Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Generalized momenta | p, P | varies with choice | varies with choice | |
Lagrangian | L |
where and p = p(t) are vectors of the generalized coords and momenta, as functions of time |
J | [M][L]2[T]−2 |
Hamiltonian | H | J | [M][L]2[T]−2 | |
Action, Hamilton's principal function | S, | J s | [M][L]2[T]−1 |
General Classical Equations
Physical quantity | Electric | Magnetic |
---|---|---|
Potential gradient and field |
|
|
Electric:Point charge
Magnetic: moment |
| |
Electric: At a point in a local array of point charges
Magnetic: N/A |
--- | |
Electric: at a point due to a continuum of charge
Magnetic: magnetic moment due to a current distribution |
||
Torque and potential energy due to non-uniform fields and dipole moments |
|
|
Name | Integral equations -- SI Units | Differential equations -- SI Units | Integral equations -- Gaussian units | Differential equations -- Gaussian units |
---|---|---|---|---|
Gauss's law | ||||
Gauss's law for magnetism | ||||
Maxwell–Faraday equation ( Faraday's law of induction) | ||||
Ampère's circuital law (with Maxwell's addition) |
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields
3D Euclidean space + time |
|
|
Potentials (any
gauge)
3D Euclidean space + time |
|
|
Potentials (
Lorenz gauge)
3D Euclidean space + time |
|
|
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields
Any space + time |
||
Potentials (any gauge)
Any space (with topological restrictions) + time |
||
Potential (Lorenz Gauge)
Any space (with topological restrictions) + time spatial metric independent of time |
ds2 is invariant under the Lorentz transformation:
The sign of the metric and the placement of the ct, ct', cdt, and cdt′ time-based terms can vary depending on the author's choice. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes η is replaced with −η, making the spatial terms produce negative contributions to the dot product or spacetime interval, while the time term makes a positive contribution.
It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace t, t′, dt, and dt′ with ct, ct', cdt, and cdt′, which has the dimensions of distance. So:
then in matrix form:
The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors. In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows:
So can be a four-vector representing position, velocity, or momentum, and the same Λ can be used when transforming between the same two frames. The most general Lorentz transformation includes boosts and rotations; the components are complicated and the transformation requires spinors.
Invariance and unification of physical quantities both arise from four-vectors. [1] The inner product of a 4-vector with itself is equal to a scalar (by definition of the inner product), and since the 4-vectors are physical quantities their magnitudes correspond to physical quantities also.
Property/effect | 3-vector | 4-vector | Invariant result |
---|---|---|---|
Space-time events | 3-position: r = (x1, x2, x3)
|
4-position: X = (ct, x1, x2, x3) |
|
Momentum-energy invariance |
3-momentum: p = (p1, p2, p3) |
4-momentum: P = (E/c, p1, p2, p3)
|
which leads to: E = total energy |
Velocity | 3-velocity: u = (u1, u2, u3)
|
4-velocity: U = (U0, U1, U2, U3)
|
Property or effect | Nomenclature | Equation |
---|---|---|
Planck–Einstein equation and de Broglie wavelength relations |
|
|
Schrödinger equation |
|
General time-dependent case:
Time-independent case: |
Heisenberg equation |
|
|
Time evolution in Heisenberg picture ( Ehrenfest theorem) |
of a particle. |
For momentum and position;
|
Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.
One particle | N particles | |
One dimension |
where the position of particle n is xn. | |
There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [2] |
for non-interacting particles
| |
Three dimensions |
where the position of the particle is r = (x, y, z). |
where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is
|
for non-interacting particles
|
Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.
One particle | N particles | |
One dimension |
where the position of particle n is xn. | |
Three dimensions | ||
This last equation is in a very high dimension, [3] so the solutions are not easy to visualize. | ||
Property/Effect | Nomenclature | Equation |
---|---|---|
Photoelectric equation |
|
|
Threshold frequency and |
|
Can only be found by experiment.
The De Broglie relations give the relation between them:
|
Photon momentum |
|
The De Broglie relations give:
|
Property or effect | Nomenclature | Equation |
---|---|---|
Heisenberg's uncertainty principles |
|
Position-momentum
Energy-time Number-phase |
Dispersion of observable |
|
|
General uncertainty relation |
|
Property or effect | Nomenclature | Equation |
---|---|---|
Density of states | ||
Fermi–Dirac distribution (fermions) |
|
|
Bose–Einstein distribution (bosons) |
Property or effect | Nomenclature | Equation |
---|---|---|
Angular momentum quantum numbers |
|
Spin projection:
Orbital:
Total: |
Angular momentum magnitudes | angular momementa:
|
Spin magnitude:
Orbital magnitude: Total magnitude:
|
Angular momentum components | Spin:
Orbital: |
In what follows, B is an applied external magnetic field and the quantum numbers above are used.
Property or effect | Nomenclature | Equation |
---|---|---|
orbital magnetic dipole moment |
|
z-component: |
spin magnetic dipole moment |
|
z-component: |
dipole moment potential |
|
Property or effect | Nomenclature | Equation |
---|---|---|
Energy level :p≈ |
|
|
Spectrum | λ = wavelength of emitted photon, during electronic transition from Ei to Ej |
These equations need to be refined such that the notation is defined as has been done for the previous sets of equations.
Name | Equations |
---|---|
Strong force | |
Electroweak interaction | :
|
Quantum electrodynamics |
see: General relativity, Einstein field equations, List of equations in gravitation
where Rμν is the Ricci curvature tensor, R is the scalar curvature, gμν is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and Tμν is the stress–energy tensor.
One can write the EFE in a more compact form by defining the Einstein tensor
which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
In standard units, each term on the left has units of 1/length2. With this choice of Einstein constant as 8πG/c4, then the stress-energy tensor on the right side of the equation must be written with each component in units of energy-density (i.e., energy per volume = pressure).
Using geometrized units where G = c = 1, this can be rewritten as
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.
These equations, together with the geodesic equation, [4] which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity.
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. [5] The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
The third sign above is related to the choice of convention for the Ricci tensor:
With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972) [6] and Peacock (1994) [7] are (+ − −), Peebles (1980) [8] and Efstathiou et al. (1990) [9] are (− + +), Rindler (1977)[ citation needed], Atwater (1974)[ citation needed], Collins Martin & Squires (1989) [10] are (− + −).
Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative
The sign of the (very small) cosmological term would change in both these versions, if the (+ − − −) metric sign convention is used rather than the MTW (− + + +) metric sign convention adopted here.
Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
---|---|---|---|---|
Generalized momenta | p, P | varies with choice | varies with choice | |
Lagrangian | L |
where and p = p(t) are vectors of the generalized coords and momenta, as functions of time |
J | [M][L]2[T]−2 |
Hamiltonian | H | J | [M][L]2[T]−2 | |
Action, Hamilton's principal function | S, | J s | [M][L]2[T]−1 |
General Classical Equations
Physical quantity | Electric | Magnetic |
---|---|---|
Potential gradient and field |
|
|
Electric:Point charge
Magnetic: moment |
| |
Electric: At a point in a local array of point charges
Magnetic: N/A |
--- | |
Electric: at a point due to a continuum of charge
Magnetic: magnetic moment due to a current distribution |
||
Torque and potential energy due to non-uniform fields and dipole moments |
|
|
Name | Integral equations -- SI Units | Differential equations -- SI Units | Integral equations -- Gaussian units | Differential equations -- Gaussian units |
---|---|---|---|---|
Gauss's law | ||||
Gauss's law for magnetism | ||||
Maxwell–Faraday equation ( Faraday's law of induction) | ||||
Ampère's circuital law (with Maxwell's addition) |
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields
3D Euclidean space + time |
|
|
Potentials (any
gauge)
3D Euclidean space + time |
|
|
Potentials (
Lorenz gauge)
3D Euclidean space + time |
|
|
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields
Any space + time |
||
Potentials (any gauge)
Any space (with topological restrictions) + time |
||
Potential (Lorenz Gauge)
Any space (with topological restrictions) + time spatial metric independent of time |
ds2 is invariant under the Lorentz transformation:
The sign of the metric and the placement of the ct, ct', cdt, and cdt′ time-based terms can vary depending on the author's choice. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes η is replaced with −η, making the spatial terms produce negative contributions to the dot product or spacetime interval, while the time term makes a positive contribution.
It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace t, t′, dt, and dt′ with ct, ct', cdt, and cdt′, which has the dimensions of distance. So:
then in matrix form:
The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors. In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows:
So can be a four-vector representing position, velocity, or momentum, and the same Λ can be used when transforming between the same two frames. The most general Lorentz transformation includes boosts and rotations; the components are complicated and the transformation requires spinors.
Invariance and unification of physical quantities both arise from four-vectors. [1] The inner product of a 4-vector with itself is equal to a scalar (by definition of the inner product), and since the 4-vectors are physical quantities their magnitudes correspond to physical quantities also.
Property/effect | 3-vector | 4-vector | Invariant result |
---|---|---|---|
Space-time events | 3-position: r = (x1, x2, x3)
|
4-position: X = (ct, x1, x2, x3) |
|
Momentum-energy invariance |
3-momentum: p = (p1, p2, p3) |
4-momentum: P = (E/c, p1, p2, p3)
|
which leads to: E = total energy |
Velocity | 3-velocity: u = (u1, u2, u3)
|
4-velocity: U = (U0, U1, U2, U3)
|
Property or effect | Nomenclature | Equation |
---|---|---|
Planck–Einstein equation and de Broglie wavelength relations |
|
|
Schrödinger equation |
|
General time-dependent case:
Time-independent case: |
Heisenberg equation |
|
|
Time evolution in Heisenberg picture ( Ehrenfest theorem) |
of a particle. |
For momentum and position;
|
Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.
One particle | N particles | |
One dimension |
where the position of particle n is xn. | |
There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [2] |
for non-interacting particles
| |
Three dimensions |
where the position of the particle is r = (x, y, z). |
where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is
|
for non-interacting particles
|
Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.
One particle | N particles | |
One dimension |
where the position of particle n is xn. | |
Three dimensions | ||
This last equation is in a very high dimension, [3] so the solutions are not easy to visualize. | ||
Property/Effect | Nomenclature | Equation |
---|---|---|
Photoelectric equation |
|
|
Threshold frequency and |
|
Can only be found by experiment.
The De Broglie relations give the relation between them:
|
Photon momentum |
|
The De Broglie relations give:
|
Property or effect | Nomenclature | Equation |
---|---|---|
Heisenberg's uncertainty principles |
|
Position-momentum
Energy-time Number-phase |
Dispersion of observable |
|
|
General uncertainty relation |
|
Property or effect | Nomenclature | Equation |
---|---|---|
Density of states | ||
Fermi–Dirac distribution (fermions) |
|
|
Bose–Einstein distribution (bosons) |
Property or effect | Nomenclature | Equation |
---|---|---|
Angular momentum quantum numbers |
|
Spin projection:
Orbital:
Total: |
Angular momentum magnitudes | angular momementa:
|
Spin magnitude:
Orbital magnitude: Total magnitude:
|
Angular momentum components | Spin:
Orbital: |
In what follows, B is an applied external magnetic field and the quantum numbers above are used.
Property or effect | Nomenclature | Equation |
---|---|---|
orbital magnetic dipole moment |
|
z-component: |
spin magnetic dipole moment |
|
z-component: |
dipole moment potential |
|
Property or effect | Nomenclature | Equation |
---|---|---|
Energy level :p≈ |
|
|
Spectrum | λ = wavelength of emitted photon, during electronic transition from Ei to Ej |
These equations need to be refined such that the notation is defined as has been done for the previous sets of equations.
Name | Equations |
---|---|
Strong force | |
Electroweak interaction | :
|
Quantum electrodynamics |
see: General relativity, Einstein field equations, List of equations in gravitation
where Rμν is the Ricci curvature tensor, R is the scalar curvature, gμν is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and Tμν is the stress–energy tensor.
One can write the EFE in a more compact form by defining the Einstein tensor
which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
In standard units, each term on the left has units of 1/length2. With this choice of Einstein constant as 8πG/c4, then the stress-energy tensor on the right side of the equation must be written with each component in units of energy-density (i.e., energy per volume = pressure).
Using geometrized units where G = c = 1, this can be rewritten as
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.
These equations, together with the geodesic equation, [4] which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity.
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. [5] The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
The third sign above is related to the choice of convention for the Ricci tensor:
With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972) [6] and Peacock (1994) [7] are (+ − −), Peebles (1980) [8] and Efstathiou et al. (1990) [9] are (− + +), Rindler (1977)[ citation needed], Atwater (1974)[ citation needed], Collins Martin & Squires (1989) [10] are (− + −).
Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative
The sign of the (very small) cosmological term would change in both these versions, if the (+ − − −) metric sign convention is used rather than the MTW (− + + +) metric sign convention adopted here.