User:JMvanDijk/Sandbox 16/Box2

Electric and Magnetic fields
General Classical Equations


 * {| class="wikitable"

! scope="col" width="20" | Physical quantity ! scope="col" width="10" | Electric ! scope="col" width="10" | Magnetic !Potential gradient and field
 * $$ \mathbf{E} = - \nabla V $$

$$ \Delta V = -\int_{r_1}^{r_2} \mathbf{E} \cdot d\mathbf{r}\,\!$$ !Electric:Point charge Magnetic: moment
 * $$ \mathbf{B} = \nabla \times \mathbf{A} $$
 * $$ \mathbf{E} = \frac{q}{4 \pi \epsilon_0 \left | \mathbf{r} \right |^2 }\mathbf{\hat{r}} \,\!$$
 * $$ \mathbf{A} = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{\left | \mathbf{r} \right |^3} $$

$$\mathbf{B}({\mathbf{r}})=\nabla\times{\mathbf{A}}=\frac{\mu_{0}}{4\pi}\left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{\left | \mathbf{r} \right |^{5}}-\frac{\left | \mathbf{r} \right |^{3}}\right)$$ !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
 * $$\mathbf{E} = \sum \mathbf{E}_i = \frac{1}{4 \pi \epsilon_0} \sum_i \frac{q_i}{\left | \mathbf{r}_i - \mathbf{r} \right |^2}\mathbf{\hat{r}}_i \,\!$$
 * $$ \mathbf{E} = \frac{1}{4\pi\epsilon_0} \int_V \frac{\mathbf{r} \rho \mathrm{d}V}{\left | \mathbf{r} \right |^3} \,\!$$
 * $$ \mathbf{m} = \frac{1}{2}\int_V \mathbf{r}\times\mathbf{J} \mathrm{d} V $$
 * $$ \boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{p} \times \mathbf{E} $$

$$ U = \int_V \mathrm{d} \mathbf{p} \cdot \mathbf{E} $$
 * $$ \boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{m} \times \mathbf{B} $$

$$ U = \int_V \mathrm{d} \mathbf{m} \cdot \mathbf{B} $$
 * }
 * }

Special Relativity: The metric and four-vectors

 * Inner product (i.e. notion of length):


 * Space-time interval


 * $$ds^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2 = \begin{pmatrix} cdt & dx & dy & dz \end{pmatrix} \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} cdt \\ dx \\ dy \\ dz \end{pmatrix}$$, :$$\eta = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0  & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ where $$\eta$$ is known as the metric tensor.  In special relativity, the metric tensor is the Minkowski metric:

ds2 is invariant under the Lorentz transformation:


 * $$ \eta ( \boldsymbol{\mathsf{a}}', \boldsymbol{\mathsf{b}}' ) = \eta \left ( \Lambda \boldsymbol{\mathsf{a}} , \Lambda \boldsymbol{\mathsf{b}} \right ) = \eta ( \boldsymbol{\mathsf{a}} , \boldsymbol{\mathsf{b}} )$$

The sign of the metric and the placement of the ct, ct', cdt, and cdt&prime; 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.

Lorentz transforms
It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace t, t&prime;, dt, and dt&prime; with ct, ct', cdt, and cdt&prime;, which has the dimensions of distance. So:


 * $$x' = \gamma x - \gamma \beta c t \,$$
 * $$y' = y \,$$
 * $$z' = z \,$$
 * $$c t' = \gamma c t - \gamma \beta x \,$$

then in matrix form:


 * $$\begin{pmatrix} c t' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} \gamma & - \gamma \beta & 0 & 0 \\ - \gamma \beta & \gamma & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} c t \\ x \\ y \\ z \end{pmatrix}$$

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:


 * $$\boldsymbol{\mathsf{a}}' = \Lambda \boldsymbol{\mathsf{a}}$$

So $$\boldsymbol{\mathsf{a}}'$$ 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.

4-vectors and frame-invariant results
Invariance and unification of physical quantities both arise from four-vectors. 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.

Non-relativistic time-independent Schrödinger equation
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.

Non-relativistic time-dependent Schrödinger equation
Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.

Angular momentum

 * Magnetic moments

In what follows, B is an applied external magnetic field and the quantum numbers above are used.

Fundamental forces
These equations need to be refined such that the notation is defined as has been done for the previous sets of equations.
 * {| class="wikitable"

! Name ! Equations !Strong force \begin{align} \mathcal{L}_\mathrm{QCD} & = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \\ & = \bar{\psi}_i (i \gamma^\mu \partial_\mu -  m )\psi_i - g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \,,\\ \end{align} \,\!$$ !Electroweak interaction
 * $$\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.\,\!$$
 * $$\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}\,\!$$
 * $$\mathcal{L}_f =  \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i \,\!$$
 * $$\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\,\!$$
 * $$\mathcal{L}_y = - y_{u\, ij} \epsilon^{ab} \,h_b^\dagger\, \overline{Q}_{ia} u_j^c - y_{d\, ij}\, h\, \overline{Q}_i d^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.\,\!$$

!Quantum electrodynamics
 * $$\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,\,\!$$
 * }

General Relativity
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
 * $$G_{\mu \nu} = R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu},$$

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
 * $$G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}.$$

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
 * $$G_{\mu \nu} + \Lambda g_{\mu \nu} = 8 \pi T_{\mu \nu}\,.$$

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, which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity.

Sign convention
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):



\begin{align} g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[6pt] {R^\mu}_{\alpha \beta \gamma} & = [S2] \times \left(\Gamma^\mu_{\alpha \gamma,\beta}-\Gamma^\mu_{\alpha \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma \alpha}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta \alpha}\right) \\[6pt] G_{\mu \nu} & = [S3] \times \frac{8 \pi G}{c^4} T_{\mu \nu} \end{align} $$

The third sign above is related to the choice of convention for the Ricci tensor:
 * $$R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} $$

With these definitions Misner, Thorne, and Wheeler classify themselves as $(+ + +)$, whereas Weinberg (1972) and Peacock (1994) are $(+ − −)$, Peebles (1980) and Efstathiou et al. (1990) are $(− + +)$, Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) 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
 * $$R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\frac{8 \pi G}{c^4} T_{\mu \nu}.$$

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.