Heisenberg Uncertainty Product for Different Photon States

Each mode of the quantized radiation field can be associated with a quantized harmonic oscillator. This Demonstration shows the dependence of Heisenberg's uncertainty product (for momentum and position ), on the quantum number , when the oscillator has the energy eigenstate (in units of ). The integer denotes the energy and the number of photons in the radiation mode, . The uncertainty product is an increasing function of . The minimal product is valid for the energy ground state, , represented by black lines in the diagrams.
In contrast to a state with a definite number of photons, a coherent state is defined in quantum optics as an eigenstate of the photon annihilation operator , , where is complex valued. The radiation of a single mode laser represents such a state; each coherent state has the minimum-uncertainty product allowed by quantum physics, . Therefore, coherent states provide the closest quantum mechanical analog to a classical single-mode field. These are not shown in this Demonstration.

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In quantum physics it is well known that two observables and can be incompatible, with . These are represented by noncommutating Hermitian operators: . As a consequence, the uncertainties and do not vanish simultaneously; denotes the variance of the operator . The variances depend on the actual state of the physical system. Therefore, no quantum state exists for the physical system considered with and . As an example, the Heisenberg uncertainty relation follows from the quantum commutation relation of the noncommutating operators momentum and position and is valid for each quantum state.
References
[1] P. Meystre and M. Sargent III, Elements of Quantum Optics, New York: Springer–Verlag, 1991.
[2] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge: Cambridge University Press, 1997.
[3] J. J. Sakurai, Modern Quantum Mechanics, Reading, MA: Addison–Wesley, 1995.
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