Extending Heisenberg Uncertainty

EXTENDING HEISENBERG UNCERTAINTY 4.7.3.1 Averaging Quantum Fluctuations... 

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. 

OBSERVABLE QUANTUM CHEMISTRY EXTENDING HEISENBERG S UNCERTAINTY... 

FIGURE 4.6 The chart of Heisenberg Uncertainty Relationship (HUR) appearance for observed and free quantum evolutions covering the complete scale of the particle to wave ratios as computed from the Eqs. (4.556) and (4.562), respectively the points Q and a correspond to wave-particle precise equivalence and to the special extended-HURs of Eqs. (4.563) and (4.564), respectively (Putz, 2010c). 

For the ultrashort events, the question arises as to whether the Heisenberg uncertainty principle has relevance. A consequence of this principle is that in measurements of the energy ( ) of a system over a time t there is a relation between the uncertainty AE (i.e., the mean square deviation) of E and that of the time (Ar), namely AE x At h/An, where h is Planck s constant. If one wishes to measure the energy E within an accuracy of A E, the measurements must be extended over a time-interval of at least hlAn A E. This fact has implications for spectroscopy, since it affects the sharpness of spectral lines. Consider the excitation by a photon of a molecule from the ground state to an upper state. The breadth (Av) of a line in the vibrational spectrum is related to AE by the equation An = AE/A, and so is related to the duration t of the excited state by Av = l/Ant. If t is very short, A V may be appreciable and the spectral lines will appear diffuse. As an example, consider the dissociation of an iodine molecule (I2 21) by a femtosecond pulse the quantities... 

These oscillations define the particle s de Broglie wavelength, and therefore its momentum. If we wanted to know the momentum exactly, we would need a perfect sine wave that extended forever consequently we would not know anything about its position. If we wanted to know its position exactly, the wavepacket would become infinitely narrow along the x axis and the wavelength (and hence momentum) would become an unknowable parameter. As the position of a quantum particle wavepacket becomes more certain, its momentum becomes more uncertain, and vice versa. We cannot know exact values of both the position and momentum of the particle at the same time. That is the qualitative version of the Heisenberg imcertainty principle. More precisely, the uncertainty... 

In the same year, Werner Heisenberg formulated the uncertainty principle, which asserts that it is impossible to simultaneously determine the position and momentum of a mass particle to a precision better than AxAp > h, where h is Planck s constant. There are several ways to imderstand this. One argument stems from the fact that that if a particle is represented by a single frequency (or wavelength), its momentum (p = h/X) is known precisely, but a plane wave extending from —oo to +oo provides no information about the position of the particle. To localize the particle, waves with slightly different frequencies must be added so their amplitudes bimch up to form a wave packet as shown in Figure 2.3. The relative width of the wave packet can be shown to be inversely proportional to the spread in frequencies or Ax/X A/AA or AxAX A. Since p = h/X, Ap= —AXh/X and Ax Ap PS h. 

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