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

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

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

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

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

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

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

More • where examples Nq2 χ(0) = 2 0mΩ • recall m(x¨ + Γx˙ + Ω2x) = qE • N ≈ 1 and Ω ≈ 1 in natural units!