Position, uncertainty principle

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 force F which has to be applied to a molecular lever requires accurate knowledge of its position x if reversible work is to be perfonned. Specifying the positional accuracy as Ax, the uncertainty principle gives the energy requirement as... 

Because of the quantum mechanical Uncertainty Principle, quantum m echanics methods treat electrons as indistinguishable particles, This leads to the Paiili Exclusion Pnn ciple, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That IS, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

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. 

Early in the twentieth century physicists established that molecules are composed of positively charged nuclei and negatively charged electrons. Given their tiny size and nonclassical behavior, exemplified by the Heisenberg uncertainty principle, it is remarkable (at least to me) that Eq. (1) can be considered exact as a description of the electrostatic forces acting between the atomic nuclei and electrons making up molecules and molecular systems. Eor those readers who are skeptical, and perhaps you should be skeptical of such a claim, I recommend the very readable introduction to Jackson s electrodynamics book [1]. 

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... 

Heisenberg, the atom-bomb man. The uncertainty principle. The more accurately you measure the position of something at a particular moment, the less accurately you can measure where it s going the velocity—the trajectory. At least, that s roughly it. My father could tell you more. ... 

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. 

In this equation, H, the Hamiltonian operator, is defined by H = — (h2/8mir2)V2 + V, where h is Planck s constant (6.6 10 34 Joules), m is the particle s mass, V2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Tint) is in terms of probability I/2 () is the probability of finding the particle between x and x + dx, at time t. 

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

The Bohr model is a determinant model of an atom. It implies that the position of the electron is exactly known at any time in the future, once that position is known at the present. The distance of the electron from the nucleus also is exactly known, as is its energy. And finally, the velocity of the electron in its orbit is exactly known. All of these exactly known quantities—position, distance from nucleus, energy, and velocity—can t, according to the Heisenberg uncertainty principle, be known with great precision simultaneously. 

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 ... 

The rest of the atom is sparsely populated but also vibrant and dynamic. The ghostly electrons are arranged in vague clouds and have no clearly defined position. Heisenberg s Uncertainty Principle (1927) tells us that we can t pin-point their positions. Instead, we have to talk in terms of the probability of there being electrons of a certain energy in certain positions (or orbits) around the nucleus at certain times. 

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. 

---