Calculation of Generalized Pauli Constraints Murat Altunbulak - PowerPoint PPT Presentation

Calculation of Generalized Pauli Constraints Murat Altunbulak Department of Mathematics Dokuz Eylul University April, 2016 Murat Altunbulak Oxford 2016

Notations Murat Altunbulak Oxford 2016

Notations Quantum States Quantum system A is described by a complex Hilbert space H A , Murat Altunbulak Oxford 2016

Notations Quantum States Quantum system A is described by a complex Hilbert space H A , called state space. Murat Altunbulak Oxford 2016

Notations Quantum States Quantum system A is described by a complex Hilbert space H A , called state space. Pure state = unit vector | ψ � ∈ H A or projector operator | ψ �� ψ | Murat Altunbulak Oxford 2016

Notations Quantum States Quantum system A is described by a complex Hilbert space H A , called state space. Pure state = unit vector | ψ � ∈ H A or projector operator | ψ �� ψ | Mixed state = classical mixture of pure states � � ρ = p i | ψ i �� ψ i | ; p i ≥ 0 ; p i = 1 i i Murat Altunbulak Oxford 2016

Notations Quantum States Quantum system A is described by a complex Hilbert space H A , called state space. Pure state = unit vector | ψ � ∈ H A or projector operator | ψ �� ψ | Mixed state = classical mixture of pure states � � ρ = p i | ψ i �� ψ i | ; p i ≥ 0 ; p i = 1 i i ρ is a non-negative ( ρ ≥ 0) Hermitian operator with Tr ρ = 1, called Density matrix. Murat Altunbulak Oxford 2016

Notations Superposition Principle implies that the state space of composite system AB splits into tensor product of its components A and B H AB = H A ⊗ H B Murat Altunbulak Oxford 2016

Notations Superposition Principle implies that the state space of composite system AB splits into tensor product of its components A and B H AB = H A ⊗ H B Density Matrix of Composite Systems Density matrix of composite system can be written as linear combination � a α L α A ⊗ L α ρ AB = B α where L α A , L α B are linear operators on H A , H B , respectively. Murat Altunbulak Oxford 2016

Notations Reduced state Its reduced matrices are defined by partial traces Murat Altunbulak Oxford 2016

Notations Reduced state Its reduced matrices are defined by partial traces � a α Tr ( L α B ) L α ρ A = A := Tr B ( ρ AB ) α Murat Altunbulak Oxford 2016

Notations Reduced state Its reduced matrices are defined by partial traces � a α Tr ( L α B ) L α ρ A = A := Tr B ( ρ AB ) α � a α Tr ( L α A ) L α ρ B = B := Tr A ( ρ AB ) α Murat Altunbulak Oxford 2016

Notations Reduced state Its reduced matrices are defined by partial traces � a α Tr ( L α B ) L α ρ A = A := Tr B ( ρ AB ) α � a α Tr ( L α A ) L α ρ B = B := Tr A ( ρ AB ) α they are called one-particle reduced density matrices. Murat Altunbulak Oxford 2016

Pauli Exclusion Principle Murat Altunbulak Oxford 2016

Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. Murat Altunbulak Oxford 2016

Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Murat Altunbulak Oxford 2016

Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H ⊗ N , H ⊗ N = state space of N electrons. Murat Altunbulak Oxford 2016

Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H ⊗ N , H ⊗ N = state space of N electrons. Define ρ = ρ N 1 + ρ N 2 + . . . + ρ N N sum of all reduced states ρ N i : H . ρ is called electron density matrix. Murat Altunbulak Oxford 2016

Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H ⊗ N , H ⊗ N = state space of N electrons. Define ρ = ρ N 1 + ρ N 2 + . . . + ρ N N sum of all reduced states ρ N i : H . ρ is called electron density matrix. In terms of electron density matrix ρ , Pauli principle amounts to � ψ | ρ | ψ � ≤ 1 ⇐ ⇒ Spec ρ ≤ 1 Murat Altunbulak Oxford 2016

Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H ⊗ N , H ⊗ N = state space of N electrons. Define ρ = ρ N 1 + ρ N 2 + . . . + ρ N N sum of all reduced states ρ N i : H . ρ is called electron density matrix. In terms of electron density matrix ρ , Pauli principle amounts to � ψ | ρ | ψ � ≤ 1 ⇐ ⇒ Spec ρ ≤ 1 Modern Version (1926) Heisenberg-Dirac replaced the Pauli exclusion principle by skew-symmetry of multi-electron wave function Murat Altunbulak Oxford 2016

Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H ⊗ N , H ⊗ N = state space of N electrons. Define ρ = ρ N 1 + ρ N 2 + . . . + ρ N N sum of all reduced states ρ N i : H . ρ is called electron density matrix. In terms of electron density matrix ρ , Pauli principle amounts to � ψ | ρ | ψ � ≤ 1 ⇐ ⇒ Spec ρ ≤ 1 Modern Version (1926) Heisenberg-Dirac replaced the Pauli exclusion principle by skew-symmetry of multi-electron wave function which implies that the state space of N electrons shrinks to ∧ N H ⊂ H ⊗ N . Murat Altunbulak Oxford 2016

Pauli exclusion principle Initial form (1925) state that no two identical fermions may occupy the same quantum state. In the language of density matrices it can be stated as follows Let ρ N : H ⊗ N , H ⊗ N = state space of N electrons. Define ρ = ρ N 1 + ρ N 2 + . . . + ρ N N sum of all reduced states ρ N i : H . ρ is called electron density matrix. In terms of electron density matrix ρ , Pauli principle amounts to � ψ | ρ | ψ � ≤ 1 ⇐ ⇒ Spec ρ ≤ 1 Modern Version (1926) Heisenberg-Dirac replaced the Pauli exclusion principle by skew-symmetry of multi-electron wave function which implies that the state space of N electrons shrinks to ∧ N H ⊂ H ⊗ N . This implies the original Pauli principle, because ψ ∧ ψ = 0. Murat Altunbulak Oxford 2016

Statement of The Problem Murat Altunbulak Oxford 2016

Statement of the problem In the latter case, electron density matrix becomes ρ = N ρ N 1 , Tr ρ = N . Murat Altunbulak Oxford 2016

Statement of the problem In the latter case, electron density matrix becomes ρ = N ρ N 1 , Tr ρ = N . Problem What are the constraints on electron density matrix ρ beyond the original Pauli principle Spec ρ ≤ 1. Murat Altunbulak Oxford 2016

Statement of the problem In the latter case, electron density matrix becomes ρ = N ρ N 1 , Tr ρ = N . Problem What are the constraints on electron density matrix ρ beyond the original Pauli principle Spec ρ ≤ 1. Pure N -representability The above problem became known as (pure) N -representability problem after A.J. Coleman (1963). Murat Altunbulak Oxford 2016

Statement of the problem In the latter case, electron density matrix becomes ρ = N ρ N 1 , Tr ρ = N . Problem What are the constraints on electron density matrix ρ beyond the original Pauli principle Spec ρ ≤ 1. Pure N -representability The above problem became known as (pure) N -representability problem after A.J. Coleman (1963). Mixed N -representability More generally we have mixed N -representability problem: “What are the constraints on the spectra of a mixed state and its reduced matrix?” Murat Altunbulak Oxford 2016

What was known before 2008? Murat Altunbulak Oxford 2016

What was known before 2008? Initial constraints Ordering inequalities λ 1 ≥ λ 2 ≥ . . . ≥ λ r ≥ 0 Murat Altunbulak Oxford 2016

What was known before 2008? Initial constraints Ordering inequalities λ 1 ≥ λ 2 ≥ . . . ≥ λ r ≥ 0 Normalization condition Tr ρ = � i λ i = N Murat Altunbulak Oxford 2016

What was known before 2008? Initial constraints Ordering inequalities λ 1 ≥ λ 2 ≥ . . . ≥ λ r ≥ 0 Normalization condition Tr ρ = � i λ i = N Two-particle ( ∧ 2 H r ) and Two-hole ( ∧ r − 2 H r ) systems Constraints on electron density matrix ρ is given by even degeneracy of its eigenvalues, i.e. λ 1 = λ 2 ≥ λ 3 = λ 4 ≥ . . . Murat Altunbulak Oxford 2016

What was known before 2008? Initial constraints Ordering inequalities λ 1 ≥ λ 2 ≥ . . . ≥ λ r ≥ 0 Normalization condition Tr ρ = � i λ i = N Two-particle ( ∧ 2 H r ) and Two-hole ( ∧ r − 2 H r ) systems Constraints on electron density matrix ρ is given by even degeneracy of its eigenvalues, i.e. λ 1 = λ 2 ≥ λ 3 = λ 4 ≥ . . . Similar results hold for two-hole system. Murat Altunbulak Oxford 2016

What was known before 2008? Borland-Dennis system For the system ∧ 3 H 6 of three electrons of rank 6, the N -representability conditions are given by the following (in)equalities: λ 1 + λ 6 = λ 2 + λ 5 = λ 3 + λ 4 = 1 , λ 4 ≤ λ 5 + λ 6 , Murat Altunbulak Oxford 2016