Heisenberg Uncertainty Relationship

All aspects of molecular shape and size are fully reflected by the molecular electron density distribution. A molecule is an arrangement of atomic nuclei surrounded by a fuzzy electron density cloud. Within the Born-Oppenheimer approximation, the location of the maxima of the density function, the actual local maximum values, and the shape of the electronic density distribution near these maxima are fully sufficient to deduce the type and relative arrangement of the nuclei within the molecule. Consequently, the electronic density itself contains all information about the molecule. As follows from the fundamental relationships of quantum mechanics, the electronic density and, in a less spectacular way, the nuclear distribution are both subject to the Heisenberg uncertainty relationship. The profound influence of quantum-mechanical uncertainty at the molecular level raises important questions concerning the legitimacy of using macroscopic analogies and concepts for the description of molecular properties. ... 

Putz, M. V. (2010). On Heisenberg uncertainty relationship, its extension, and the quantum issue of wave-particle duality. International Journal of Molecular Sciences 11, 4124 139 (DOI 10.3390/ijmslll04124). 

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). 

Because the form of each sine wave was defined by Equation (3.4), the quantity x/2 must behave in the same manner as vt. Thus, Ax A( i IX) > % Finally, substitution of Equation (3.20) affords Equation (3.25), which is nearly identical to the Heisenberg uncertainty relationship in Equation (3.22), the main difference being in the exact manner in which the uncertainty is defined. 

The fact that the lifetime t of an excited nuclear state is finite means that its energy has a certain distribution of width F, which, due to the Heisenberg uncertainty relationship, is connected to T by... 

Without reference to electron spin, this result may also be deduced for atoms from Bohr circular orbit theory + Heisenberg uncertainty relationship . For principal quantum... 

This relationship is known as the Heisenberg uncertainty principle. 

Ultrafast time-resolved resonance Raman (TR ) spectroscopy experiments need to consider the relationship of the laser pulse bandwidth to its temporal pulse width since the bandwidth of the laser should not be broader than the bandwidth of the Raman bands of interest. The change in energy versus the change in time Heisenberg uncertainty principle relationship can be applied to ultrafast laser pulses and the relationship between the spectral and temporal widths of ultrafast transform-limited Gaussian laser pulse can be expressed as... 

The laser used to generate the pump and probe pulses must have appropriate characteristics in both the time and the frequency domains as well as suitable pulse power and repetition rates. The time and frequency domains are related through the Fourier transform relationship that hmits the shortness of the laser pulse time duration and the spectral resolution in reciprocal centimeters. The limitation has its basis in the Heisenberg uncertainty principle. The shorter pulse that has better time resolution has a broader band of wavelengths associated with it, and therefore a poorer spectral resolution. For a 1-ps, sech -shaped pulse, the minimum spectral width is 10.5 cm. The pulse width cannot be <10 ps for a spectral resolution of 1 cm . An optimal choice of time duration and spectral bandwidth are 3.2 ps and 3.5 cm. The pump pulse typically is in the UV region. The probe pulse may also be in the UV region if the signal/noise enhancements of resonance Raman... 

The natural line width is determined by Heisenberg s uncertainty relationship ... 

The heavy superstructure of modern quantum mechanics rests largely upon a set of mathematical relationships published in 1927 by Heisenberg. These relationships are now usually referred to collectively as the uncertainty principle. Heisenberg showed that in any quantum-mechanical system, pairs of dynamical variables for particles can be simultaneously and sharply defined only if their operators commute. This means only if their operators H and K satisfy the equation... 

Footnote The Wave Nature of the Electron. So far the electron has been considered as a particle, with clearly quantised energy levels, that can be precisely measured, as in the emission lines of the spectrum of hydrogen. Because the electron is so small and light, the accuracy with which it can be measured is very uncertain. This is associated with the Heisenberg Uncertainty Principle, which states that it is impossible to determine both the position and momentum of an electron simultaneously , i.e. Ax Ap = hl2it, where Ax is the uncertainty in measuring the position of the electron and Ap is the uncertainty in measuring the momentum (p = mass X velocity) of the electron. The two uncertainties bear an inverse relationship to each other. Consequently, if the position of the... 

There is a relationship between the lifetime of an excited state and the bandwidth of the absorption band associated with the transition to the excited state. This relationship is a consequence of the Heisenberg uncertainty principle, which states that at the microscopic level the... 

Zewail. Ahmed H. Chemistry at the Uncertainty Limit. Angewandte Chemie International Edition 40 (2001) 4,371-75. This is a nonmath-ematical presentation of the relationship between femtosecond (10" s) spectroscopy [and even attosecond (10 s) spectroscopy] with the Heisenberg uncertainty principle. 

Although currently (as a matter of fact even since its publication by Heisenberg in 1927) are heated discussions and attempts to dismantle the dogma imposed by limiting/Heisenberg uncertainty in the Planck constant, the utility of this relationship (even borderline) is incontestable, which will be illustrated also by application to the Hydrogen atom (Bohr model), immediately below, and latter in a more elaborate framework. 

Finally, for spectroscopic analysis, one could ask upon the corresponding time-energy uncertainty relationship (Busch, 2008) within the actual approach. Firstly, the correctness of such problem is conceptually guaranteed by the Heisenberg representation of a quantum evolution, where, for a cyclic vector of state viz. the present periodical paths or orbits) and an unitary transformation U, the cyclic Hamiltonian is accompanied by... 

De Broglie s relationship suggests that electrons are matter waves and thus should display wavelike properties. A consequence of this wave-particle duality is the limited precision in determining an electron s position and momentum imposed hy the Heisenberg uncertainty principle. How then are we to view electrons in atoms To answer this question, we must begin by identifying two types of waves. 

Incidentally, the uncertainty principle associated with the name of Heisenberg, well known in quantum mechanics, follows from the expression given here when de Broglie s relationship connecting the momentum of a particle with its wavelength is included. 

While for some purposes it may be necessary to have accurate frequency definition, for others good time discrimination is useful. These are opposite requirements. Because of the Fourier relationship between frequency and time, the more precisely the time of a signal is known, the greater bandwidth of frequencies is necessary (there is a close analogy here with Heisenberg s uncertainty principle). Approximately, the time resolution t is the reciprocal of the bandwidth Bw, so that their product Bwr 1. 

The Heisenberg relationship says that the uncertainty in an object s position, Ax, times the uncertainty in its momentum, Amv, is equal to or greater than the quantity h/4ir. 

The imprecise nature of Schrodinger s model was supported shortly afterwards by a principle proposed by Werner Heisenberg, in 1927. Heisenberg demonstrated that it is impossible to know both an electron s pathway and its exact location. Heisenberg s uncertainty principle is a mathematical relationship that shows that you can never know both the position and the momentum of an object beyond a certain measure of precision. 

Electrons moving in circles around the nucleus, as in Bohr s theory, can be thought of as forming standing waves that can be described by the de Broglie equation. However, we no longer believe that it is possible to describe the motion of an electron in an atom so precisely. This is a consequence of another fundamental principle of modern physics, Heisenberg s uncertainty principle, which states that there is a relationship... 

Foundations are laid in the introductory chapter, which deals with fundamental particles, electromagnetic radiation and Heisenberg s uncertainty principle. Atomic orbitals are then described, using a minimum of mathematics, followed by a discussion of the electron configurations of the elements. Further chapters reveal the relationships between the electronic configurations of the elements and some properties of their atoms and the variations in the properties of their fluorides and oxides across the periods and down the groups of the Periodic Table. 

Heisenberg s Uncertainty Principle Relationship between a Particle s Uncertainty in Position [Ax) and Uncertainty in Velocity (Av) (7.4)... 

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