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= Quantum mechanics =

Wavefunctions
A fundamental physical constant occurring in quantum mechanics is the Planck constant, h. A common abbreviation is ħ = h/2π, also known as the reduced Planck constant or Dirac constant.

The general form of wavefunction for a system of particles, each with position ri and z-component of spin sz i. Sums are over the discrete variable sz, integrals over continuous positions r.

For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necessary). Following are general mathematical results, used in calculations.

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.

The Hydrogen atom
= Thermodynamics =

Definitions
Many of the definitions below are also used in the thermodynamics of chemical reactions.

Entropy

 * $$ S = k_\mathrm{B} \ln \Omega $$, where kB is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
 * $$ dS = \frac{\delta Q}{T} $$, for reversible processes only

Statistical physics
Below are useful results from the Maxwell–Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity. The distribution is valid for atoms or molecules constituting ideal gases. Corollaries of the non-relativistic Maxwell–Boltzmann distribution are below.

Quasi-static and reversible processes
For quasi-static and reversible processes, the first law of thermodynamics is:


 * $$dU=\delta Q - \delta W$$

where δQ is the heat supplied to the system and δW is the work done by the system.

Thermodynamic potentials
The following energies are called the thermodynamic potentials.

and the corresponding fundamental thermodynamic relations or "master equations" are:

Maxwell's relations
The four most common Maxwell's relations are: More relations include the following. Other differential equations are:

Quantum properties

 * $$ U = N k_\text{B} T^2 \left(\frac{\partial \ln Z}{\partial T}\right)_V $$
 * $$ S = \frac{U}{T} + N k_\text{B} \ln Z - N k \ln N + Nk $$  Indistinguishable Particles

where N is number of particles, h is that Planck constant, I is moment of inertia, and Z is the partition function, in various forms:

Thermal efficiencies
= Classical mechanics =

Derived dynamic quantities
[[File:Classical_angular_momentum.svg|thumb|350x350px|Angular momenta of a classical object.Left: intrinsic "spin" angular momentum S is really orbital angular momentum of the object at every point,

right: extrinsic orbital angular momentum L about an axis,

top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω)

bottom: momentum p and its radial position r from the axis.

The total angular momentum (spin + orbital) is J.]]

General energy definitions
Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:


 * Wherever the force is zero, its potential energy is defined to be zero as well.
 * Whenever the force does work, potential energy is lost.

Kinematics
In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector

$$\mathbf{\hat{n}} = \mathbf{\hat{e}}_r\times\mathbf{\hat{e}}_\theta $$

defines the axis of rotation, $$ \scriptstyle \mathbf{\hat{e}}_r $$ = unit vector in direction of $r$, $$ \scriptstyle \mathbf{\hat{e}}_\theta $$ = unit vector tangential to the angle.

Precession
The precession angular speed of a spinning top is given by:

$$ \boldsymbol{\Omega} = \frac{wr}{I\boldsymbol{\omega}} $$

where w is the weight of the spinning flywheel.

Energy
The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:

General work-energy theorem (translation and rotation)
The work done W by an external agent which exerts a force F (at r) and torque τ on an object along a curved path C is:

$$ W = \Delta T = \int_C \left ( \mathbf{F} \cdot \mathrm{d} \mathbf{r} + \boldsymbol{\tau} \cdot \mathbf{n} \, {\mathrm{d} \theta} \right ) $$

where θ is the angle of rotation about an axis defined by a unit vector n.

Kinetic energy
The change in kinetic energy for an object initially traveling at speed $$v_0$$ and later at speed $$v$$ is: $$ \Delta E_k = W = \frac{1}{2} m(v^2 - {v_0}^2) $$

Elastic potential energy
For a stretched spring fixed at one end obeying Hooke's law, the elastic potential energy is

$$ \Delta E_p = \frac{1}{2} k(r_2-r_1)^2 $$

where r2 and r1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.

Euler's equations for rigid body dynamics
Euler also worked out analogous laws of motion to those of Newton, see Euler's laws of motion. These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is:

$$ \mathbf{I} \cdot \boldsymbol{\alpha} + \boldsymbol{\omega} \times \left ( \mathbf{I} \cdot \boldsymbol{\omega} \right ) = \boldsymbol{\tau} $$

where I is the moment of inertia tensor.

General planar motion
The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane,

$$ \mathbf{r} = \mathbf{r}(t) = r\hat\mathbf r $$

the following general results apply to the particle.

Central force motion
For a massive body moving in a central potential due to another object, which depends only on the radial separation between the centers of masses of the two objects, the equation of motion is:

$$\frac{d^2}{d\theta^2}\left(\frac{1}{\mathbf{r}}\right) + \frac{1}{\mathbf{r}} = -\frac{\mu\mathbf{r}^2}{\mathbf{l}^2}\mathbf{F}(\mathbf{r})$$

Equations of motion (constant acceleration)
These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).

Galilean frame transforms
For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

Mechanical oscillators
SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.

= Wave theory =

General fundamental quantities
A wave can be longitudinal where the oscillations are parallel (or antiparallel) to the propagation direction, or transverse where the oscillations are perpendicular to the propagation direction. These oscillations are characterized by a periodically time-varying displacement in the parallel or perpendicular direction, and so the instantaneous velocity and acceleration are also periodic and time varying in these directions. (the apparent motion of the wave due to the successive oscillations of particles or fields about their equilibrium positions) propagates at the phase and group velocities parallel or antiparallel to the propagation direction, which is common to longitudinal and transverse waves. Below oscillatory displacement, velocity and acceleration refer to the kinematics in the oscillating directions of the wave - transverse or longitudinal (mathematical description is identical), the group and phase velocities are separate.

General derived quantities
Relation between space, time, angle analogues used to describe the phase:

$$ \frac{\Delta r}{\lambda} = \frac{\Delta t}{T} = \frac{\phi}{2\pi} \,\!$$

Equations
In what follows n, m are any integers (Z = set of integers); $$n, m \in \mathbf{Z} \,\!$$.

Gravitational waves
Gravitational radiation for two orbiting bodies in the low-speed limit.

Wave propagation
A common misconception occurs between phase velocity and group velocity (analogous to centres of mass and gravity). They happen to be equal in non-dispersive media. In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency.


 * The phase velocity is the rate at which the phase of the wave propagates in space.
 * The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.

Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile". Each propagates at generally different speeds determined by the important function called the dispersion relation. The use of the explicit form ω(k) is standard, since the phase velocity ω/k and the group velocity dω/dk usually have convenient representations by this function.

Sinusoidal solutions to the 3d wave equation

 * N different sinusoidal waves

Complex amplitude of wave n

$$ A_n = \left | A_n \right | e^{i \left ( \mathbf{k}_\mathrm{n}\cdot\mathbf{r} - \omega_n t + \phi_n \right )} \,\!$$

Resultant complex amplitude of all N waves

$$ A = \sum_{n=1}^{N} A_n \,\!$$

Modulus of amplitude

$$ A = \sqrt{AA^{*}} = \sqrt{\sum_{n=1}^N \sum_{m=1}^N \left | A_n \right | \left | A_m \right | \cos \left [ \left ( \mathbf{k}_n - \mathbf{k}_m \right ) \cdot \mathbf{r} + \left ( \omega_n - \omega_m \right ) t + \left ( \phi_n - \phi_m \right ) \right ]} \,\!$$

The transverse displacements are simply the real parts of the complex amplitudes.

1-dimensional corollaries for two sinusoidal waves

The following may be deduced by applying the principle of superposition to two sinusoidal waves, using trigonometric identities. The angle addition and sum-to-product trigonometric formulae are useful; in more advanced work complex numbers and fourier series and transforms are used.

= Electromagnetism =

Definitions
Here subscripts e and m are used to differ between electric and magnetic charges. The definitions for monopoles are of theoretical interest, although real magnetic dipoles can be described using pole strengths. There are two possible units for monopole strength, Wb (Weber) and A m (Ampere metre). Dimensional analysis shows that magnetic charges relate by qm(Wb) = μ0 qm(Am).

Electric quantities
Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or "centre of electrostatic attraction" analogues.

Electric transport Electric fields

Magnetic quantities
Magnetic transport Magnetic fields

Electric circuits
DC circuits, general definitions AC circuits

Electric fields
General Classical Equations

Magnetic fields and moments
General classical equations

Electric circuits and electronics
Below N = number of conductors or circuit components. Subscript net refers to the equivalent and resultant property value.

= Special relativity =