Uncertainty principle position-momentum

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. 

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 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... 

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. 

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

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... 

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 ... 

What is the lowest possible energy for the harmonic oscillator defined in Eq. (5.10) Using classical mechanics, the answer is quite simple it is the equilibrium state with x 0, zero kinetic energy and potential energy E0. The quantum mechanical answer cannot be quite so simple because of the Heisenberg uncertainty principle, which says (roughly) that the position and momentum of a particle cannot both be known with arbitrary precision. Because the classical minimum energy state specifies both the momentum and position of the oscillator exactly (as zero), it is not a valid quantum... 

The electron and momentum densities are just marginal probability functions of the density matrix in the Wigner representation even though the latter, by the Heisenberg uncertainty principle, cannot be and is not a true joint position-momentum probability density. However, it is possible to project the Wigner density matrix onto a set of physically realizable states that optimally fulfill the uncertainty condition. One such representation is the Husimi function [122,133-135]. This seductive line of thought takes us too far away from the focus of this... 

D) The Heisenberg Uncertainty Principle says that it is impossible to determine the exact position and momentum of an electron at the same time. It is a fundamental principle of quantum mechanics. 

The uncertainty principle rationalizes our inability to observe the momentum and position of an atomic particle simultaneously. The act of observation interferes with atomic particles so that their momenta and positions are altered. 

This lies at the heart of the difference between classical and quantum physics. Classically, at any instant in time we can characterize a particle by its exact position, x, and exact momentum, p, at least in principle. Quantum mechanically, on the other hand, if we know the position x, with a high degree of certainty, then there will be a large uncertainty in its momentum, and vice versa. 

J s are the operators corresponding to spin around the z-axis. In other words, we can measure the z-spin states of the two particles simultaneously. In contrast, it is impossible to measure both the x- and z-spins of a single particle simultaneously nor is it possible to measure both the position and the momentum of a dynamical particle simultaneously, by Heisenberg s uncertainty principle. The independence of our two measurements is crucial. 

If the entering particle was in a mixed state (relative to the r-spin measurement), then the act of measurement changes the state of the particle. No one understands how this happens, but it is an essential feature of the quantum mechanical model. For example, this phenomenon contributes to Heisenberg s uncertainty principle, whose most famous implication is that one cannot measure both the position and the momentum of a particle exactly. The point is that a position measurement changes the state of tlie particle in a way that erases information about the momentum, and vice versa. 

Our understanding of the basic nature of matter is limited by Heisenberg s uncertainty principle. Stated simply, this principle implies that our measurements of the position and momentum of a particle of subatomic mass arc always in error when radiation is used to study matter. If x... 

You may have heard of the uncertainty principle, but if you have not studied chemical physics you may have little idea of its possible importance to organic chemistry. The usual statement of the principle is that there are limits to how precisely we can specify the momentum and the position of a particle at the same time. An alternative statement has more relevance to spectroscopy and chemistry, namely, that the precision with which we can define the energy of a state depends on the lifetime of the state. The shorter the lifetime, the less the certainty with which we can define the energy.2... 

The symbol h, which is read h bar, means h/ln, a useful combination that occurs widely in quantum mechanics. From inside the back cover, we see that ti = 1.054 X 10-34 J-s. Equation 6 tells us that if the uncertainty in position is very small (Ax very small), then the uncertainty in linear momentum must be large, and vice versa (Fig. 1.11). The uncertainty principle has negligible practical consequences for macroscopic objects, but it is of profound importance for electrons in atoms and for a scientific understanding of the nature of the world. 

In mathematical terms, Heisenberg s principle states that the uncertainty in the electron s position, Ax, times the uncertainty in its momentum, Amv, is equal to or greater than the quantity h/4ir ... 

The uncertainty principle shows that the classical trajectory of a particle, with a precisely determined position and momentum, is really an illusion. It is a very good approximation, however, for macroscopic bodies. Consider a particle with mass I Xg, and position known to an accuracy of 1 pm. Equation 2.41 shows that the uncertainty in momentum is at least 5 x 10 29 kg m s-1, corresponding to a velocity of 5 x 10 JO m s l. This is totally negligible for any practical purpose, and it illustrates that in the macroscopic world, even with very light objects, the uncertainty principle is irrelevant. If we wanted to, we could describe these objects by wave packets and use the quantum theory, but classical mechanics gives essentially the same answer, and is much easier. At the atomic and molecular level, however, especially with electrons, which are very light, we must abandon the idea of a classical trajectory. The statistical predictions provided by Bom s interpretation of the wavefunction are the best that can be obtained. 

Figure 5.11 shows one of the most counterintuitive results of quantum mechanics. There are fundamental limits to our ability to make certain measurements—the act of determining the state of a system intrinsically perturbs it. For example, it is impossible to measure position and momentum simultaneously to arbitrarily high accuracy any attempt to measure position automatically introduces uncertainty into the momentum. Similarly, a molecule which is excited for a finite period of time cannot have a perfectly well-defined energy. As a result, classical determinism fails. It is not possible, even in principle, to completely specify the state of the universe at any instant, hence the future need not be completely defined by the past. These results are usually phrased something like ... 

Instead, if you measure Sz again after measuring Sx, you find that Sz has been randomized and Sz = —h/2 is just as likely as Sz = A-Ti/l (Figure 5.16). Only one component of the angular momentum can be specified at a time, and the act of measuring this component completely randomizes the others—just as measurements of position and momentum were limited by the Heisenberg Uncertainty Principle. 

Notice that the lowest energy state j = 0 has E = 0, but it does not correspond to a bond which is merely pointing in one direction in space just like an s orbital, it is simultaneously pointing in all directions. This is yet another manifestation of the Uncertainty Principle. If a bond is known to point in one direction, the positional uncertainty perpendicular to the bond direction is zero, and the momentum uncertainty is infinite ... 

As with many of the paradoxical results of quantum mechanics, the Uncertainty Principle comes to the rescue. You cannot localize the position of the electron inside the nucleus (very small Ax) without creating a huge uncertainty in the momentum, and thus losing any knowledge of the orbital you are in. 

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