For any

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

written in polar form (such as ), the phase factor is the complex exponential (), where the variable is the ''phase'' of a wave or other periodic function. The phase factor is a unit complex number, i.e. a complex number of

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

1. It is commonly used in

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

and

optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

. It is a special case of ''

phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude and initial phase are time-invariant and whose angular frequency is fixed. It is related to a mor ...

s'', which may have arbitrary magnitude (i.e. not necessarily on the unit circle in the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

). Multiplying the equation of a

plane wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...

by a phase factor shifts the phase of the wave by :

e^ A\,e^ = A\,e^.

In quantum mechanics, a phase factor is a complex coefficient that multiplies a ket

, \psi\rangle

or bra

\langle\phi,

. It does not, in itself, have any physical meaning, since the introduction of a phase factor does not change the expectation values of a

Hermitian operator In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...

. That is, the values of

\langle\phi,  A , \phi\rangle

and

\langle\phi,  A e^ , \phi\rangle

are the same. However, ''differences'' in phase factors between two interacting

quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...

s can sometimes be measurable (such as in the Berry phase) and this can have important consequences. In optics, the phase factor is an important quantity in the treatment of

interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...

Notes

References

* {{DEFAULTSORT:Phase Factor Information theory Quantum information science Notation

See also

Notes

References