Momentum, uncertainty principle

Of the variety of quantum effects which are present at low temperatures we focus here mainly on delocalization effects due to the position-momentum uncertainty principle. Compared to purely classical systems, the quantum delocalization introduces fluctuations in addition to the thermal fluctuations. This may result in a decrease of phase transition temperatures as compared to a purely classical system under otherwise unchanged conditions. The ground state order may decrease as well. From the experimental point of view it is rather difficult to extract the amount of quantumness of the system. The delocahzation can become so pronounced that certain phases are stable in contrast to the case in classical systems. We analyze these effects in Sec. V, in particular the phase transitions in adsorbed N2, H2 and D2 layers. 

The Heisenberg position-momentum uncertainty principle (3.82) agrees with equation (2.26), which was derived by a different, but mathematically... 

Dr. Bernard L. Cohen [29] employs the energy-time uncertainty principle in discussing quantum fluctuations. Dr. Robert Comer [30] (cited by Dr. Cohen [29]) shows how the position-momentum uncertainty principle can be employed in more limited circumstances. 

Gomer R. Field Emission and Field Ionization. Cambridge, Mass Harvard University Press 1961, pp. 6-11, 66-70, and 181-183. The application of the position-momentum uncertainty principle per se is discussed on pp. 6-8. 

Describe the wave-particle duality of matter and energy and the theories and experiments that led to it (particle wavelength, electron diffraction, photon momentum, uncertainty principle) ( 7.3) (SP 7.3) (EPs 7.27-7.34)... 

Heisenburg uncertainty principle For small particles which possess both wave and particle. properties, it is impossible to determine accurately both the position and momentum of the particle simultaneously. Mathematically the uncertainty in the position A.v and momentum Ap are related by the equation... 

One feature of this inequality warrants special attention. In the previous paragraph it was shown that the precise measurement of A made possible when v is an eigenfiinction of A necessarily results in some uncertainty in a simultaneous measurement of B when the operators /land fido not conmuite. However, the mathematical statement of the uncertainty principle tells us that measurement of B is in fact completely uncertain one can say nothing at all about B apart from the fact that any and all values of B are equally probable A specific example is provided by associating A and B with the position and momentum of a particle moving along the v-axis. It is rather easy to demonstrate that [p, x]=- ih, so that If... 

The uncertainty principle, according to which either the position of a confined microscopic particle or its momentum, but not both, can be precisely measured, requires an increase in the carrier energy. In quantum wells having abmpt barriers (square wells) the carrier energy increases in inverse proportion to its effective mass (the mass of a carrier in a semiconductor is not the same as that of the free carrier) and the square of the well width. The confined carriers are allowed only a few discrete energy levels (confined states), each described by a quantum number, as is illustrated in Eigure 5. Stimulated emission is allowed to occur only as transitions between the confined electron and hole states described by the same quantum number. 

The difficulty will not go away. Wave-particle duality denies the possibility of specifying the location if the linear momentum is known, and so we cannot specify the trajectory of particles. If we know that a particle is here at one instant, we can say nothing about where it will be an instant later The impossibility of knowing the precise position if the linear momentum is known precisely is an aspect of the complementarity of location and momentum—if one property is known the other cannot be known simultaneously. The Heisenberg uncertainty principle, which was formulated by the German scientist Werner Heisenberg in 1927, expresses this complementarity quantitatively. It states that, if the location of a particle is known to within an uncertainty Ax, then the linear momentum, p, parallel to the x-axis can be known simultaneously only to within an uncertainty Ap, where... 

FIGURE 1.22 A representation of the uncertainty principle, (a) The location ot the particle is ill defined and so the momentum of the particle (represented by the arrow) can be specified reasonably precisely, (b) The location of the particle is well defined, and so the momentum cannot be specified very precisely. 

The location and momentum of a particle are complementary that is, both the location and the momentum cannot be known simultaneously with arbitrary precision. The quantitative relation between the precision of each measurement is described by the Heisenberg uncertainty principle. 

Heisenberg uncertainty principle If the location of a particle is known to within an uncertainty Ax, then the linear momentum parallel to the x-axis can he known only to within an uncertainty Ap, where ApAx > till. Henderson-Hasselbalch equation An approximate equation for estimating the pH of a solution containing a conjugate acid and base. See also Section 11.2. Henry s constant The constant kH that appears in Henry s law. 

Uncertainty principle The principle developed by Werner Heisenberg that it is not possible to know the momentum and position of a particle with unlimited accuracy. 

The Heisenberg uncertainty principle is a consequence of the stipulation that a quantum particle is a wave packet. The mathematical construction of a wave packet from plane waves of varying wave numbers dictates the relation (1.44). It is not the situation that while the position and the momentum of the particle are well-defined, they cannot be measured simultaneously to any desired degree of accuracy. The position and momentum are, in fact, not simultaneously precisely defined. The more precisely one is defined, the less precisely is the other, in accordance with equation (1.44). This situation is in contrast to classical-mechanical behavior, where both the position and the momentum can, in principle, be specified simultaneously as precisely as one wishes. 

Werner Heisenberg (1901-1976 Nobel Prize for physics 1932) developed quantum mechanics, which allowed an accurate description of the atom. Together with his teacher and friend Niels Bohr, he elaborated the consequences in the "Copenhagen Interpretation" — a new world view. He found that the classical laws of physics are not valid at the atomic level. Coincidence and probability replaced cause and effect. According to the Heisenberg Uncertainty Principle, the location and momentum of atomic particles cannot be determined simultaneously. If the value of one is measured, the other is necessarily changed. 

If we increase the accuracy with which the position of the electron is determined by decreasing the wavelength of the light that is used to observe the electron, then the photon has a greater momentum, since p = hiA. The photon can then transfer a larger amount of momentum to the electron, and so the uncertainty in the momentum of the electron increases. Thus any reduction in the uncertainty in the position of the electron is accompanied by an increase in the uncertainty in the momentum of the electron, in accordance with the uncertainty principle relationship. We may summarize by saying that there is no way of accurately measuring simultaneously both the position and velocity of an electron the more closely we attempt to measure its position, the more we disturb its motion and the less accurately therefore we are able to define its velocity. 

Uncertainty principle Quantum mechanics restricts the knowledge of certain pairs of variables, notably time and energy and position and momentum, so that complete... 

For matter waves hk is the particle momentum and the uncertainty relation AxAp > h/2, known as the Heisenberg uncertainty principle. 

The fact that there is no zero-point energy may seem to conflict with the uncertainty principle. However, when the rotor stops, it could happen anywhere on the circle and that accounts for the uncertainty. The expression L2 = 2El = h2k2 shows that the angular momentum, L is restricted to integral multiples of h. 

The Heisenberg Uncertainty Principle, describing a dispersion in location and momentum of material particles that depends inversely on their mass, gives rise to vibrational zero-point energy differences between molecules that differ only isotopically. These zero-point energy differences are the main origin of equilibrium chemical isotope effects, i.e., non-unit isotopic ratios of equilibrium constants such as K /Kj) for a reaction of molecules bearing a protium (H) atom or a deuterium (D) atom. 

The residual energy (designated of a harmonic oscillator in the ground state. The Heisenberg Uncertainty Principle does not permit any state of completely defined position and momentum. A one-dimensional harmonic oscillator has energy levels corresponding to ... 

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