Identical particles Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: March 19, 2015)
In classical mechanics, identical particles (such as electrons) do not lose their "individuality", despite the identity of their physical properties. In quantum mechanics, the situation is entirely different. Because of the Heisenberg's principle of uncertainty, the concept of the path of an electron ceases to have any meaning. Thus, there is in principle no possibility of separately following each of a number of similar particles and thereby distinguishing them. In quantum mechanics, identical particles entirely lose their individuality. The principle of the in-distinguishability of similar particles plays a fundamental part in the quantum theory of system composed of identical particles. Here we start to consider a system of only two particles. Because of the identity of the particles, the states of the system obtained from each other by interchanging the two particles must be completely equivalent physically.
1 System of particles 1 and 2
k' , k"
k' k" k" k' a 1 2 b 1 2
Even though the two particles are indistinguishable, mathematically a and b are distinct kets for
k' k" . In fact we have a b 0 . Suppose we make a measurement on the two particle system.
k' : state of one particle and k" : state of the other particle.
We do not know a priori whether the state ket is k' k" or k" k' or -for that matter- any a 1 2 b 1 2 linear combination of the two: ca a cb b .
2 Exchange degeneracy A specification of the eigenvalue of a complete set of observables does not completely determine the state ket.
Mathematics of permutation symmetry:
Pˆ Pˆ k' k" k" k' . 12 a 12 1 2 1 2 b
Clearly
ˆ ˆ ˆ 2 P21 P12 , and P12 1.
ˆ Under P12 , particle 1 having k' becomes particle 1 having k" ; particle 2 having k" becomes particle 1 having k' . In other words, it has the effect of interchanging 1 and 2.
((Note)) Identical particles 1 11/10/2019 Matrix element of Pˆ Pˆ in terms of k' k" and k" k' 21 12 a 1 2 b 1 2
Pˆ Pˆ k' k" k" k' , 12 a 12 1 2 1 2 b
Pˆ Pˆ k" k' k' k" . 12 b 12 1 2 1 2 a
ˆ ˆ Matrix element of P21 P12
0 1 ˆ P12 1 0 or
a b
a 0 1
b 1 0
ˆ Eigensystem[ P12 ]
= 1 (symmetric):
P12 symm symm
1 1 ( k' k" k" k )' (1ˆ Pˆ k') k" symm 2 1 2 1 2 2 12 1 2 where the symmetrizer is defined by
1 Sˆ (1ˆ Pˆ ) 2 12
= -1 (anti-symmetric)
P12 anti anti
1 1 k'( k" k" k )' (1ˆ Pˆ k') k" anti 2 1 2 1 2 2 12 1 2 where the antisymmetrizer is defined by
1 Aˆ (1ˆ Pˆ ) 2 12
Identical particles 2 11/10/2019 Our consideration can be extended to a system made up of many identical particles. A transposition is a permutation which simply exchange the role of two of the particles, without touching others.
Pˆ k' k" ..... k i k i1 ..... k j ..... k' k" ..... k j k i1 ..... k i ..... ij 1 2 i i1 j 1 2 i i1 j
ˆ ˆ ˆ The transposition operators Pij are Hermitian ( Pij Pij )
ˆ 2 Pij 1
ˆ ˆ So that Pij is an unitary operator. The allowed eigenvalues of Pij are ±1. It is important to note, however, that in general
ˆ ˆ [Pij , Pkl ] 0 .
(i) The first example
ˆ Now we consider a permutation operator P123 for
Pˆ k' k" k '" k' k" k '" k '" k' k" 123 1 2 3 2 3 1 1 2 3 for the system of 3 identical particles ( k' k" k ''' ) 1 2 3
1 2 3 P 123 2 3 1
This means replacement of 1 2, 2 3, 3 1.
1 2 3 1 2 32 1 3 P P P 123 2 3 1 2 1 32 3 1 12 13
Quantum mechanically this is not correct. The correct one is
ˆ ˆ ˆ P123 P13 P12
(ii) The second example
1 2 3 1 2 31 3 2 P P P 123 2 3 1 1 3 22 3 1 23 12 or quantum mechanically
ˆ ˆ ˆ P123 P12P23
(iii) The third example Identical particles 3 11/10/2019
1 2 3 1 2 33 2 1 P P P 132 3 1 2 3 2 13 1 2 13 12 or quantum mechanically
ˆ ˆ ˆ P132 P12 P13
((Proof))
Pˆ Pˆ k' k" k ''' Pˆ k ''' k" k' k" k ''' k' 12 13 1 2 3 12 1 2 3 1 2 3 and
Pˆ k' k" k ''' k' k" k ''' k" k ''' k' 132 1 2 3 3 1 2 1 2 3
ˆ ˆ ˆ Therefore we have P132 P12 P13 .
Any permutation operators can be broken into a product of transposition operators.
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 P132 P23 P12 P13 P23 P12 P13 P23 P12 P23 ...
The decomposition is not unique. However, for a given permutation, it can be shown that the parity of the number of transposition into which it can be broken down is always the same: it is called the parity of the permutation.
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 P132 P23P12 P13P23 P12P13 P23P12 P23 even permutation
ˆ ˆ ˆ ˆ ˆ P123 P12 P23 P13 P12 : even permutation
ˆ P23 odd permutation
ˆ P12 odd permutation
ˆ P31 odd permutation
((Note))
ˆ ˆ ˆ P132 P321 P213
3 Symmetrizer and antisymmetrizer Now we consider the two Hermitian operators
ˆ 1 ˆ S P : symmetrizer N!
Identical particles 4 11/10/2019
ˆ 1 ˆ A P : antisymmetrizer N!
ˆ ˆ where =1 if P is an even permutation and = -1 if P is an odd permutation.
Sˆ Sˆ and Aˆ Aˆ .
((Theorem-1)) ˆ If P 0 is an arbitrary permutation operator, we have
ˆ ˆ ˆ ˆ ˆ P 0 S PS 0 S
ˆ ˆ ˆ ˆ P 0 A AP 0 0 A (1)
((Proof)) This is due to the fact that
ˆ ˆ ˆ P 0P P such that
0 or
2 0 0
ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ P 0S P 0P P S N! N!
ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ P 0 A P 0 P 0 P 0 A N! N!
From Eq.(1) we see the following theorem
((Theorem-2))
Sˆ 2 Sˆ
Aˆ 2 Aˆ and
ˆSA ˆ ˆAS ˆ 0
Identical particles 5 11/10/2019 ((Proof))
ˆ 2 1 ˆˆ 1 ˆ ˆ S P S S S N! N!
ˆ 2 1 ˆˆ 1 2 ˆ ˆ A P A A A N! N!
ˆˆ ˆ 1 ˆ ˆ 1 ˆ SA A P S S 0 N! N!
ˆ ˆ since half the are equal to 1 and half the equal to -1. S and A are therefore projectors. Their action on any ket of the state space yields a completely symmetric or completely antisymmetric ket.
ˆ ˆ ˆ P 0S S
Pˆ Aˆ Aˆ 0 0
ˆ S S
ˆ A A
(( Example ))
For N = 3,
1 Sˆ [1ˆ Pˆ Pˆ Pˆ Pˆ Pˆ ] 6 12 23 31 123 132 and
1 Aˆ [1ˆ Pˆ Pˆ Pˆ Pˆ Pˆ ] 6 12 23 31 123 132 where
ˆ ˆ ˆ ˆ ˆ ˆ P123 P12 P23 , P132 P12 P13
1 Sˆ Aˆ (1ˆ Pˆ Pˆ ) 1ˆ 3 123 132
4 Symmetrization postulate The system containing N identical particles are either totally symmetrical under the interchange of any pair (boson ), or totally antisymmetrical under the interchange of any pair ( fermion ).
Identical particles 6 11/10/2019 ˆ Pij ,BN ,BN ,
ˆ Pij ,FN ,FN ,
where ,BN is the eigenket of N identical boson systems and ,FN is the eigenket of N identical fermion systems.
(( Note )) It is an empirical fact that a mixed symmetry does not occur.
Even more remarkable is that there is a connection between the spin of a particle and the statistics obeyed by it:
Half-integer spin particles are fermion, while integer-spin particles are bosons.
5 Pauli exclusion principle Wolfgang Ernst Pauli (April 25, 1900 – December 15, 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or Pauli principle," involving spin theory, underpinning the structure of matter and the whole of chemistry.
http://en.wikipedia.org/wiki/Wolfgang_Pauli
Electron is a fermion. No two electrons can occupy the same state. We discuss the framatic difference between fermions and bosons. Let us consider two particles. Each of which can occupy only two states k' and k" . For a system of two fermions, we have no choice
1 1 k' k' k'[ k" k' k ]" 1 2 1 2 2 1 k '' k" 2 2 1 2
For bosons, there are three states.
Identical particles 7 11/10/2019 k' k' , 1 2
k" k" , 1 2
1 k'( k" k" k )' . 2 1 2 1 2
In contrast, for “classical particles” satisfying Maxwell-Boltzmann (M-B) statitics with no restriction on symmetry, we have altogether four independent states.
k' k" , k" k' , k' k' and k" k" 1 2 1 2 1 2 1 2
We see that in the fermion case, it is impossible for both particles to occupy the same state.
6 Transformation of observables by permutation For simplicity, we consider a specific case where the two particle state ket is completely specified by the eigenvalues of a single observable Aˆ for each of the particle.
Aˆ a' a" a a'' a" 1 1 2 1 2 and
Aˆ a' a" a" a' a" 2 1 2 1 2
Since
Pˆ A Pˆˆ 1Pˆ a' a" Pˆ A Pˆˆ 1 a" a' 12 121 12 1 2 12 121 1 2 or
Pˆ Aˆ Pˆ 1Pˆ a' a" Pˆ Aˆ a' a" 12 1 12 12 1 2 12 1 1 2 a' Pˆ a' a" 12 1 2 a' a" a' Aˆ a" a' Aˆ Pˆ a' a" 1 2 2 1 2 2 12 1 1 we obtain
ˆ ˆ ˆ 1 ˆ P12 A1 P12 A2 .
Similarly, we have
ˆ ˆ ˆ 1 ˆ P21 A2 P21 A1 .
ˆ It follows that P12 must change the particle label of observables.
Identical particles 8 11/10/2019 ˆ ˆ ˆ ˆ There are also observables, such as A1 B2 , A1 B2 , which involve both indices simultaneously.
ˆ ˆ ˆ ˆ 1 ˆ ˆ P12 (A1 B2 )P12 A2 B1
ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ 1 ˆ ˆ ˆ 1 ˆ ˆ P12 A1 B2 P12 P12 A1 P12 P12 B2 P12 A2 B1
These results can be generalized to all observables which can be expressed in terms of observables which can be ˆ ˆ ˆ expressed in terms of observables of the type of A1 and B2 , to be denoted by O(1,2).
ˆ ˆ ˆ 1 ˆ P12 O(1,2) P12 O(2,1) where Oˆ (2,1) is the observable obtained from Oˆ (1,2) by exchanging indices 1 and 2 throughout.
ˆ Os (1,2) is said to be symmetric if
ˆ ˆ Os (1,2) Os (2,1) or
ˆ ˆ [Os (1,2),P12 ] 0
Symbolic observables commute with the permutation operator.
ˆ In general. the observables Os (1,2,3,..., N) which are completely symmetric under exchange of indices 1, 2, ..., N commute with all the transposition operators, and with all the permutation operators
7 Example Let us now consider a Hamiltonian of a system of two identical particles.
1 1 Hˆ pˆ 2 pˆ 2 V ( rˆ rˆ ) V (rˆ ) V (rˆ ) 2m 1 2m 2 pair 1 2 ext 1 ext 2
Clearly we have
ˆ ˆ ˆ 1 ˆ P12 H P12 H or
ˆ ˆ [P12, ,H ] 0
ˆ ˆ 2 ˆ P12 is a constant of the motion. Since P12 1, the eigenvalue of P12 allowed are ±1.
Hˆ E
ˆ P12
Identical particles 9 11/10/2019
ˆ 2 ˆ 2 P12 P12 or
= ±1.
It therefore follows that if the two-particle state ket is symmteric (antisymmettric) to start with, it remains so at all times.
(i) N = 2 case We can define the symmetrizer and antisymmetrtizer as follows.
1 1 Sˆ 1( Pˆ ) Aˆ 1( Pˆ ) 2 12 2 12
Sˆ Aˆ 1ˆ
1 1 1 1 Sˆ 2 (1ˆ Pˆ ) (1ˆ Pˆ ) (1ˆ 2Pˆ ˆ)1 (1ˆ Pˆ ) Sˆ 2 12 2 12 4 12 2 12
1 1 1 1 Aˆ 2 (1ˆ Pˆ ) (1ˆ Pˆ ) (1ˆ 2Pˆ ˆ)1 (1ˆ Pˆ ) Aˆ 2 12 2 12 4 12 2 12
Then we have
1 k'( k" k" k )' S 2 1 2 1 2 and
1 k'( k" k" k )' A 2 1 2 1 2 where
1 1 ˆ kS ' k" 1( Pˆ k') k" k'( k" k" k )' 1 2 2 12 1 2 2 1 2 1 2
1 1 ˆ kA ' k" 1( Pˆ k') k" k'( k" k" k )' 1 2 2 12 1 2 2 1 2 1 2
(ii) N = 3 Cases
1 Sˆ (1 Pˆ Pˆ Pˆ Pˆ Pˆ ) 6 12 23 31 123 132
Identical particles 10 11/10/2019 1 Aˆ (1 Pˆ Pˆ Pˆ Pˆ Pˆ ) 6 12 23 31 123 132
1 Sˆ Aˆ (1 Pˆ Pˆ ) 1 3 123 132
k' k" k ''' 1 1 1 1 ˆ kA ' k" k ''' k' k" k ''' 1 2 3 !3 2 2 2 k' 3 k" 3 k ''' 3
Slater determinant ˆ kA ' k" k ''' is zero if two of individual states coincide. We obtain Pauli’s exclusion principle. 1 2 3
8 Generalized method (Tomonaga) We now consider a system consisting of many spins.
Sˆ Sˆ Sˆ Sˆ Sˆ 1 2 3 ... N
Sˆ Sˆ Sˆ Sˆ Sˆ Sˆ Sˆ Sˆ Sˆ Sˆ ( 1 2 3 ... N () 1 2 3 ... N ) or
N ℏ2 N 1 Sˆ 2 Sˆ 2 Sˆ Sˆ σˆ 2 ℏ2 σˆ σˆ n 2( n n' ) n ( n n') n1 nn ' 4 n1 2 nn ' or
Sˆ 2 3N 1 1ˆ (σˆ σˆ ) ℏ2 n n' 4 2 nn '
Here we define an operator
ˆ 2 ˆ O Pnn' N(N )1 nn '
Oˆ is Hermitian and [Pˆ,Oˆ] 0. We assume that
1 Pˆ (1ˆ σˆ σˆ ) (Dirac exchange interaction) nn' 2 n n'
2 1 2 1 N(N )1 1 ˆ ˆ σˆ σˆ ˆ σˆ σˆ O (1 n n') [ 1 n n' ] N(N )1 nn ' 2 N(N )1 2 2 2 nn ' or Identical particles 11 11/10/2019
1 2 ˆ ˆ σˆ σˆ O [1 n n'] 2 N(N )1 nn '
Using the relation,
Sˆ 2 3N 1 1ˆ (σˆ σˆ ) , ℏ2 n n' 4 2 nn ' we get
1 2 2 3N 1 N 4 4 1 Oˆ [1ˆ ( Sˆ 2 )] [ 1ˆ Sˆ 2 ] 2 N (N )1 ℏ2 2 2 (N )1 N (N )1 ℏ2
[Sˆ 2 ,Oˆ] 0 . When the eigenvalue of Sˆ 2 is given by ℏ2S(S )1 , the eigenvalue of Oˆ is equal to
1 N 4 4S(S )1 [ ] 2 (N )1 N(N )1
The eigenvalue of Oˆ ,)( , specifies the symmetry.
(i) For N = 2,
D1/2 x D1/2 = D1 + D0
1 [2 2S(S )]1 1 S(S )1 2
When S = 1, = 1 (symmetric). When S = 0, = -1 (anti-symmetric).
(ii) For N = 3,
D1/2 x D1/2 x D1/2 = D3/2 + 2 D 1/2
1 1 2 [ S(S )]1 2 2 3
When S = 3/2, = 1 (symmetric). When S = 1/2, = 0.
(iii) For N = 4,
D1/2 x D1/2 x D1/2 x D1/2 = D 2 + 3D 1 + 2D 0
Identical particles 12 11/10/2019 S(S )1 6
For S = 2, = 1 (symmetric). For S = 1, = 1/3. For S = 0, = 0.
9. Summary: fermion and boson The Hamiltonian H of the system (in the absence of a magnetic field) does not contain the spin operators, and hence, when it is applied to the wave function, it has no effect on the spin variables. The wave function of the system of particles can be written in the form of product,
space spin ,
where space depends only on the coordinate of the particles and spin only on their spins. space spin should be anti-symmetric since electrons are fermions. The system containing N identical particles are either totally symmetrical under the interchange of any pair (boson ), or totally antisymmetrical under the interchange of any pair ( fermion ).
ˆ Pij ,BN ,BN ,
ˆ Pij ,FN ,FN ,
where ,BN is the eigenket of N identical boson systems and ,FN is the eigenket of N identical fermion systms.
(( Note )) It is an empirical fact that a mixed symmetry does not occur.
Even more remarkable is that there is a connection between the spin of a particle and the statistics obeyed by it:
Half-integer spin particles are fermion, obeying the Fermi-Dirac statistic, while integer-spin particles are bosons, obeying the Bose-Einstein statistics.
10. Ground state for systems with many electrons We construct the form of wavefunctions for the system with two, three, four, and five electrons (identical particles, fermion) using the Slater determinant. We assume that each electron has spin states
, .
We also assume the orbital state: a n 1 b n ,2 c n 3 .., where n is the principal quantum number. It is reasonable to consider that the ground state can be expressed by electrons occupied by
a , a
Identical particles 13 11/10/2019 taking into account of the Pauli’s exclusion principle. For the system with three electrons, the ground state can be expressed by
a , a , b
For the system with the four electrons, the ground state can be expressed by
a , a , b , b
For the system with the five electrons, the ground state can be expressed by
a , a , b , b , ` c
(a) Two particles state For the two states,
a a , and a a we have the Slater determinant
1 a 11 a 11 1 aa 21 ( 21 21 ) 2 a 22 a 22 2 where index 1, 2 denotes the particle 1 and particle 2. Clearly, the state vector is antisymmetric under the exchange of two particles.
(b) Three particles state For the three states,
a a , a a , and b b we have the Slater determinant
a 11 a 11 b11 1 a a b 6 22 22 22 a 33 a 33 b 33 1 [ aa ( )b aa ( )b aa ( )b ] 6 32 32 32 11 31 31 31 22 21 21 21 33
This state vector is antisymmetric under the exchange between any two particles (such that 1 2 ). Identical particles 14 11/10/2019
(c) Four particle state For the four states,
a a , a a , b b , and b b we have the Slater determinant
a 11 a 11 b11 b 11 1 a a b b 22 22 22 22 24 a 33 a 33 b 33 b 33
a 44 a 44 b 44 b 44
bb 21 ( 21 21 ) aa 43 ( 43 43 )
bb 32 ( 32 32 ) aa 41 ( 41 41 )
bb 42 ( 42 42 ) aa 41 ( 41 41 )
bb 31 ( 31 31 ) aa 42 ( 42 42 )
bb 41 ( 41 41 ) aa 32 ( 32 32 )
bb 43 ( 43 43 ) aa 21 ( 21 21 )
This state vector is antisymmetric under the exchange between any two particles (such that 1 2 ), but symmetric under the two types exchange (such that 1 2 and 3 4 )
(d) Five particle state For the five states
a a , a a , b b , b b , c c we have the Slater determinant
a 11 a 11 b11 b 11 c 11
a 22 a 22 b 22 b 22 c 22 1 a a b b c 120 33 33 33 33 33 a 44 a 44 b 44 b 44 c 44
a 55 a 55 b 55 b 55 c 55 1 [ aa ( ) bb ( )c ...] 120 21 21 21 43 43 34 55
(( Mathematica ))
Identical particles 15 11/10/2019 a1 1 a1 1 Clear "Global` " ; A1 ; a2 2 a2 2 Det A1 FullSimplify a1 a2 2 1 1 2
a1 1 a1 1 b1 1 B1 a2 2 a2 2 b2 2 ; a3 3 a3 3 b3 3 Det B1 FullSimplify a1 a3 b2 a2 b3 2 3 1 a2 a3 b1 a1 b3 1 3 2 a3 a2 b1 a1 b2 1 2 3
a1 1 a1 1 b1 1 b1 1 a2 2 a2 2 b2 2 b2 2 C1 ; a3 3 a3 3 b3 3 b3 3 a4 4 a4 4 b4 4 b4 4 Det C1 FullSimplify b2 a3 a4 b1 2 1 1 2 4 3 3 4 a1 a4 b3 3 2 2 3 4 1 1 4 a3 b4 3 1 1 3 4 2 2 4 a2 a4 b1 b3 3 1 1 3 4 2 2 4 b4 a3 b1 3 2 2 3 4 1 1 4 a1 b3 2 1 1 2 4 3 3 4
______REFERENCES S. Tomonaga Angular momentum and spin (Misuzo Syobo, Tokyo, 1989) [in Japanese]. J.J. Sakurai Modern Quantum Mechanics , Revised Edition (Addison-Wesley, Reading Massachusetts, 1994). L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1977). John S. Townsend , A Modern Approach to Quantum Mechanics , second edition (University Science Books, 2012). David H. McIntyre, Quantum Mechanics A Paradigms Approach (Pearson Education, Inc., 2012).
APPENDIX: r (( Townsend textbook ))
Identical particles 16 11/10/2019 1. Formula-1 For a symmetric spatial state under the exchange of two particles, we have
dr' dr r r ;",'" S r r ;",' S S S dr' dr"Sˆ r r",' r r",' Sˆ S dr' dr r r",'" r r",' S and for an anti-symmetric spatial state under the exchange of two particles, we have
dr' dr r r ;",'" A r r ;",' A A A dr' dr"Aˆ r r",' r r",' Aˆ A dr' dr r r",'" r r",' A where
r r",' r' r" 1 2
1 1 Sˆ (1ˆ Pˆ ) , Aˆ (1ˆ Pˆ ) 2 12 2 12
(( Proof ))
dr' dr r r ;",'" S r r ;",' S S S dr' dr"Sˆ r r",' r r",' Sˆ S 1 dr' dr" (1ˆ Pˆ r r",') r r",' 2 12 S 1 dr' dr (" 1ˆ Pˆ r r",') r r",' 2 12 S 1 dr' dr r r",'(" r r r r",')'," 2 S 1 dr' dr r r",'(" r r",' r r'," r r",' Pˆ 2 S 12 S 1 dr' dr r r",'(" r r",' r r'," r r'," ) 2 S S dr' dr r r",'(" r r",' S where
ˆ 2 ˆ ˆ S S S S S
ˆ S S S , Identical particles 17 11/10/2019
r r r r ˆ r r ",' S ",' P12 S '," S .
ˆ P12 S S
Similarly,
dr' dr"Aˆ r r ;",' A r r ;",' A A A dr' dr"Aˆ r r",' r r",' Aˆ A 1 dr' dr (" 1ˆ Pˆ r r",') r r",' 2 12 A 1 dr' dr r r",'(" r r'," r r",' 2 A 1 dr' dr r r",'(" r r",' r r'," r r",' ) 2 A A 1 dr' dr r r",'(" r r",' r r'," r r",' Pˆ 2 A 12 A 1 dr' dr r r",'(" r r",' r r'," r r'," ) 2 A A dr' dr r r",'" r r",' A where
ˆ 2 ˆ ˆ A A A A A
r r r r ˆ r r ",' A ",' P12 A '," A
ˆ P12 A A
2. Formula-2: symmetric case
1 k k ;",' S (1ˆ Pˆ k') k" ˆ kS k",' 2 12 1 2
1 x x ;",' S 1( Pˆ x') x" ˆ xS x",' 2 12 1 2
Identical particles 18 11/10/2019 x x ;",' S k k ;",' S x x",' Sˆ k k ;",' S x x",' ˆ kS k ;",' S x x",' k k ;",' S 1 1 [ x k'' x" k" x' k" x" k ]' 2 2 or
2 x x ;",' S k k ;",' S S x x )",'( 1 [ x k'' x" k" x' k" x" k ]' 2
3. Formula-3: antisymmetric case
1 k k ;",' A (1ˆ Pˆ )[ k' k" ˆ kA k",' 2 12 1 2
x x ;",' kA k ;",' A x x",' Aˆ k k ;",' A x x",' ˆ kA k ;",' A x x",' Aˆ 2 k ,' k x x",' ˆ kA ,' k x x",' k k ;",' A 1 [ x k'' x" k" x k"' x" k ]' 2 1 1 [ x k'' x" k" x' k" x" k ]' 2 2 or
2 x x ;",' A k k ;",' A A x x )",'( 1 x k'' x' k" 2 x" k' x" k"
4. Inner product for the symmetric and antisymmetric cases We consider the inner product for the symmetric case
Identical particles 19 11/10/2019 x x ;",' S y y ;",' S xx "' Sˆ y y ;",' S xx "' Sˆ y y ;",' S xx "' Sˆ 2 y y",' xx "' Sˆ y y",' xx "' y y ;",' S 1 [ x y'' x" y" x' y" x" y ]' 2 1 x'([ y x"()' y )" x'( y x"()" y )]' 2 where
Sˆ Sˆ (Hermitian operator)
Sˆ 2 Sˆ
y y ;",' S ˆ yS y",'
We also consider the inner product for the antisymmetric case,
x x ;",' A y y ;",' A xx "' Aˆ y y ;",' A xx "' Aˆ y y ;",' A xx "' Aˆ 2 y y",' xx "' Aˆ y y",' xx "' y y ;",' A 1 [ x y'' x" y" x' y" x" y ]' 2 1 x'([ y x"()' y )" x'( y x"()" y )]' 2 where
Aˆ Aˆ (Hermitian operator)
Aˆ 2 Aˆ
y y ;",' A Aˆ y y",'
5. Closure relation
(i) Symmetric case
Identical particles 20 11/10/2019
d d rrr r ;",'' S r r ;",' S 1ˆ (completeness)
(( Proof ))
dr' dr" s s ;",' S r r ;",' S r r ;",' S S 1 dr' dr" s'([ r s"()' r )" s'( r s"()" r )]' r r ;",' S S 2 1 s s ;",'[ S s s ;'," S 2 S S s s ;",' S S since
1 s s ;",' S r r ;",' S s'([ r s"()' r )" s'( r r"()" r )]' 2
s s s s ˆ 2 s s ˆ s s ;",' S S ",' S ",' S ",' S
s s s s s s ˆ s s ;'," S S '," S ",' P12 S ",' S or
s s s s ;",' S S ;'," S S
(ii) Antisymmetric case
d d rrr r ;",'' A r r ;",' A 1ˆ (completeness)
(( Proof ))
dr' dr" s s ;",' A r r ;",' A r r ;",' A S 1 dr' dr" s'([ r s"()' r )" s'( r r"()" r )]' r r ;",' A A 2 1 s s ;",'[ A s s ;'," A ] 2 A A s s ;",' A S since
1 s s ;",' A r r ;",' A s'([ r s"()' r )" s'( r r"()" r )]' 2
Identical particles 21 11/10/2019 s s s s ˆ 2 s s ˆ s s ;",' A A ",' A ",' A ",' A
s s s s s s ˆ s s ;'," A A '," A ",' P12 A ",' A or
s s s s ;'," A A ;",' A A
6. Wave function representation (symmetric case) We have the relation for the symmetric case, such that
1 x x ;",' S [ x x",' x x'," S 2 S S and
ˆ 2 ˆ x x ;",' S S x x",' S x x",' S x x",' S
Noting that
ˆ x x ;",' S S x x",' S x x'," P12 S x x'," S
ˆ P12 S S we get
x x ;",' S S x x",' S x x'," S
7. Wave function representation (anti-symmetric case) We have the relation for the anti-symmetric case, such that
1 x x ;",' A [ x x",' x x'," A 2 A A and
ˆ 2 ˆ x x ;",' A A x x",' A x x",' A x x",' A
Noting tht
ˆ x x ;",' A A x x",' A x x'," P12 A x x'," A
ˆ P12 A A
Identical particles 22 11/10/2019 we have
x x ;",' A A x x",' A x x'," A
REFERENCES R. Shankar, Principles of Quantum Mechanics (Kluwer Academic, 1980). J.S.. Townsend, A Modern Approach to Quantum Mechanics, second edition (University Science Book, 2012).
Identical particles 23 11/10/2019