equations
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| ====== Equations ====== | ====== Equations ====== | ||
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| | | **Important in:** | **Relationship:** | **Used For:** | | | | | **Important in:** | **Relationship:** | **Used For:** | | | ||
| - | | [[equations:schroedinger_equation|Schrödinger Equation]] | [[theories:quantum_mechanics|Quantum Mechanics]], [[theories:quantum_field_theory|Quantum Field Theory]] | non-relativistic limit of the Klein-Gordon Equation | Describes time evolution | linear | | + | | [[equations:schroedinger_equation|Schrödinger Equation]] | [[theories:quantum_mechanics:canonical|Quantum Mechanics]], [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | non-relativistic limit of the Klein-Gordon Equation | Describes time evolution | linear | |
| - | | [[equations:klein-gordon_equation|Klein-Gordon Equation]] | [[theories:quantum_field_theory|Quantum Field Theory]] | | Equation of motion for particles with [[basic_notions:spin|spin]] 0 | linear | | + | | [[equations:klein-gordon_equation|Klein-Gordon Equation]] | [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | | Equation of motion for particles with [[basic_notions:spin|spin]] 0 | linear | |
| - | | [[equations:pauli_equation|Pauli Equation]] | [[theories:quantum_mechanics|Quantum Mechanics]] | non-relativistic limit of the Dirac Equation | Equation of motion for particles with spin 1/2 | linear | | + | | [[equations:pauli_equation|Pauli Equation]] | [[theories:quantum_mechanics:canonical|Quantum Mechanics]] | non-relativistic limit of the Dirac Equation | Equation of motion for particles with spin 1/2 | linear | |
| - | | [[equations:dirac_equation|Dirac Equation]] | [[theories:quantum_field_theory|Quantum Field Theory]] | | Equation of motion for particles with spin 1/2 | linear | | + | | [[equations:dirac_equation|Dirac Equation]] | [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | | Equation of motion for particles with spin 1/2 | linear | |
| - | | [[equations:maxwell_equations|Maxwell Equations]] | [[models:classical_electrodynamics|Classical Electrodynamics]], [[theories:quantum_field_theory|Quantum Field Theory]] | special case of the Yang-Mills equation for a \\ non-abelian gauge theory | Equation of motion for particles with spin 1 in abelian gauge theories | linear | | + | | [[equations:maxwell_equations|Maxwell Equations]] | [[models:classical_electrodynamics|Classical Electrodynamics]], [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | special case of the Yang-Mills equation for a \\ non-abelian gauge theory | Equation of motion for particles with spin 1 in abelian gauge theories | linear | |
| | [[equations:einstein_equation|Einstein Equation]] | [[models:general_relativity|General Relativity]] | | Describes how spacetime gets curved through energy and matter | non-linear | | | [[equations:einstein_equation|Einstein Equation]] | [[models:general_relativity|General Relativity]] | | Describes how spacetime gets curved through energy and matter | non-linear | | ||
| - | | [[equations:yang_mills_equations|Yang-Mills Equation]] | [[theories:quantum_field_theory|Quantum Field Theory]] | | Equation of motion for particles with spin 1 in non-abelian gauge theories | non-linear | | + | | [[equations:yang_mills_equations|Yang-Mills Equation]] | [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | | Equation of motion for particles with spin 1 in non-abelian gauge theories | non-linear | |
| | [[equations:navier_stokes]] | [[theories:classical_theories:hydrodynamics|Hydrodynamics]] | | Describe the flow of fluids | non-linear | | | [[equations:navier_stokes]] | [[theories:classical_theories:hydrodynamics|Hydrodynamics]] | | Describe the flow of fluids | non-linear | | ||
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| The equations of motion are usually not enough to describe a system. Especially in the Newtonian framework, we need additional equations that give us, for example, the correct formulas which describe a force that acts on the object in question. For example, | The equations of motion are usually not enough to describe a system. Especially in the Newtonian framework, we need additional equations that give us, for example, the correct formulas which describe a force that acts on the object in question. For example, | ||
| - | * [[equations:newtons_law|Newton's law of gravity]] | + | * [[formulas:newtons_law|Newton's law of gravity]] |
| - | * [[equations:lorentz_force_law|Lorentz' force law]] | + | * [[formulas:lorentz_force_law|Lorentz' force law]] |
| - | * [[equations:coulombs_law|Coulomb's force law]] | + | * [[formulas:coulombs_law|Coulomb's force law]] |
| In addition, we always need to specify the [[basic_notions:boundary_conditions]] for the system in question. | In addition, we always need to specify the [[basic_notions:boundary_conditions]] for the system in question. | ||
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| **The Equations of Motion yield the Most Important Paths** | **The Equations of Motion yield the Most Important Paths** | ||
| - | In the [[theories:path_integral_quantum_mechanics|path integral formuation]] of [[theories:quantum_mechanics|quantum mechanics]], particles do not follow one individual path but instead all of them. Hence there can't be one equation whose solution yields the correct particle trajectory. | + | In the [[theories:quantum_mechanics:path_integral|path integral formuation]] of [[theories:quantum_mechanics:canonical|quantum mechanics]], particles do not follow one individual path but instead all of them. Hence there can't be one equation whose solution yields the correct particle trajectory. |
| However, the equations of motion are still important and the path integral formalism tells us why. | However, the equations of motion are still important and the path integral formalism tells us why. | ||
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| When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism. | When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism. | ||
| - | //In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as [[equations:maxwell_equations|Maxwell equations]]// in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with [[models:quantum_electrodynamics|quantum electro dynamics - QED]]. | + | //In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as [[equations:maxwell_equations|Maxwell equations]]// in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with [[models:standard_model:qed|quantum electro dynamics - QED]]. |
| <cite>http://axelmaas.blogspot.de/2010/10/electromagnetism-photons-and-symmetry.html</cite></blockquote> | <cite>http://axelmaas.blogspot.de/2010/10/electromagnetism-photons-and-symmetry.html</cite></blockquote> | ||
| <-- | <-- | ||
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