# Heisenberg Uncertainty Product for Different Photon States

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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.

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Contributed by: Reinhard Tiebel (March 2011)

Open content licensed under CC BY-NC-SA

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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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