1 Heisenberg Uncertainty Principle in Quantum theories:

If X and P are, respectively, the coordinate and conjugate momentum quantum operators then they satisfy the Heisenberg non-commutation/ ’uncertainty’ relation, or the Heisenberg Uncertainty Principle:

[X,P] ≥ iℏI,

(1)

where the identity operator I is employed to simplify notation. This is also called sometimes the ‘Principle of Indetermination’ for obvious reasons, as explained next (see also p. 30 in ref [1]). Stated in words, in quantum physics, the Heisenberg uncertainty principle says that one cannot simultaneously measure with any desired precision both the position and momentum of a quantum particle; thus, “locating a particle in a small region of space makes the momentum of the particle uncertain, and conversely, measuring precisely the momentum of a particle makes the position uncertain–in inverse proportion to the precision of the particle momentum measurement”. The principle applies also to energy and time, where it takes however a somewhat different form in quantum mechanics and QFT. More generally, it applies to many, but not all quantum operators, as there are certain pairs of quantum operators that do commute–those that belong to the same set of eigenvalues.

1.1 ‘Derivation’ from Harmonic Analysis

The following is a related, interesting ‘derivation’ of a general Uncertainty Principle based on Harmonic Anlaysis/Fourier transforms that may hold for all dual (or conjugate) Fourier pairs such as (time, FREQUENCY),

(space,RECIPROCAL < orEuler∕scattering∕diffraction > space),

(2)

or, more generally:

(quantum group element,Hopf algebra < dual > element) := (qG,qG = aH)

(3)

This is further explained in the related attachment as follows:

“If t is the time and f is the action of a force on a system of oscillators with their natural frequencies, then in the formula:

f(t) = ∫ −∞∞F(ω)eiωt dω

(4)

of the inverse Fourier transform, F(ω) represents the amplitude of the oscillator with angular frequency ω. One can infer from the above equation (4) that the more localised is the external force in time (smaller Δt), the more spread out is its spectrum of frequencies (greater Δω), i.e. the greater is the amount of the oscillators that the force has excited with roughly the same amplitude. If, conversely, one wants to achieve better selectivity, i.e. to compress the spectrum to a narrower range of frequencies, then one has to spread out the external action in time. The impossibility to simultaneously localise the action in time and also enhance the selectivity of the action is one of the manifestations of the quantum-mechanical uncertainty principle, which has a fundamental role in modern physics.”

Note: A quantum ‘particle’ is also subject to the de Broglie wave-particle duality principle, which establishes the relation between the associated wavelength and the momentum of a quantum particle.

References