The Susskind–Glogower operator, first proposed by

Leonard Susskind Leonard Susskind (; born June 16, 1940)his 60th birthday was celebrated with a special symposium at Stanford University.in Geoffrey West's introduction, he gives Suskind's current age as 74 and says his birthday was recent. is an American physicis ...

and J. Glogower, refers to the operator where the phase is introduced as an approximate polar decomposition of the

creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...

. It is defined as :

V=\fraca

, and its adjoint :

V^=a^\frac

. Their commutation relation is :

, 0\rangle\langle 0,

, where

, 0\rangle

is the vacuum state of the

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...

. They may be regarded as a (exponential of)

phase operator Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...

because :

Va^a V^=a^a+1

, where

a^a

is the number operator. So the exponential of the phase operator displaces the

number operator In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The number operator acts on Fock space. Let :, \Psi\rangle_\nu=, \phi_1,\p ...

in the same fashion as the

momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensio ...

acts as the generator of translations in quantum mechanics:

\exp\left(i\frac\right)\hat\exp\left(-i\frac\right)=\hat+x_0

. They may be used to solve problems such as atom-field interactions, level-crossings or to define some class of

non-linear coherent states Coherent states are quasi-classical states that may be defined in different ways, for instance as eigenstates of the annihilation operator : a, \alpha\rangle=\alpha, \alpha\rangle, or as a displacement from the vacuum : , \alpha\rangle=D(\alpha) ...

, among others.

References

Quantum mechanics {{quantum-stub