Heisenberg Uncertainty Theorem

THEOREM Two non-commutative hermitic operators cannot provide simultaneous observable measurements with the same precision on a given (prepared) eigen-state. 

To proceed with the proof, note that two operators are said to be non-commutative if their commutator 

considering two hermitic non-commutative operators one would be interested in behavior their standard deviation forms, namely the behavior of the new operators 

However, this resulted state has to be characterized at least by the positive probability of existence, that is  

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

A critical difference between the Fourier transform defined in Equation 10.9 and the wavelet transform defined in Equation 10.22 is the fact that the latter permits localization in both frequency and time that is, we can use the equation to determine what frequencies are active at a specific time interval in a sequence. However, we cannot get exact frequency information and exact time information simultaneously because of the Heisenberg uncertainty principle, a theorem that says that for a given signal, the variance of the signal in the time domain a2, and the variance of the signal in the frequency (e.g., Fourier) domain c2p are related... 

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