Peter Hertel
Overview Normal matter Natural units Atomic units
Length and energy
Electric and Peter Hertel magnetic fields
More University of Osnabr¨uck,Germany examples Lecture presented at APS, Nankai University, China
http://www.home.uni-osnabrueck.de/phertel
October/November 2011 • Normal matter • Atomic units • SI to AU conversion
Natural units Overview Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • Atomic units • SI to AU conversion
Natural units Overview Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields • Normal matter More examples • SI to AU conversion
Natural units Overview Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields • Normal matter More examples • Atomic units Natural units Overview Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields • Normal matter More examples • Atomic units • SI to AU conversion • normal matter is governed by electrostatic forces and non-relativistic quantum mechanics • heavy nuclei are surrounded by electrons neutral atoms, molecules, liquids and solids • Planck’s constant ~ • unit charge e • electron mass m
• 4π0 from Coulomb’s law • typical values for normal matter are reasonable numbers • times products of powers of these constants
Natural units Normal matter Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • heavy nuclei are surrounded by electrons neutral atoms, molecules, liquids and solids • Planck’s constant ~ • unit charge e • electron mass m
• 4π0 from Coulomb’s law • typical values for normal matter are reasonable numbers • times products of powers of these constants
Natural units Normal matter Peter Hertel
Overview
Normal matter Atomic units • normal matter is governed by electrostatic forces and Length and non-relativistic quantum mechanics energy
Electric and magnetic fields
More examples • Planck’s constant ~ • unit charge e • electron mass m
• 4π0 from Coulomb’s law • typical values for normal matter are reasonable numbers • times products of powers of these constants
Natural units Normal matter Peter Hertel
Overview
Normal matter Atomic units • normal matter is governed by electrostatic forces and Length and non-relativistic quantum mechanics energy Electric and • heavy nuclei are surrounded by electrons neutral atoms, magnetic fields molecules, liquids and solids
More examples • unit charge e • electron mass m
• 4π0 from Coulomb’s law • typical values for normal matter are reasonable numbers • times products of powers of these constants
Natural units Normal matter Peter Hertel
Overview
Normal matter Atomic units • normal matter is governed by electrostatic forces and Length and non-relativistic quantum mechanics energy Electric and • heavy nuclei are surrounded by electrons neutral atoms, magnetic fields molecules, liquids and solids More • examples Planck’s constant ~ • electron mass m
• 4π0 from Coulomb’s law • typical values for normal matter are reasonable numbers • times products of powers of these constants
Natural units Normal matter Peter Hertel
Overview
Normal matter Atomic units • normal matter is governed by electrostatic forces and Length and non-relativistic quantum mechanics energy Electric and • heavy nuclei are surrounded by electrons neutral atoms, magnetic fields molecules, liquids and solids More • examples Planck’s constant ~ • unit charge e • 4π0 from Coulomb’s law • typical values for normal matter are reasonable numbers • times products of powers of these constants
Natural units Normal matter Peter Hertel
Overview
Normal matter Atomic units • normal matter is governed by electrostatic forces and Length and non-relativistic quantum mechanics energy Electric and • heavy nuclei are surrounded by electrons neutral atoms, magnetic fields molecules, liquids and solids More • examples Planck’s constant ~ • unit charge e • electron mass m • typical values for normal matter are reasonable numbers • times products of powers of these constants
Natural units Normal matter Peter Hertel
Overview
Normal matter Atomic units • normal matter is governed by electrostatic forces and Length and non-relativistic quantum mechanics energy Electric and • heavy nuclei are surrounded by electrons neutral atoms, magnetic fields molecules, liquids and solids More • examples Planck’s constant ~ • unit charge e • electron mass m
• 4π0 from Coulomb’s law • times products of powers of these constants
Natural units Normal matter Peter Hertel
Overview
Normal matter Atomic units • normal matter is governed by electrostatic forces and Length and non-relativistic quantum mechanics energy Electric and • heavy nuclei are surrounded by electrons neutral atoms, magnetic fields molecules, liquids and solids More • examples Planck’s constant ~ • unit charge e • electron mass m
• 4π0 from Coulomb’s law • typical values for normal matter are reasonable numbers Natural units Normal matter Peter Hertel
Overview
Normal matter Atomic units • normal matter is governed by electrostatic forces and Length and non-relativistic quantum mechanics energy Electric and • heavy nuclei are surrounded by electrons neutral atoms, magnetic fields molecules, liquids and solids More • examples Planck’s constant ~ • unit charge e • electron mass m
• 4π0 from Coulomb’s law • typical values for normal matter are reasonable numbers • times products of powers of these constants • today, the Syst`emeinternational d’unit´es (SI) is in common use, in particular in science and engineering • also called MKSA • MKSA = meter (m), kilogram (kg), second (s) and ampere (A) • other SI units are derived from it, like volt (V) −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A
Natural units Constants of nature Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • also called MKSA • MKSA = meter (m), kilogram (kg), second (s) and ampere (A) • other SI units are derived from it, like volt (V) −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A
Natural units Constants of nature Peter Hertel
Overview
Normal matter Atomic units • today, the Syst`emeinternational d’unit´es (SI) is in Length and energy common use, in particular in science and engineering
Electric and magnetic fields
More examples • MKSA = meter (m), kilogram (kg), second (s) and ampere (A) • other SI units are derived from it, like volt (V) −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A
Natural units Constants of nature Peter Hertel
Overview
Normal matter Atomic units • today, the Syst`emeinternational d’unit´es (SI) is in Length and energy common use, in particular in science and engineering Electric and • magnetic also called MKSA fields
More examples • other SI units are derived from it, like volt (V) −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A
Natural units Constants of nature Peter Hertel
Overview
Normal matter Atomic units • today, the Syst`emeinternational d’unit´es (SI) is in Length and energy common use, in particular in science and engineering Electric and • magnetic also called MKSA fields • MKSA = meter (m), kilogram (kg), second (s) and More examples ampere (A) −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A
Natural units Constants of nature Peter Hertel
Overview
Normal matter Atomic units • today, the Syst`emeinternational d’unit´es (SI) is in Length and energy common use, in particular in science and engineering Electric and • magnetic also called MKSA fields • MKSA = meter (m), kilogram (kg), second (s) and More examples ampere (A) • other SI units are derived from it, like volt (V) • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A
Natural units Constants of nature Peter Hertel
Overview
Normal matter Atomic units • today, the Syst`emeinternational d’unit´es (SI) is in Length and energy common use, in particular in science and engineering Electric and • magnetic also called MKSA fields • MKSA = meter (m), kilogram (kg), second (s) and More examples ampere (A) • other SI units are derived from it, like volt (V) −34 2 −1 • ~ = 1.054572 ×10 kg m s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A
Natural units Constants of nature Peter Hertel
Overview
Normal matter Atomic units • today, the Syst`emeinternational d’unit´es (SI) is in Length and energy common use, in particular in science and engineering Electric and • magnetic also called MKSA fields • MKSA = meter (m), kilogram (kg), second (s) and More examples ampere (A) • other SI units are derived from it, like volt (V) −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A
Natural units Constants of nature Peter Hertel
Overview
Normal matter Atomic units • today, the Syst`emeinternational d’unit´es (SI) is in Length and energy common use, in particular in science and engineering Electric and • magnetic also called MKSA fields • MKSA = meter (m), kilogram (kg), second (s) and More examples ampere (A) • other SI units are derived from it, like volt (V) −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg Natural units Constants of nature Peter Hertel
Overview
Normal matter Atomic units • today, the Syst`emeinternational d’unit´es (SI) is in Length and energy common use, in particular in science and engineering Electric and • magnetic also called MKSA fields • MKSA = meter (m), kilogram (kg), second (s) and More examples ampere (A) • other SI units are derived from it, like volt (V) −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A −34 2 −1 • ~ = 1.054572 ×10 kg m s • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A • the powers of SI units m kg s A ~ 2 1 -1 0 e 0 0 1 1 m 0 1 0 0 4π0 -3 -1 4 2
Natural units Powers of MKSA Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • e = 1.602177 ×10−19 A s • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A • the powers of SI units m kg s A ~ 2 1 -1 0 e 0 0 1 1 m 0 1 0 0 4π0 -3 -1 4 2
Natural units Powers of MKSA Peter Hertel
Overview
Normal matter Atomic units −34 2 −1 • ~ = 1.054572 ×10 kg m s Length and energy
Electric and magnetic fields
More examples • m = 9.10938 ×10−31 kg −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A • the powers of SI units m kg s A ~ 2 1 -1 0 e 0 0 1 1 m 0 1 0 0 4π0 -3 -1 4 2
Natural units Powers of MKSA Peter Hertel
Overview
Normal matter Atomic units −34 2 −1 • ~ = 1.054572 ×10 kg m s Length and energy • e = 1.602177 ×10−19 A s Electric and magnetic fields
More examples −10 −1 −3 2 • 4π0 = 1.112650 ×10 kg m A • the powers of SI units m kg s A ~ 2 1 -1 0 e 0 0 1 1 m 0 1 0 0 4π0 -3 -1 4 2
Natural units Powers of MKSA Peter Hertel
Overview
Normal matter Atomic units −34 2 −1 • ~ = 1.054572 ×10 kg m s Length and energy • e = 1.602177 ×10−19 A s Electric and −31 magnetic • m = 9.10938 ×10 kg fields
More examples • the powers of SI units m kg s A ~ 2 1 -1 0 e 0 0 1 1 m 0 1 0 0 4π0 -3 -1 4 2
Natural units Powers of MKSA Peter Hertel
Overview
Normal matter Atomic units −34 2 −1 • ~ = 1.054572 ×10 kg m s Length and energy • e = 1.602177 ×10−19 A s Electric and −31 magnetic • m = 9.10938 ×10 kg fields • 4π = 1.112650 ×10−10 kg−1 m−3 A2 More 0 examples Natural units Powers of MKSA Peter Hertel
Overview
Normal matter Atomic units −34 2 −1 • ~ = 1.054572 ×10 kg m s Length and energy • e = 1.602177 ×10−19 A s Electric and −31 magnetic • m = 9.10938 ×10 kg fields • 4π = 1.112650 ×10−10 kg−1 m−3 A2 More 0 examples • the powers of SI units m kg s A ~ 2 1 -1 0 e 0 0 1 1 m 0 1 0 0 4π0 -3 -1 4 2 • the former 4 × 4 matrix can be inverted • SI units can be expressed as a product of powers of the involved constants of nature
~ e m 4π0 m 2 -2 -1 1 kg 0 0 1 0 s 3 -4 -1 2 A -3 5 1 -2 3 −4 −1 2 • read: s = number multiplied by ~ e m (4π0) etc. • find out number and exponents for arbitrary SI unit
Natural units Powers of natural constants Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • SI units can be expressed as a product of powers of the involved constants of nature
~ e m 4π0 m 2 -2 -1 1 kg 0 0 1 0 s 3 -4 -1 2 A -3 5 1 -2 3 −4 −1 2 • read: s = number multiplied by ~ e m (4π0) etc. • find out number and exponents for arbitrary SI unit
Natural units Powers of natural constants Peter Hertel
Overview
Normal matter Atomic units • the former 4 × 4 matrix can be inverted Length and energy
Electric and magnetic fields
More examples 3 −4 −1 2 • read: s = number multiplied by ~ e m (4π0) etc. • find out number and exponents for arbitrary SI unit
Natural units Powers of natural constants Peter Hertel
Overview
Normal matter Atomic units • the former 4 × 4 matrix can be inverted Length and energy • SI units can be expressed as a product of powers of the Electric and magnetic involved constants of nature fields e m 4π0 More ~ examples m 2 -2 -1 1 kg 0 0 1 0 s 3 -4 -1 2 A -3 5 1 -2 • find out number and exponents for arbitrary SI unit
Natural units Powers of natural constants Peter Hertel
Overview
Normal matter Atomic units • the former 4 × 4 matrix can be inverted Length and energy • SI units can be expressed as a product of powers of the Electric and magnetic involved constants of nature fields e m 4π0 More ~ examples m 2 -2 -1 1 kg 0 0 1 0 s 3 -4 -1 2 A -3 5 1 -2 3 −4 −1 2 • read: s = number multiplied by ~ e m (4π0) etc. Natural units Powers of natural constants Peter Hertel
Overview
Normal matter Atomic units • the former 4 × 4 matrix can be inverted Length and energy • SI units can be expressed as a product of powers of the Electric and magnetic involved constants of nature fields e m 4π0 More ~ examples m 2 -2 -1 1 kg 0 0 1 0 s 3 -4 -1 2 A -3 5 1 -2 3 −4 −1 2 • read: s = number multiplied by ~ e m (4π0) etc. • find out number and exponents for arbitrary SI unit Natural units
Peter Hertel function [value,power]=atomic_unit(si) Overview val=[1.05457e-34,1.60218e-19,9.10938e-31,4*pi*8.85419E-12]; Normal matter dim=[2 1 -1 0; 0 0 1 1; 0 1 0 0; -3 -1 4 2]; Atomic units mid=round(inv(dim)); Length and energy power=si*mid; Electric and value=prod(val.^power); magnetic fields
More examples
>> length = [0 1 0 0]; >> length = [1 0 0 0]; >> [astar,apow]=atomic_unit(length); >> astar 5.2917e-11 >> apow 2 -2 -1 1 Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • recall length=[1 0 0 0] • recall [astar,apow]=atomic unit(length) • recall astar = 5.2971e-11 • recall apow = [2 -2 -1 1] • we have just calculated the atomic unit of length • i. e. Bohr’s radius 4π 2 a = 0~ = 5.2917 × 10−11 m ∗ me2
Natural units e. g. atomic length unit Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • recall [astar,apow]=atomic unit(length) • recall astar = 5.2971e-11 • recall apow = [2 -2 -1 1] • we have just calculated the atomic unit of length • i. e. Bohr’s radius 4π 2 a = 0~ = 5.2917 × 10−11 m ∗ me2
Natural units e. g. atomic length unit Peter Hertel
Overview
Normal matter
Atomic units Length and • recall energy length=[1 0 0 0]
Electric and magnetic fields
More examples • recall astar = 5.2971e-11 • recall apow = [2 -2 -1 1] • we have just calculated the atomic unit of length • i. e. Bohr’s radius 4π 2 a = 0~ = 5.2917 × 10−11 m ∗ me2
Natural units e. g. atomic length unit Peter Hertel
Overview
Normal matter
Atomic units Length and • recall energy length=[1 0 0 0] Electric and • recall [astar,apow]=atomic unit(length) magnetic fields
More examples • recall apow = [2 -2 -1 1] • we have just calculated the atomic unit of length • i. e. Bohr’s radius 4π 2 a = 0~ = 5.2917 × 10−11 m ∗ me2
Natural units e. g. atomic length unit Peter Hertel
Overview
Normal matter
Atomic units Length and • recall energy length=[1 0 0 0] Electric and • recall [astar,apow]=atomic unit(length) magnetic fields • recall astar = 5.2971e-11 More examples • we have just calculated the atomic unit of length • i. e. Bohr’s radius 4π 2 a = 0~ = 5.2917 × 10−11 m ∗ me2
Natural units e. g. atomic length unit Peter Hertel
Overview
Normal matter
Atomic units Length and • recall energy length=[1 0 0 0] Electric and • recall [astar,apow]=atomic unit(length) magnetic fields • recall astar = 5.2971e-11 More examples • recall apow = [2 -2 -1 1] • i. e. Bohr’s radius 4π 2 a = 0~ = 5.2917 × 10−11 m ∗ me2
Natural units e. g. atomic length unit Peter Hertel
Overview
Normal matter
Atomic units Length and • recall energy length=[1 0 0 0] Electric and • recall [astar,apow]=atomic unit(length) magnetic fields • recall astar = 5.2971e-11 More examples • recall apow = [2 -2 -1 1] • we have just calculated the atomic unit of length Natural units e. g. atomic length unit Peter Hertel
Overview
Normal matter
Atomic units Length and • recall energy length=[1 0 0 0] Electric and • recall [astar,apow]=atomic unit(length) magnetic fields • recall astar = 5.2971e-11 More examples • recall apow = [2 -2 -1 1] • we have just calculated the atomic unit of length • i. e. Bohr’s radius 4π 2 a = 0~ = 5.2917 × 10−11 m ∗ me2 • Schr¨odingerequation for hydrogen atom 2 1 −e2 − ~ ∆φ + φ = Eφ 2m 4π0 r • in atomic units 1 1 − ∆φ − φ = Eφ 2 r • ground state −r 1 φ ∝ e with E = − 2 • radius −r −r R ∞ dr r2 e r e hRi = 0 = 1 R ∞ 2 −r −r 0 dr r e 1 e • Bohr’s result 4π 2 hRi = a = 0~ ∗ me2
Natural units Bohr’s radius Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • in atomic units 1 1 − ∆φ − φ = Eφ 2 r • ground state −r 1 φ ∝ e with E = − 2 • radius −r −r R ∞ dr r2 e r e hRi = 0 = 1 R ∞ 2 −r −r 0 dr r e 1 e • Bohr’s result 4π 2 hRi = a = 0~ ∗ me2
Natural units Bohr’s radius Peter Hertel
Overview • Schr¨odingerequation for hydrogen atom Normal matter 2 2 ~ 1 −e Atomic units − ∆φ + φ = Eφ 2m 4π r Length and 0 energy
Electric and magnetic fields
More examples • ground state −r 1 φ ∝ e with E = − 2 • radius −r −r R ∞ dr r2 e r e hRi = 0 = 1 R ∞ 2 −r −r 0 dr r e 1 e • Bohr’s result 4π 2 hRi = a = 0~ ∗ me2
Natural units Bohr’s radius Peter Hertel
Overview • Schr¨odingerequation for hydrogen atom Normal matter 2 2 ~ 1 −e Atomic units − ∆φ + φ = Eφ 2m 4π r Length and 0 energy • in atomic units Electric and 1 1 magnetic − ∆φ − φ = Eφ fields 2 r More examples • radius −r −r R ∞ dr r2 e r e hRi = 0 = 1 R ∞ 2 −r −r 0 dr r e 1 e • Bohr’s result 4π 2 hRi = a = 0~ ∗ me2
Natural units Bohr’s radius Peter Hertel
Overview • Schr¨odingerequation for hydrogen atom Normal matter 2 2 ~ 1 −e Atomic units − ∆φ + φ = Eφ 2m 4π r Length and 0 energy • in atomic units Electric and 1 1 magnetic − ∆φ − φ = Eφ fields 2 r More • ground state examples −r 1 φ ∝ e with E = − 2 • Bohr’s result 4π 2 hRi = a = 0~ ∗ me2
Natural units Bohr’s radius Peter Hertel
Overview • Schr¨odingerequation for hydrogen atom Normal matter 2 2 ~ 1 −e Atomic units − ∆φ + φ = Eφ 2m 4π r Length and 0 energy • in atomic units Electric and 1 1 magnetic − ∆φ − φ = Eφ fields 2 r More • ground state examples −r 1 φ ∝ e with E = − 2 • radius −r −r R ∞ dr r2 e r e hRi = 0 = 1 R ∞ 2 −r −r 0 dr r e 1 e Natural units Bohr’s radius Peter Hertel
Overview • Schr¨odingerequation for hydrogen atom Normal matter 2 2 ~ 1 −e Atomic units − ∆φ + φ = Eφ 2m 4π r Length and 0 energy • in atomic units Electric and 1 1 magnetic − ∆φ − φ = Eφ fields 2 r More • ground state examples −r 1 φ ∝ e with E = − 2 • radius −r −r R ∞ dr r2 e r e hRi = 0 = 1 R ∞ 2 −r −r 0 dr r e 1 e • Bohr’s result 4π 2 hRi = a = 0~ ∗ me2 • MKSA for energy is [2 1 -2 0] • [Estar,Epow]=atomic unit(energy) −18 • E∗ = 4.3598 × 10 J = 27.21 eV • atomic energy unit Hartree • H atom ground state energy is E = −13.6 eV • photon energies are in the eV range • He-Ne laser: λ = 633 nm
• ~ω = 0.072 E∗ = 1.96 eV
Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • [Estar,Epow]=atomic unit(energy) −18 • E∗ = 4.3598 × 10 J = 27.21 eV • atomic energy unit Hartree • H atom ground state energy is E = −13.6 eV • photon energies are in the eV range • He-Ne laser: λ = 633 nm
• ~ω = 0.072 E∗ = 1.96 eV
Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • MKSA for energy is [2 1 -2 0]
Electric and magnetic fields
More examples −18 • E∗ = 4.3598 × 10 J = 27.21 eV • atomic energy unit Hartree • H atom ground state energy is E = −13.6 eV • photon energies are in the eV range • He-Ne laser: λ = 633 nm
• ~ω = 0.072 E∗ = 1.96 eV
Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • MKSA for energy is [2 1 -2 0] Electric and • [Estar,Epow]=atomic unit(energy) magnetic fields
More examples • atomic energy unit Hartree • H atom ground state energy is E = −13.6 eV • photon energies are in the eV range • He-Ne laser: λ = 633 nm
• ~ω = 0.072 E∗ = 1.96 eV
Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • MKSA for energy is [2 1 -2 0] Electric and • [Estar,Epow]=atomic unit(energy) magnetic fields −18 • E∗ = 4.3598 × 10 J = 27.21 eV More examples • H atom ground state energy is E = −13.6 eV • photon energies are in the eV range • He-Ne laser: λ = 633 nm
• ~ω = 0.072 E∗ = 1.96 eV
Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • MKSA for energy is [2 1 -2 0] Electric and • [Estar,Epow]=atomic unit(energy) magnetic fields −18 • E∗ = 4.3598 × 10 J = 27.21 eV More examples • atomic energy unit Hartree • photon energies are in the eV range • He-Ne laser: λ = 633 nm
• ~ω = 0.072 E∗ = 1.96 eV
Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • MKSA for energy is [2 1 -2 0] Electric and • [Estar,Epow]=atomic unit(energy) magnetic fields −18 • E∗ = 4.3598 × 10 J = 27.21 eV More examples • atomic energy unit Hartree • H atom ground state energy is E = −13.6 eV • He-Ne laser: λ = 633 nm
• ~ω = 0.072 E∗ = 1.96 eV
Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • MKSA for energy is [2 1 -2 0] Electric and • [Estar,Epow]=atomic unit(energy) magnetic fields −18 • E∗ = 4.3598 × 10 J = 27.21 eV More examples • atomic energy unit Hartree • H atom ground state energy is E = −13.6 eV • photon energies are in the eV range • ~ω = 0.072 E∗ = 1.96 eV
Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • MKSA for energy is [2 1 -2 0] Electric and • [Estar,Epow]=atomic unit(energy) magnetic fields −18 • E∗ = 4.3598 × 10 J = 27.21 eV More examples • atomic energy unit Hartree • H atom ground state energy is E = −13.6 eV • photon energies are in the eV range • He-Ne laser: λ = 633 nm Natural units Hartree Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • MKSA for energy is [2 1 -2 0] Electric and • [Estar,Epow]=atomic unit(energy) magnetic fields −18 • E∗ = 4.3598 × 10 J = 27.21 eV More examples • atomic energy unit Hartree • H atom ground state energy is E = −13.6 eV • photon energies are in the eV range • He-Ne laser: λ = 633 nm
• ~ω = 0.072 E∗ = 1.96 eV • voltage=[2 1 -3 -1] • el field str=[1 1 -3 -1] • the atomic (natural) unit of electric field strength is 2 5 m e 11 −1 E∗ = 3 4 = 5.1423 × 10 V m (4π0) ~ • Ohms law, Pockels effect, Starck effect . . . • external field strength is always very small
Natural units Electric field strength Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • el field str=[1 1 -3 -1] • the atomic (natural) unit of electric field strength is 2 5 m e 11 −1 E∗ = 3 4 = 5.1423 × 10 V m (4π0) ~ • Ohms law, Pockels effect, Starck effect . . . • external field strength is always very small
Natural units Electric field strength Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • voltage=[2 1 -3 -1] Electric and magnetic fields
More examples • the atomic (natural) unit of electric field strength is 2 5 m e 11 −1 E∗ = 3 4 = 5.1423 × 10 V m (4π0) ~ • Ohms law, Pockels effect, Starck effect . . . • external field strength is always very small
Natural units Electric field strength Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • voltage=[2 1 -3 -1] Electric and magnetic • el field str=[1 1 -3 -1] fields
More examples • Ohms law, Pockels effect, Starck effect . . . • external field strength is always very small
Natural units Electric field strength Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • voltage=[2 1 -3 -1] Electric and magnetic • el field str=[1 1 -3 -1] fields
More • the atomic (natural) unit of electric field strength is examples 2 5 m e 11 −1 E∗ = 3 4 = 5.1423 × 10 V m (4π0) ~ • external field strength is always very small
Natural units Electric field strength Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • voltage=[2 1 -3 -1] Electric and magnetic • el field str=[1 1 -3 -1] fields
More • the atomic (natural) unit of electric field strength is examples 2 5 m e 11 −1 E∗ = 3 4 = 5.1423 × 10 V m (4π0) ~ • Ohms law, Pockels effect, Starck effect . . . Natural units Electric field strength Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • voltage=[2 1 -3 -1] Electric and magnetic • el field str=[1 1 -3 -1] fields
More • the atomic (natural) unit of electric field strength is examples 2 5 m e 11 −1 E∗ = 3 4 = 5.1423 × 10 V m (4π0) ~ • Ohms law, Pockels effect, Starck effect . . . • external field strength is always very small • induction=[0 1 -2 -1] • the atomic (natural) unit of magnetic induction is 2 3 m e 5 B∗ = 2 3 = 2.3505 × 10 T (4π0) ~ • Faraday effect, Hall effect . . . • external field induction is always very small
Natural units Magnetic induction Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • the atomic (natural) unit of magnetic induction is 2 3 m e 5 B∗ = 2 3 = 2.3505 × 10 T (4π0) ~ • Faraday effect, Hall effect . . . • external field induction is always very small
Natural units Magnetic induction Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and • induction=[0 1 -2 -1] magnetic fields
More examples • Faraday effect, Hall effect . . . • external field induction is always very small
Natural units Magnetic induction Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and • induction=[0 1 -2 -1] magnetic fields • the atomic (natural) unit of magnetic induction is More 2 3 examples m e 5 B∗ = 2 3 = 2.3505 × 10 T (4π0) ~ • external field induction is always very small
Natural units Magnetic induction Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and • induction=[0 1 -2 -1] magnetic fields • the atomic (natural) unit of magnetic induction is More 2 3 examples m e 5 B∗ = 2 3 = 2.3505 × 10 T (4π0) ~ • Faraday effect, Hall effect . . . Natural units Magnetic induction Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and • induction=[0 1 -2 -1] magnetic fields • the atomic (natural) unit of magnetic induction is More 2 3 examples m e 5 B∗ = 2 3 = 2.3505 × 10 T (4π0) ~ • Faraday effect, Hall effect . . . • external field induction is always very small • rho=[-3 1 0 0] • [rhoval,rhopow]=atomic unit(rho) • rhoval = 6.174 • rhopow=[-6 6 4 -3] • 4 6 m e m −3 %∗ = 3 6 = 3 = 6.147 kg m (4π0) ~ a∗ • mass is mass of nucleons, therefore must be multiplied by approximately 4000
Natural units Mass density Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • [rhoval,rhopow]=atomic unit(rho) • rhoval = 6.174 • rhopow=[-6 6 4 -3] • 4 6 m e m −3 %∗ = 3 6 = 3 = 6.147 kg m (4π0) ~ a∗ • mass is mass of nucleons, therefore must be multiplied by approximately 4000
Natural units Mass density Peter Hertel
Overview
Normal matter
Atomic units
Length and • rho=[-3 1 0 0] energy
Electric and magnetic fields
More examples • rhoval = 6.174 • rhopow=[-6 6 4 -3] • 4 6 m e m −3 %∗ = 3 6 = 3 = 6.147 kg m (4π0) ~ a∗ • mass is mass of nucleons, therefore must be multiplied by approximately 4000
Natural units Mass density Peter Hertel
Overview
Normal matter
Atomic units
Length and • rho=[-3 1 0 0] energy • [rhoval,rhopow]=atomic unit(rho) Electric and magnetic fields
More examples • rhopow=[-6 6 4 -3] • 4 6 m e m −3 %∗ = 3 6 = 3 = 6.147 kg m (4π0) ~ a∗ • mass is mass of nucleons, therefore must be multiplied by approximately 4000
Natural units Mass density Peter Hertel
Overview
Normal matter
Atomic units
Length and • rho=[-3 1 0 0] energy • [rhoval,rhopow]=atomic unit(rho) Electric and magnetic fields • rhoval = 6.174
More examples • 4 6 m e m −3 %∗ = 3 6 = 3 = 6.147 kg m (4π0) ~ a∗ • mass is mass of nucleons, therefore must be multiplied by approximately 4000
Natural units Mass density Peter Hertel
Overview
Normal matter
Atomic units
Length and • rho=[-3 1 0 0] energy • [rhoval,rhopow]=atomic unit(rho) Electric and magnetic fields • rhoval = 6.174 More • rhopow=[-6 6 4 -3] examples • mass is mass of nucleons, therefore must be multiplied by approximately 4000
Natural units Mass density Peter Hertel
Overview
Normal matter
Atomic units
Length and • rho=[-3 1 0 0] energy • [rhoval,rhopow]=atomic unit(rho) Electric and magnetic fields • rhoval = 6.174 More • rhopow=[-6 6 4 -3] examples • 4 6 m e m −3 %∗ = 3 6 = 3 = 6.147 kg m (4π0) ~ a∗ Natural units Mass density Peter Hertel
Overview
Normal matter
Atomic units
Length and • rho=[-3 1 0 0] energy • [rhoval,rhopow]=atomic unit(rho) Electric and magnetic fields • rhoval = 6.174 More • rhopow=[-6 6 4 -3] examples • 4 6 m e m −3 %∗ = 3 6 = 3 = 6.147 kg m (4π0) ~ a∗ • mass is mass of nucleons, therefore must be multiplied by approximately 4000 • velocity=[1 0 -1 0] • atomic unit of velocity is 2 e 6 −1 v∗ = = 2.1877 × 10 m s 4π0~ −1 • velocity of light is c = α v∗ • fine structure constant α = 1/137.035999074 • can be measured directly (quantum Hall effect) • relativistic corrections
Natural units Light velocity Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • atomic unit of velocity is 2 e 6 −1 v∗ = = 2.1877 × 10 m s 4π0~ −1 • velocity of light is c = α v∗ • fine structure constant α = 1/137.035999074 • can be measured directly (quantum Hall effect) • relativistic corrections
Natural units Light velocity Peter Hertel
Overview
Normal matter
Atomic units Length and • velocity=[1 0 -1 0] energy
Electric and magnetic fields
More examples −1 • velocity of light is c = α v∗ • fine structure constant α = 1/137.035999074 • can be measured directly (quantum Hall effect) • relativistic corrections
Natural units Light velocity Peter Hertel
Overview
Normal matter
Atomic units Length and • velocity=[1 0 -1 0] energy • Electric and atomic unit of velocity is magnetic 2 fields e 6 −1 v∗ = = 2.1877 × 10 m s More 4π0~ examples • fine structure constant α = 1/137.035999074 • can be measured directly (quantum Hall effect) • relativistic corrections
Natural units Light velocity Peter Hertel
Overview
Normal matter
Atomic units Length and • velocity=[1 0 -1 0] energy • Electric and atomic unit of velocity is magnetic 2 fields e 6 −1 v∗ = = 2.1877 × 10 m s More 4π0~ examples −1 • velocity of light is c = α v∗ • can be measured directly (quantum Hall effect) • relativistic corrections
Natural units Light velocity Peter Hertel
Overview
Normal matter
Atomic units Length and • velocity=[1 0 -1 0] energy • Electric and atomic unit of velocity is magnetic 2 fields e 6 −1 v∗ = = 2.1877 × 10 m s More 4π0~ examples −1 • velocity of light is c = α v∗ • fine structure constant α = 1/137.035999074 • relativistic corrections
Natural units Light velocity Peter Hertel
Overview
Normal matter
Atomic units Length and • velocity=[1 0 -1 0] energy • Electric and atomic unit of velocity is magnetic 2 fields e 6 −1 v∗ = = 2.1877 × 10 m s More 4π0~ examples −1 • velocity of light is c = α v∗ • fine structure constant α = 1/137.035999074 • can be measured directly (quantum Hall effect) Natural units Light velocity Peter Hertel
Overview
Normal matter
Atomic units Length and • velocity=[1 0 -1 0] energy • Electric and atomic unit of velocity is magnetic 2 fields e 6 −1 v∗ = = 2.1877 × 10 m s More 4π0~ examples −1 • velocity of light is c = α v∗ • fine structure constant α = 1/137.035999074 • can be measured directly (quantum Hall effect) • relativistic corrections • external electric field E • permittivity change −1 −1 ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek • Pockels coefficients vanish if crystal has an inversion center • lithium niobate has no inversion center −1 • largest coefficient r333 = 30 pm V • small or large?
• r333 E∗ ≈ 16 • lithium niobate is rather resilient to electrical fields
Natural units Pockels coefficients Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • permittivity change −1 −1 ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek • Pockels coefficients vanish if crystal has an inversion center • lithium niobate has no inversion center −1 • largest coefficient r333 = 30 pm V • small or large?
• r333 E∗ ≈ 16 • lithium niobate is rather resilient to electrical fields
Natural units Pockels coefficients Peter Hertel
Overview
Normal matter Atomic units • external electric field E Length and energy
Electric and magnetic fields
More examples • Pockels coefficients vanish if crystal has an inversion center • lithium niobate has no inversion center −1 • largest coefficient r333 = 30 pm V • small or large?
• r333 E∗ ≈ 16 • lithium niobate is rather resilient to electrical fields
Natural units Pockels coefficients Peter Hertel
Overview
Normal matter Atomic units • external electric field E Length and energy • permittivity change Electric and −1 −1 magnetic ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek fields
More examples • lithium niobate has no inversion center −1 • largest coefficient r333 = 30 pm V • small or large?
• r333 E∗ ≈ 16 • lithium niobate is rather resilient to electrical fields
Natural units Pockels coefficients Peter Hertel
Overview
Normal matter Atomic units • external electric field E Length and energy • permittivity change Electric and −1 −1 magnetic ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek fields • More Pockels coefficients vanish if crystal has an examples inversion center −1 • largest coefficient r333 = 30 pm V • small or large?
• r333 E∗ ≈ 16 • lithium niobate is rather resilient to electrical fields
Natural units Pockels coefficients Peter Hertel
Overview
Normal matter Atomic units • external electric field E Length and energy • permittivity change Electric and −1 −1 magnetic ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek fields • More Pockels coefficients vanish if crystal has an examples inversion center • lithium niobate has no inversion center • small or large?
• r333 E∗ ≈ 16 • lithium niobate is rather resilient to electrical fields
Natural units Pockels coefficients Peter Hertel
Overview
Normal matter Atomic units • external electric field E Length and energy • permittivity change Electric and −1 −1 magnetic ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek fields • More Pockels coefficients vanish if crystal has an examples inversion center • lithium niobate has no inversion center −1 • largest coefficient r333 = 30 pm V • r333 E∗ ≈ 16 • lithium niobate is rather resilient to electrical fields
Natural units Pockels coefficients Peter Hertel
Overview
Normal matter Atomic units • external electric field E Length and energy • permittivity change Electric and −1 −1 magnetic ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek fields • More Pockels coefficients vanish if crystal has an examples inversion center • lithium niobate has no inversion center −1 • largest coefficient r333 = 30 pm V • small or large? • lithium niobate is rather resilient to electrical fields
Natural units Pockels coefficients Peter Hertel
Overview
Normal matter Atomic units • external electric field E Length and energy • permittivity change Electric and −1 −1 magnetic ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek fields • More Pockels coefficients vanish if crystal has an examples inversion center • lithium niobate has no inversion center −1 • largest coefficient r333 = 30 pm V • small or large?
• r333 E∗ ≈ 16 Natural units Pockels coefficients Peter Hertel
Overview
Normal matter Atomic units • external electric field E Length and energy • permittivity change Electric and −1 −1 magnetic ((ω; E) )ij = ((ω; 0) )ij + rijk(ω)Ek fields • More Pockels coefficients vanish if crystal has an examples inversion center • lithium niobate has no inversion center −1 • largest coefficient r333 = 30 pm V • small or large?
• r333 E∗ ≈ 16 • lithium niobate is rather resilient to electrical fields • An elastic medium is described by stress and strain
• strain tensor Sij describes deformation
• stress tensor Tij describes force on area element
• dFj = dAiTij • Hooke’s law E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • strain tensor Sij describes deformation
• stress tensor Tij describes force on area element
• dFj = dAiTij • Hooke’s law E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units
Length and energy
Electric and magnetic fields
More examples • stress tensor Tij describes force on area element
• dFj = dAiTij • Hooke’s law E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy
Electric and magnetic fields
More examples • dFj = dAiTij • Hooke’s law E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy • stress tensor Tij describes force on area element Electric and magnetic fields
More examples • Hooke’s law E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy • stress tensor Tij describes force on area element Electric and magnetic • dF = dA T fields j i ij
More examples • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy • stress tensor Tij describes force on area element Electric and magnetic • dF = dA T fields j i ij
More • Hooke’s law examples E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy • stress tensor Tij describes force on area element Electric and magnetic • dF = dA T fields j i ij
More • Hooke’s law examples E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy • stress tensor Tij describes force on area element Electric and magnetic • dF = dA T fields j i ij
More • Hooke’s law examples E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E • . . . as order of magnitude • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy • stress tensor Tij describes force on area element Electric and magnetic • dF = dA T fields j i ij
More • Hooke’s law examples E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • steel : E = 200 GPa
Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy • stress tensor Tij describes force on area element Electric and magnetic • dF = dA T fields j i ij
More • Hooke’s law examples E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude Natural units Elasticity module Peter Hertel
Overview Normal matter • An elastic medium is described by stress and strain Atomic units • strain tensor S describes deformation Length and ij energy • stress tensor Tij describes force on area element Electric and magnetic • dF = dA T fields j i ij
More • Hooke’s law examples E ν T = S + δ S ij 1 + ν ij 1 − 2ν ij kk • Poisson’s ratio between 0 and 1/2 • elasticity module E −3 • expect E = eV A˚ = 160 GPa • . . . as order of magnitude • steel : E = 200 GPa • Recall the Drude model result Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω • where Nq2 χ(0) = 2 0mΩ • recall m(x¨ + Γx˙ + Ω2x) = qE • N ≈ 1 and Ω ≈ 1 in natural units!
Natural units Susceptibility Peter Hertel
Overview
Normal matter
Atomic units
Length and energy
Electric and magnetic fields
More examples • where Nq2 χ(0) = 2 0mΩ • recall m(x¨ + Γx˙ + Ω2x) = qE • N ≈ 1 and Ω ≈ 1 in natural units!
Natural units Susceptibility Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • Recall the Drude model result 2 Electric and Ω magnetic χ(ω) = χ(0) 2 2 fields Ω − ω − iΓω
More examples • recall m(x¨ + Γx˙ + Ω2x) = qE • N ≈ 1 and Ω ≈ 1 in natural units!
Natural units Susceptibility Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • Recall the Drude model result 2 Electric and Ω magnetic χ(ω) = χ(0) 2 2 fields Ω − ω − iΓω
More • where examples Nq2 χ(0) = 2 0mΩ • N ≈ 1 and Ω ≈ 1 in natural units!
Natural units Susceptibility Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • Recall the Drude model result 2 Electric and Ω magnetic χ(ω) = χ(0) 2 2 fields Ω − ω − iΓω
More • where examples Nq2 χ(0) = 2 0mΩ • recall m(x¨ + Γx˙ + Ω2x) = qE Natural units Susceptibility Peter Hertel
Overview
Normal matter
Atomic units
Length and energy • Recall the Drude model result 2 Electric and Ω magnetic χ(ω) = χ(0) 2 2 fields Ω − ω − iΓω
More • where examples Nq2 χ(0) = 2 0mΩ • recall m(x¨ + Γx˙ + Ω2x) = qE • N ≈ 1 and Ω ≈ 1 in natural units!