Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 9916.1.6 Quantum Coherent StatesRecall that a quantum coherent state is a specific kind of quantum stateof the quantum harmonic oscillator whose dynamics most closely resemblethe oscillating behavior of a classical harmonic oscillator. It was the firstexample of quantum dynamics when Erwin Schrödinger derived it in 1926while searching for solutions of the Schrödinger equation that satisfy thecorrespondence principle. The quantum harmonic oscillator and hence, thecoherent state, arise in the quantum theory of a wide range of physicalsystems. For instance, a coherent state describes the oscillating motion ofthe particle in a quadratic potential well. In the quantum electrodynamicsand other bosonic quantum field theories they were introduced by the2005 Nobel Prize winning work of Roy Glauber in 1963 [Glauber (1963a);Glauber (1963b)]. Here the coherent state of a field describes an oscillatingfield, the closest quantum state to a classical sinusoidal wave such as acontinuous laser wave.In classical optics, light is thought of as electromagnetic waves radiatingfrom a source. Specifically, coherent light is thought of as light that isemitted by many such sources that are in phase. For instance, a lightbulb radiates light that is the result of waves being emitted at all thepoints along the filament. Such light is incoherent because the process ishighly random in space and time. On the other hand, in a laser, light isemitted by a carefully controlled system in processes that are not randombut interconnected by stimulation and the resulting light is highly ordered,or coherent. Therefore a coherent state corresponds closely to the quantumstate of light emitted by an ideal laser. Semi–classically we describe sucha state by an electric field oscillating as a stable wave. Contrary to thecoherent state, which is the most wave–like quantum state, the Fock state(e.g., a single photon) is the most particle–like state. It is indivisible andcontains only one quanta of energy. These two states are examples of theopposite extremes in the concept of wave–particle duality. A coherent statedistributes its quantum–mechanical uncertainty equally, which means thatthe phase and amplitude uncertainty are approximately equal. Conversely,in a single–particle state the phase is completely uncertain.Formally, the coherent state |α〉 is defined to be the eigenstate of theannihilation operator a, i.e., a|α〉 = α|α〉. Note that since a is not Hermitian,α = |α|e iθ is complex. |α| and θ are called the amplitude and phaseof the state.Physically, a|α〉 = α|α〉 means that a coherent state is left unchanged