...αN |α1� α2� ...αN
�=�
..
..
..
.
.
�
.
.
.
.
�
� �αN |α� � �αN |α� � . . . �αN |α� �
1
2
N
Identical
Particles
�
�
�
�
�
�.
�
�
�
Normalized N-particle wave function
• Called a Slater determinant
�
� �x1 |α1 � . . . �xN |α1 �
�
�
1 � �x1 |α2 � . . . �xN |α2 �
ψα1 α2 ...αN (x1 x2 ...xN ) = √ �
..
..
..
�
N! �
.
.
.
� �x1 |αN � . . . �xN |αN �
• Hard to work with Slater determinants
�
�
�
�
�
�.
�
�
�
• Use occupation number representation or second quantization
Identical
Particles
Second quantization
• Motivation:
– Slater determinants tedious to work with
– Relevant operators change only the quantum numbers of one or two
particles (and in exceptional cases three)
• Consider states that are labeled by the # of particles occupying
sp states ⇒ occupation number representation
• Allow states in CVS with different # of particles ⇒ Fock space
• Includes new state: the vacuum
– all sp states
– all antisymmetric two-particle (tp) states
– ..
– all antisymmetric N-particle states
– up to infinite number of particles
|0�
{|α�}
{|α1 α2 �}
{|α1 α2 ...αN �}
…..
Identical
Particles