Phase factor
Encyclopedia
For any complex number
written in polar form (such as r eiθ), the phase factor is the exponential part, i.e. eiθ. As such, the term "phase factor" is similar to the term phasor
, although the former term is more common in quantum mechanics. This phase factor is itself a complex number of absolute value
1. The variable θ appearing in such an expression is generally referred to as the phase
. Multiplying the equation of a plane wave
Aei(k·r−ωt) by a phase factor shifts the phase of the wave by θ:
.
In quantum mechanics
, a phase factor is a complex coefficient eiθ that multiplies a ket
or bra
. 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. That is, the values of 
and 
are the same. However, differences in phase factors between two interacting quantum states 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.
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
written in polar form (such as r eiθ), the phase factor is the exponential part, i.e. eiθ. As such, the term "phase factor" is similar to the term phasor
Phasor
Phasor is a phase vector representing a sine wave.Phasor may also be:* Phasor , a stereo music, sound and speech synthesizer for the Apple II computer* Phasor measurement unit, a device that measures phasors on an electricity grid...
, although the former term is more common in quantum mechanics. This phase factor is itself a complex number of absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
1. The variable θ appearing in such an expression is generally referred to as the phase
Phase
-In physics:*Phase , a physically distinctive form of a substance, such as the solid, liquid, and gaseous states of ordinary matter**Phase transition is the transformation of a thermodynamic system from one phase to another*Phase...
. Multiplying the equation of a plane wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
Aei(k·r−ωt) by a phase factor shifts the phase of the wave by θ:
.In quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, a phase factor is a complex coefficient eiθ that multiplies a ket
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
or braBra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
. 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. That is, the values of and are the same. However, differences in phase factors between two interacting quantum states can sometimes be measurable (such as in the Berry phase) and this can have important consequences.In optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, the phase factor is an important quantity in the treatment of interference.
See also
- Berry phase
- Bra-ket notationBra-ket notationBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
- Euler's formulaEuler's formulaEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
- PhasorPhasorPhasor is a phase vector representing a sine wave.Phasor may also be:* Phasor , a stereo music, sound and speech synthesizer for the Apple II computer* Phasor measurement unit, a device that measures phasors on an electricity grid...
- Plane wavePlane waveIn the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....