overview natural units - uni-osnabrueck.deelds more examples overview normal matter atomic units si...
TRANSCRIPT
![Page 1: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/1.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Natural units
Peter Hertel
University of Osnabruck, Germany
Lecture presented at APS, Nankai University, China
http://www.home.uni-osnabrueck.de/phertel
October/November 2011
![Page 2: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/2.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Overview
• Normal matter
• Atomic units
• SI to AU conversion
![Page 3: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/3.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Overview
• Normal matter
• Atomic units
• SI to AU conversion
![Page 4: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/4.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Overview
• Normal matter
• Atomic units
• SI to AU conversion
![Page 5: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/5.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Overview
• Normal matter
• Atomic units
• SI to AU conversion
![Page 6: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/6.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 7: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/7.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 8: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/8.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 9: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/9.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~
• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 10: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/10.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 11: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/11.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 12: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/12.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 13: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/13.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 14: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/14.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Normal matter
• normal matter is governed by electrostatic forces andnon-relativistic quantum mechanics
• heavy nuclei are surrounded by electrons neutral atoms,molecules, liquids and solids
• Planck’s constant ~• unit charge e
• electron mass m
• 4πε0 from Coulomb’s law
• typical values for normal matter are reasonable numbers
• times products of powers of these constants
![Page 15: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/15.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 16: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/16.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 17: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/17.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 18: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/18.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 19: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/19.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 20: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/20.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 21: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/21.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 22: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/22.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 23: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/23.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Constants of nature
• today, the Systeme international d’unites (SI) is incommon use, in particular in science and engineering
• also called MKSA
• MKSA = meter (m), kilogram (kg), second (s) andampere (A)
• other SI units are derived from it, like volt (V)
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
![Page 24: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/24.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of MKSA
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
• the powers of SI units
m kg s A
~ 2 1 -1 0e 0 0 1 1m 0 1 0 0
4πε0 -3 -1 4 2
![Page 25: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/25.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of MKSA
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
• the powers of SI units
m kg s A
~ 2 1 -1 0e 0 0 1 1m 0 1 0 0
4πε0 -3 -1 4 2
![Page 26: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/26.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of MKSA
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
• the powers of SI units
m kg s A
~ 2 1 -1 0e 0 0 1 1m 0 1 0 0
4πε0 -3 -1 4 2
![Page 27: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/27.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of MKSA
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
• the powers of SI units
m kg s A
~ 2 1 -1 0e 0 0 1 1m 0 1 0 0
4πε0 -3 -1 4 2
![Page 28: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/28.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of MKSA
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
• the powers of SI units
m kg s A
~ 2 1 -1 0e 0 0 1 1m 0 1 0 0
4πε0 -3 -1 4 2
![Page 29: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/29.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of MKSA
• ~ = 1.054572×10−34 kg m2 s−1
• e = 1.602177×10−19 A s
• m = 9.10938×10−31 kg
• 4πε0 = 1.112650×10−10 kg−1 m−3 A2
• the powers of SI units
m kg s A
~ 2 1 -1 0e 0 0 1 1m 0 1 0 0
4πε0 -3 -1 4 2
![Page 30: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/30.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of natural constants
• the former 4× 4 matrix can be inverted
• SI units can be expressed as a product of powers of theinvolved constants of nature
~ e m 4πε0
m 2 -2 -1 1kg 0 0 1 0s 3 -4 -1 2A -3 5 1 -2
• read: s = number multiplied by ~3e−4m−1(4πε0)2 etc.
• find out number and exponents for arbitrary SI unit
![Page 31: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/31.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of natural constants
• the former 4× 4 matrix can be inverted
• SI units can be expressed as a product of powers of theinvolved constants of nature
~ e m 4πε0
m 2 -2 -1 1kg 0 0 1 0s 3 -4 -1 2A -3 5 1 -2
• read: s = number multiplied by ~3e−4m−1(4πε0)2 etc.
• find out number and exponents for arbitrary SI unit
![Page 32: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/32.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of natural constants
• the former 4× 4 matrix can be inverted
• SI units can be expressed as a product of powers of theinvolved constants of nature
~ e m 4πε0
m 2 -2 -1 1kg 0 0 1 0s 3 -4 -1 2A -3 5 1 -2
• read: s = number multiplied by ~3e−4m−1(4πε0)2 etc.
• find out number and exponents for arbitrary SI unit
![Page 33: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/33.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of natural constants
• the former 4× 4 matrix can be inverted
• SI units can be expressed as a product of powers of theinvolved constants of nature
~ e m 4πε0
m 2 -2 -1 1kg 0 0 1 0s 3 -4 -1 2A -3 5 1 -2
• read: s = number multiplied by ~3e−4m−1(4πε0)2 etc.
• find out number and exponents for arbitrary SI unit
![Page 34: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/34.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Powers of natural constants
• the former 4× 4 matrix can be inverted
• SI units can be expressed as a product of powers of theinvolved constants of nature
~ e m 4πε0
m 2 -2 -1 1kg 0 0 1 0s 3 -4 -1 2A -3 5 1 -2
• read: s = number multiplied by ~3e−4m−1(4πε0)2 etc.
• find out number and exponents for arbitrary SI unit
![Page 35: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/35.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
function [value,power]=atomic_unit(si)
val=[1.05457e-34,1.60218e-19,9.10938e-31,4*pi*8.85419E-12];
dim=[2 1 -1 0; 0 0 1 1; 0 1 0 0; -3 -1 4 2];
mid=round(inv(dim));
power=si*mid;
value=prod(val.^power);
>> length = [0 1 0 0];
>> length = [1 0 0 0];
>> [astar,apow]=atomic_unit(length);
>> astar
5.2917e-11
>> apow
2 -2 -1 1
![Page 36: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/36.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
![Page 37: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/37.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
e. g. atomic length unit
• 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
a∗ =4πε0~2
me2= 5.2917 × 10−11 m
![Page 38: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/38.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
e. g. atomic length unit
• 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
a∗ =4πε0~2
me2= 5.2917 × 10−11 m
![Page 39: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/39.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
e. g. atomic length unit
• 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
a∗ =4πε0~2
me2= 5.2917 × 10−11 m
![Page 40: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/40.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
e. g. atomic length unit
• 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
a∗ =4πε0~2
me2= 5.2917 × 10−11 m
![Page 41: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/41.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
e. g. atomic length unit
• 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
a∗ =4πε0~2
me2= 5.2917 × 10−11 m
![Page 42: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/42.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
e. g. atomic length unit
• 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
a∗ =4πε0~2
me2= 5.2917 × 10−11 m
![Page 43: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/43.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
e. g. atomic length unit
• 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
a∗ =4πε0~2
me2= 5.2917 × 10−11 m
![Page 44: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/44.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Bohr’s radius
• Schrodinger equation for hydrogen atom
− ~2
2m∆φ+
1
4πε0
−e2
rφ = Eφ
• in atomic units
−1
2∆φ− 1
rφ = Eφ
• ground state
φ ∝ e−r
with E = −1
2• radius
〈R〉 =
∫∞0 dr r2 e
−rr e−r∫∞
0 dr r2 e−r
1 e−r = 1
• Bohr’s result
〈R〉 = a∗ =4πε0~2
me2
![Page 45: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/45.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Bohr’s radius
• Schrodinger equation for hydrogen atom
− ~2
2m∆φ+
1
4πε0
−e2
rφ = Eφ
• in atomic units
−1
2∆φ− 1
rφ = Eφ
• ground state
φ ∝ e−r
with E = −1
2• radius
〈R〉 =
∫∞0 dr r2 e
−rr e−r∫∞
0 dr r2 e−r
1 e−r = 1
• Bohr’s result
〈R〉 = a∗ =4πε0~2
me2
![Page 46: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/46.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Bohr’s radius
• Schrodinger equation for hydrogen atom
− ~2
2m∆φ+
1
4πε0
−e2
rφ = Eφ
• in atomic units
−1
2∆φ− 1
rφ = Eφ
• ground state
φ ∝ e−r
with E = −1
2• radius
〈R〉 =
∫∞0 dr r2 e
−rr e−r∫∞
0 dr r2 e−r
1 e−r = 1
• Bohr’s result
〈R〉 = a∗ =4πε0~2
me2
![Page 47: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/47.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Bohr’s radius
• Schrodinger equation for hydrogen atom
− ~2
2m∆φ+
1
4πε0
−e2
rφ = Eφ
• in atomic units
−1
2∆φ− 1
rφ = Eφ
• ground state
φ ∝ e−r
with E = −1
2
• radius
〈R〉 =
∫∞0 dr r2 e
−rr e−r∫∞
0 dr r2 e−r
1 e−r = 1
• Bohr’s result
〈R〉 = a∗ =4πε0~2
me2
![Page 48: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/48.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Bohr’s radius
• Schrodinger equation for hydrogen atom
− ~2
2m∆φ+
1
4πε0
−e2
rφ = Eφ
• in atomic units
−1
2∆φ− 1
rφ = Eφ
• ground state
φ ∝ e−r
with E = −1
2• radius
〈R〉 =
∫∞0 dr r2 e
−rr e−r∫∞
0 dr r2 e−r
1 e−r = 1
• Bohr’s result
〈R〉 = a∗ =4πε0~2
me2
![Page 49: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/49.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Bohr’s radius
• Schrodinger equation for hydrogen atom
− ~2
2m∆φ+
1
4πε0
−e2
rφ = Eφ
• in atomic units
−1
2∆φ− 1
rφ = Eφ
• ground state
φ ∝ e−r
with E = −1
2• radius
〈R〉 =
∫∞0 dr r2 e
−rr e−r∫∞
0 dr r2 e−r
1 e−r = 1
• Bohr’s result
〈R〉 = a∗ =4πε0~2
me2
![Page 50: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/50.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 51: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/51.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 52: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/52.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 53: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/53.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 54: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/54.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 55: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/55.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 56: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/56.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 57: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/57.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 58: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/58.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Hartree
• MKSA for energy is [2 1 -2 0]
• [Estar,Epow]=atomic unit(energy)
• E∗ = 4.3598 × 10−18 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 = 0.072E∗ = 1.96 eV
![Page 59: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/59.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Electric field strength
• voltage=[2 1 -3 -1]
• el field str=[1 1 -3 -1]
• the atomic (natural) unit of electric field strength is
E∗ =m2e5
(4πε0)3~4= 5.1423 × 1011 V m−1
• Ohms law, Pockels effect, Starck effect . . .
• external field strength is always very small
![Page 60: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/60.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Electric field strength
• voltage=[2 1 -3 -1]
• el field str=[1 1 -3 -1]
• the atomic (natural) unit of electric field strength is
E∗ =m2e5
(4πε0)3~4= 5.1423 × 1011 V m−1
• Ohms law, Pockels effect, Starck effect . . .
• external field strength is always very small
![Page 61: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/61.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Electric field strength
• voltage=[2 1 -3 -1]
• el field str=[1 1 -3 -1]
• the atomic (natural) unit of electric field strength is
E∗ =m2e5
(4πε0)3~4= 5.1423 × 1011 V m−1
• Ohms law, Pockels effect, Starck effect . . .
• external field strength is always very small
![Page 62: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/62.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Electric field strength
• voltage=[2 1 -3 -1]
• el field str=[1 1 -3 -1]
• the atomic (natural) unit of electric field strength is
E∗ =m2e5
(4πε0)3~4= 5.1423 × 1011 V m−1
• Ohms law, Pockels effect, Starck effect . . .
• external field strength is always very small
![Page 63: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/63.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Electric field strength
• voltage=[2 1 -3 -1]
• el field str=[1 1 -3 -1]
• the atomic (natural) unit of electric field strength is
E∗ =m2e5
(4πε0)3~4= 5.1423 × 1011 V m−1
• Ohms law, Pockels effect, Starck effect . . .
• external field strength is always very small
![Page 64: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/64.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Electric field strength
• voltage=[2 1 -3 -1]
• el field str=[1 1 -3 -1]
• the atomic (natural) unit of electric field strength is
E∗ =m2e5
(4πε0)3~4= 5.1423 × 1011 V m−1
• Ohms law, Pockels effect, Starck effect . . .
• external field strength is always very small
![Page 65: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/65.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Magnetic induction
• induction=[0 1 -2 -1]
• the atomic (natural) unit of magnetic induction is
B∗ =m2e3
(4πε0)2~3= 2.3505 × 105 T
• Faraday effect, Hall effect . . .
• external field induction is always very small
![Page 66: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/66.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Magnetic induction
• induction=[0 1 -2 -1]
• the atomic (natural) unit of magnetic induction is
B∗ =m2e3
(4πε0)2~3= 2.3505 × 105 T
• Faraday effect, Hall effect . . .
• external field induction is always very small
![Page 67: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/67.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Magnetic induction
• induction=[0 1 -2 -1]
• the atomic (natural) unit of magnetic induction is
B∗ =m2e3
(4πε0)2~3= 2.3505 × 105 T
• Faraday effect, Hall effect . . .
• external field induction is always very small
![Page 68: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/68.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Magnetic induction
• induction=[0 1 -2 -1]
• the atomic (natural) unit of magnetic induction is
B∗ =m2e3
(4πε0)2~3= 2.3505 × 105 T
• Faraday effect, Hall effect . . .
• external field induction is always very small
![Page 69: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/69.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Magnetic induction
• induction=[0 1 -2 -1]
• the atomic (natural) unit of magnetic induction is
B∗ =m2e3
(4πε0)2~3= 2.3505 × 105 T
• Faraday effect, Hall effect . . .
• external field induction is always very small
![Page 70: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/70.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Mass density
• rho=[-3 1 0 0]
• [rhoval,rhopow]=atomic unit(rho)
• rhoval = 6.174
• rhopow=[-6 6 4 -3]
•
%∗ =m4e6
(4πε0)3~6=m
a3∗= 6.147 kg m−3
• mass is mass of nucleons, therefore must be multiplied byapproximately 4000
![Page 71: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/71.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Mass density
• rho=[-3 1 0 0]
• [rhoval,rhopow]=atomic unit(rho)
• rhoval = 6.174
• rhopow=[-6 6 4 -3]
•
%∗ =m4e6
(4πε0)3~6=m
a3∗= 6.147 kg m−3
• mass is mass of nucleons, therefore must be multiplied byapproximately 4000
![Page 72: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/72.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Mass density
• rho=[-3 1 0 0]
• [rhoval,rhopow]=atomic unit(rho)
• rhoval = 6.174
• rhopow=[-6 6 4 -3]
•
%∗ =m4e6
(4πε0)3~6=m
a3∗= 6.147 kg m−3
• mass is mass of nucleons, therefore must be multiplied byapproximately 4000
![Page 73: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/73.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Mass density
• rho=[-3 1 0 0]
• [rhoval,rhopow]=atomic unit(rho)
• rhoval = 6.174
• rhopow=[-6 6 4 -3]
•
%∗ =m4e6
(4πε0)3~6=m
a3∗= 6.147 kg m−3
• mass is mass of nucleons, therefore must be multiplied byapproximately 4000
![Page 74: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/74.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Mass density
• rho=[-3 1 0 0]
• [rhoval,rhopow]=atomic unit(rho)
• rhoval = 6.174
• rhopow=[-6 6 4 -3]
•
%∗ =m4e6
(4πε0)3~6=m
a3∗= 6.147 kg m−3
• mass is mass of nucleons, therefore must be multiplied byapproximately 4000
![Page 75: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/75.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Mass density
• rho=[-3 1 0 0]
• [rhoval,rhopow]=atomic unit(rho)
• rhoval = 6.174
• rhopow=[-6 6 4 -3]
•
%∗ =m4e6
(4πε0)3~6=m
a3∗= 6.147 kg m−3
• mass is mass of nucleons, therefore must be multiplied byapproximately 4000
![Page 76: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/76.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Mass density
• rho=[-3 1 0 0]
• [rhoval,rhopow]=atomic unit(rho)
• rhoval = 6.174
• rhopow=[-6 6 4 -3]
•
%∗ =m4e6
(4πε0)3~6=m
a3∗= 6.147 kg m−3
• mass is mass of nucleons, therefore must be multiplied byapproximately 4000
![Page 77: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/77.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Light velocity
• velocity=[1 0 -1 0]
• atomic unit of velocity is
v∗ =e2
4πε0~= 2.1877 × 106 m s−1
• velocity of light is c = α−1v∗
• fine structure constant
α = 1/137.035999074
• can be measured directly (quantum Hall effect)
• relativistic corrections
![Page 78: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/78.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Light velocity
• velocity=[1 0 -1 0]
• atomic unit of velocity is
v∗ =e2
4πε0~= 2.1877 × 106 m s−1
• velocity of light is c = α−1v∗
• fine structure constant
α = 1/137.035999074
• can be measured directly (quantum Hall effect)
• relativistic corrections
![Page 79: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/79.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Light velocity
• velocity=[1 0 -1 0]
• atomic unit of velocity is
v∗ =e2
4πε0~= 2.1877 × 106 m s−1
• velocity of light is c = α−1v∗
• fine structure constant
α = 1/137.035999074
• can be measured directly (quantum Hall effect)
• relativistic corrections
![Page 80: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/80.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Light velocity
• velocity=[1 0 -1 0]
• atomic unit of velocity is
v∗ =e2
4πε0~= 2.1877 × 106 m s−1
• velocity of light is c = α−1v∗
• fine structure constant
α = 1/137.035999074
• can be measured directly (quantum Hall effect)
• relativistic corrections
![Page 81: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/81.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Light velocity
• velocity=[1 0 -1 0]
• atomic unit of velocity is
v∗ =e2
4πε0~= 2.1877 × 106 m s−1
• velocity of light is c = α−1v∗
• fine structure constant
α = 1/137.035999074
• can be measured directly (quantum Hall effect)
• relativistic corrections
![Page 82: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/82.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Light velocity
• velocity=[1 0 -1 0]
• atomic unit of velocity is
v∗ =e2
4πε0~= 2.1877 × 106 m s−1
• velocity of light is c = α−1v∗
• fine structure constant
α = 1/137.035999074
• can be measured directly (quantum Hall effect)
• relativistic corrections
![Page 83: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/83.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Light velocity
• velocity=[1 0 -1 0]
• atomic unit of velocity is
v∗ =e2
4πε0~= 2.1877 × 106 m s−1
• velocity of light is c = α−1v∗
• fine structure constant
α = 1/137.035999074
• can be measured directly (quantum Hall effect)
• relativistic corrections
![Page 84: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/84.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek• Pockels coefficients vanish if crystal has an
inversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 85: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/85.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E
• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek• Pockels coefficients vanish if crystal has an
inversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 86: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/86.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek
• Pockels coefficients vanish if crystal has aninversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 87: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/87.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek• Pockels coefficients vanish if crystal has an
inversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 88: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/88.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek• Pockels coefficients vanish if crystal has an
inversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 89: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/89.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek• Pockels coefficients vanish if crystal has an
inversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 90: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/90.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek• Pockels coefficients vanish if crystal has an
inversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 91: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/91.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek• Pockels coefficients vanish if crystal has an
inversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 92: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/92.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Pockels coefficients
• external electric field E• permittivity change
(ε(ω;E)−1)ij = (ε(ω; 0)−1)ij + rijk(ω)Ek• Pockels coefficients vanish if crystal has an
inversion center
• lithium niobate has no inversion center
• largest coefficient r333 = 30 pm V−1
• small or large?
• r333 E∗ ≈ 16
• lithium niobate is rather resilient to electrical fields
![Page 93: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/93.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 94: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/94.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 95: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/95.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 96: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/96.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 97: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/97.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 98: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/98.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 99: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/99.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 100: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/100.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 101: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/101.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 102: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/102.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 103: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/103.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Elasticity module
• 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
Tij =E
1 + ν
Sij +
ν
1− 2νδijSkk
• Poisson’s ratio between 0 and 1/2
• elasticity module E
• expect E = eV A−3
= 160 GPa
• . . . as order of magnitude
• steel : E = 200 GPa
![Page 104: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/104.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Susceptibility
• Recall the Drude model result
χ(ω) = χ(0)Ω2
Ω2 − ω2 − iΓω• where
χ(0) =Nq2
ε0mΩ2
• recall
m(x + Γx + Ω2x) = qE
• N ≈ 1 and Ω ≈ 1 in natural units!
![Page 105: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/105.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Susceptibility
• Recall the Drude model result
χ(ω) = χ(0)Ω2
Ω2 − ω2 − iΓω
• where
χ(0) =Nq2
ε0mΩ2
• recall
m(x + Γx + Ω2x) = qE
• N ≈ 1 and Ω ≈ 1 in natural units!
![Page 106: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/106.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Susceptibility
• Recall the Drude model result
χ(ω) = χ(0)Ω2
Ω2 − ω2 − iΓω• where
χ(0) =Nq2
ε0mΩ2
• recall
m(x + Γx + Ω2x) = qE
• N ≈ 1 and Ω ≈ 1 in natural units!
![Page 107: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/107.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Susceptibility
• Recall the Drude model result
χ(ω) = χ(0)Ω2
Ω2 − ω2 − iΓω• where
χ(0) =Nq2
ε0mΩ2
• recall
m(x + Γx + Ω2x) = qE
• N ≈ 1 and Ω ≈ 1 in natural units!
![Page 108: Overview Natural units - uni-osnabrueck.deelds More examples Overview Normal matter Atomic units SI to AU conversion. Natural units Peter Hertel Overview Normal matter Atomic units](https://reader033.vdocuments.net/reader033/viewer/2022052011/60261faa477f344f7f2ff2ec/html5/thumbnails/108.jpg)
Natural units
Peter Hertel
Overview
Normal matter
Atomic units
Length andenergy
Electric andmagneticfields
Moreexamples
Susceptibility
• Recall the Drude model result
χ(ω) = χ(0)Ω2
Ω2 − ω2 − iΓω• where
χ(0) =Nq2
ε0mΩ2
• recall
m(x + Γx + Ω2x) = qE
• N ≈ 1 and Ω ≈ 1 in natural units!