Introduction to Second-quantization I

Transcription

1 Introduction to Second-quantization I Jeppe Olsen Lundbeck Foundation Center for Theoretical Chemistry Department of Chemistry, University of Aarhus September 19, 2011 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Contents, lecture I Brief motivation Occupation number vectors and the Fock space Definition of creation operators and the properties Annihilation operators and their properties Products of creation and annihilation operators Summary I Operators in second quantization One-electron operators Two-electron operators Conclusion and summary Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

2 What is Second Quantization? The idea Operators and wavefunctions are described by a common set of elementary operators (creation and annihilation operators) Another representation of Quantum Mechanics Advantages Antisymmetry is built automatically in The use of a common set of elementary operators for wavefunctions and operators allows manipulations not easily realized in the standard formulation Disadvantages Yet another formalism to learn Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Occupation number vectors Slater-determinants and occupation number vectors Slater-determinant A Slater-determinant is an antisymmetric combination of some spin-orbitals Occupation number vector (ONV) Assume a space of M (orthonormal) spin-orbitals are given An ONV is a vector of M integers, each integer may be 0 or 1 k = k 1, k 2,, k M, k P = 0, 1 for P = 1, 2,, M Each occupation number vector gives the occupation of a given Slater-determinant Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

3 Occupation number vectors Slater-determinant ONV Assume we have a given Slater-determinant, how do we find the corresponding ONV? Entry P in the ONV 1 if spin-orbital P is occupied 0 if spin-orbital P is unoccupied (The total number of spin-orbitals n must also be known) Example(M=4) Slater-determinant: 1 2! φ 2 (1) φ 4 (1) φ 2 (2) φ 4 (2) ONV: 0, 1, 0, 1 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Occupation number vectors Other Examples (still M = 4) Slater-determinant ONV # of elecs. φ 3 (1) 0, 0, 1, φ 1 (1) φ 3 (1) 2! φ 1 (2) φ 3 (2) φ 1 (1) φ 3 (1) φ 4 (1) 1 φ 3! 1 (2) φ 3 (2) φ 4 (2) φ 1 (3) φ 3 (3) φ 4 (3) 1, 0, 1, 0 2 1, 0, 1, vac = 0, 0, 0, 0 0 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

4 Occupation number vectors Example: H 2 in a minimal basis Two molecular orbitals and four spinorbitals MO s: 1σ g, 1σ u Spinorbitals: 1σ g α, 1σ g β, 1σ u α, 1σ u β Groundstate of H + 2 One electron in a 1s-spin-orbital, say 1sα occupied 1, 0, 0, 0 Groundstate of H 2 (single ONV approximation) One electron in 1sα, one electron in 1sβ 1, 1, 0, 0 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Occupation number vectors Nomenclature: uses correspondence between ONV and SD k P is occupation number for spin-orbital P If k P = 1, spin-orbital/level P is occupied If k P = 0, spin-orbital/level P is unoccupied P=1,M k P is the total number of electrons in the ONV The ONV with zero electrons is the vacuum state vac = 0, 0,, 0 ONV s represents SD s, but are not SD s We have a one-to-one mapping between ONV s and SD s But they are not identical SD s are functions of space-coordinates of electrons In SD s there is explicit reference to electron 1, electron 2,... ONV s a just vectors in an abstract vector space Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

5 Occupation number vectors In other words This place is to small for two of us / John Wayne To be or not to be, that is the question / Hamlet I am you and you are me / John Lennon Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 The Fock-space Basic definitions Defined by the total number of spin-orbitals, M Abstract vector space The ONV s are unit-vectors Each unit-vector corresponds thus to a SD, combination of several determinants corresponds to c = k c k k Contains ONV s with 0,1,... M electrons We thus map SD s in real Cartesian space to ONV s in the Fock-space. Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

6 The Fock-space Inner product between ONV s We will define an inner product between ONV s, so it includes the standard inner product of SD s for orthonormal orbitals k m = δ k,m = M P=1 δ kp,m P = 1 if the two ONV s are identical = 0 if the two ONV s differ in one or more occupation numbers Inner product also defined for ONV s with different number of electrons Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Creation-operators a P Definition One creation operator a P for each spin-orbital P a P k 1,, 0 P,, k M = Γ k P k 1,, 1 P,, k M a P k 1,, 1 P,, k M = 0 Γ k P = ( 1)( P 1 Q=1 k Q) Comments Creates an electron in spin-orbital P if this spin-orbital is unoccupied in k Gives zero if spin-orbital P is occupied in k Phase-factor Γ k P count the number of electrons in k before P Even number Γ k P = 1, odd Γk P = 1 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

7 Creation-operators a P Examples, M=4 1 a 1 1, 0, 0, 0 = 0 2 a 1 0, 1, 0, 0 = 1, 1, 0, 0 3 a 2 1, 0, 0, 0 = 1, 1, 0, 0 All ONV s may be obtained from vac k 1, k 2,, k M = (a p) k p vac (Â) 0 = ˆ1 P=1,M Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Creation-operators a P Example: H 2 in a minimal basis Two molecular orbitals and four spinorbitals Spinorbitals: 1σ g α, 1σ g β, 1σ u α, 1σ u β Four creation operators (M = 4) 1: a 1σ g α, 2: a 1σ g β, 3: a 1σ u α, 4: a 1σ u β Groundstate of H + 2 1, 0, 0, 0 = a 1σ g α vac Groundstate of H 2 1, 1, 0, 0 = a 1σ g α a 1σ g β vac Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

8 Creation-operators a P Products of two creation-operators a P a P Consider the action of a P a P on an arbitrary ONV, two cases 1 a P a P, 1 P, = a P 0 = 0 2 a P a P, 0 P, = Γ k P a P, 1 P, = 0 a P a P working on any ONV gives 0, so a P a P = 0 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Creation-operators a P Products of two creation-operators a P a Q and a Q a P, P < Q Only non vanishing if both spin-orbitals P, Q are unoccupied a P a Q 0 P 0 Q = Γ k Q a P 0 P 1 Q = Γ(k) P Γ(k) Q 1 P 1 Q a Q a P 0 P 0 Q = Γ k P a Q 1 P 0 Q = Γ k P Γk Q 1 P 1 Q (1) so (a P a Q + a Q a P ) 0 P 0 Q = 0 Same relation holds(trivially) for other ONV S so a P a Q + a Q a P = 0 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

9 Creation-operators a P Products of two creation-operators Conclusion: a P a Q + a Q a P = 0 holds for all P, Q Creation-operators anti commute The anti-commutation arise from the definition of the phase-factor Γ Also written as [a P, a Q ] + = 0 ([A, B] + = AB + BA) Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Annihilation-operators a P Definition The operators obtained by conjugating a P : a P = (a P ) Properties From the definition of the creation-operators it may be shown a P 1 P = Γ k P 0 P a P 0 P = 0 (2) 1 a P annihilates an electron in spin-orbital P if possible Examples a 1 0, 0, 0, 0 = 0 a 1 1, 1, 0, 0 = 0, 1, 0, 0 a 2 1, 1, 0, 0 = 1, 0, 0, 0 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

10 Annihilation-operators a P Products Product of two annihilation operators From the anti-commutation of the creation operators a P a Q + a Q a P = 0 We obtain by conjugation (a P a Q + a Q a P ) = a Q a P + a P a Q = 0 (3) Thus annihilation-operators are also anti-commuting [a P, a Q ] + = 0 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Products of operators Anti-commutation of creation- and annihilation-operator a P a Q + a Q a P =? From examining the action on various ONV s one obtains 1 P Q : a P a Q + a Q a P = 0 2 P = Q : a P a P + a P a P = 1 Collecting the two gives a P a Q + a Q a P = δ PQ δ PQ is the Kronecker delta-function δ PQ = { 1 for P = Q 0 for P Q Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

11 Products of operators The operator a P a P Two cases Combined expression a P a P 1 P = Γ k P a P 0 P a P a P 0 P = 0 = (Γ n P )2 1 P = 1 P a P a P k P = k P k P When a P a P works on a ONV, it thus gives the ONV multiplied with the occupation number ˆN P = a P a P is thus the number-operator for spin-orbital P ˆN = ˆN P P gives the total number of electrons, P k P of an ONV Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Products of operators Excitation-operators Definition of one-electron excitation operator Comments ˆX Q P = a P a Q (P Q) a P a Q 0 P 1 Q = ± 1 P 0 Q Removes one electron in spin-orbital Q and creates one electron P In other words: Excites one electron from Q to P a P a Q (P Q) is therefore a single-electron excitation Two-electron excitations may be obtained as a P a Q a R a S Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

12 Elements of second quantization Basis vectors k = k 1, k 2,, k M, k P = 0, 1 Inner product k m = n P=1 δ k P m P Creation operators a P k 1,, 0 P,, k M = Γ k P k 1,, 1 P,, k M Γ k P = P 1 1( Q=1 k Q) a P k 1,, 1 P,, k P = 0 Annihilation operators a P k 1,, 1 P,, k M = Γ k P k 1,, 0 P,, k M a P k 1,, 0 P,, k M = 0 Anti-commutation a P a Q + a Q a P = δ PQ relations a P a Q + a Q a P = 0 a P a Q + a Q a P = 0 Number operators a P a P k = k P k ( P a P a P ) k = ( P k P) k Vacuum state vac = 0 1, 0 2,, 0 M vac vac = 1 a P vac = 0 Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Operators in second quantization The problem In the Fock-space, we could define very simple operators like the number-operator We need representations of quantum mechanical operators, for example the kinetic-energy operator Procedure 1 We know the mapping SD i ONV i 2 We know the mapping SD j ONV j 3 Obtain the second quantization representation, ˆf of a given first-quantization operator f c by requiring ONV i ˆf ONV j = SD i f c SD j It is only then we have obtained an alternative representation of quantum mechanics Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

13 Operators in second quantization One-electron operators Examples kinetic energy operator, nuclear-electron attraction operator First-quantization form Form: f c = i=1,n f c (x i ), N: number of electrons Properties: 1 Works on SD s with at least one electron 2 Connects Slater-determinants differing in at most one set of occupations Implies form of second quantization operator ˆf = PQ f PQa P a Q where f PQ pt are unknowns Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Operators in second quantization One-electron operators f PQ =? Setting f PQ = dx φ P (x) f c (x)φ Q (x) gives the same matrix-elements in first- and second quantization Example: Diagonal elements First quantization (from elementary QM) SD k f c SD k = P k P dx φp (x) f c (x)φ P (x) Second quantization: k PQ k P = P f PQ a P a Q k = f PP a P a P k = k f PP ˆN P k P k P dx φ P (x) f c (x)φ P (x) The phase-factor Γ k is essential for agreement Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

14 Operators in second quantization Example: one-electron operator in the H 2 minimal basis The operator ĥ = h 1σg α1σ g α a 1σ g α a 1σ g α + h 1σ g β1σ g β a 1σ g β a 1σ g β + h 1σu α1σ u α a 1σ u α a 1σ u α + h 1σ u β1σ u β a 1σ u β a 1σ u β Integrals between 1σ g, 1σ u vanish by symmetry Integrals between α and β vanish as h c does no change spin Some matrix elements vac a 1σg β a 1σ g α ĥ a 1σ g α a 1σ g β vac = h 1σ g α1σ g α + h 1σg β1σ g β Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Operators in second quantization Two-electron operators Examples Electron-electron repulsion, two-electron spin-orbit First-quantization form Form: g c = 1 2 i,j=1,n g c (i, j), Properties: 1 Works on SD s with at least two electron 2 Connects Slater-determinants differing in at most two sets of occupations Implies form of second quantization operator ĝ = 1 2 PQRS g PQRSa P a R a S a Q where g PQRS pt are unknowns Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

15 Operators in second quantization Two-electron operators g PQRS =? Equivalence is obtained between first and second quantization by setting g PQRS = dxdx φ P (x)φ R (x )g c (x, x )φ Q (x)φ S (x ) Shown by going through the various cases The phase-factor Γ k is essential for obtaining agreement Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Operators in second quantization The Hamiltonian in SQ In spin-orbital form Ĥ = PQ PQ h PQ a P a Q attraction 1 2 PQRS h PQ a P a Q PQRS g PQRS a P a R a S a Q + h nuc contains kinetic energy and electron-nuclear g PQRS a P a R a S a Q is electron-electron repulsion The Hamiltonian is a weighted sum of single and double excitations Simplifications when we consider orbitals ( instead of spin-orbitals) Jeppe Olsen (Aarhus) Second quantization I September 19, / 32

16 Conclusion We have now Obtained a new way of representing antisymmetric wave-functions Obtained a new way of representing operators So that Expectation values are identical to the standard formulation In other words We have obtained a new representation of quantum mechanics for electrons Jeppe Olsen (Aarhus) Second quantization I September 19, / 32 Operators in first and second quantization First quantization One-electron operator i f (x i) Second quantization One-electron operator PQ f PQa P a Q Two-electron operator Two-electron operator 1 2 i j g(x 1 i, x j ) 2 PQRS g PQRSa P a R a S a Q Operators are independent of spin-orbital basis Operators depend in the number of electrons Exact operators Operators depend on spin-orbital basis Operators are independent of the number of electrons Projected operators Jeppe Olsen (Aarhus) Second quantization I September 19, / 32