China
Africa
Explore chapters and articles related to this topic
Electrons and anti-symmetry
Published in David K Ferry, Quantum Mechanics, 2001
In classical mechanics, it is quite easy to follow the individual trajectories of each and every electron. However, the Heisenberg uncertainty principle prevents this from being possible in quantum mechanics. If we completely understand the position of a given particle, we can say nothing about its momentum and most other dynamical variables. This principle of the indistinguishability of identical particles leads to other more complex forms of the wave functions. These more complicated forms can lead to effects in quantum mechanics that have no analogue in classical mechanics. We use the word identical to indicate particles that can be freely interchanged with one another with no change in the physical system. While these particles may be distinguished from one another in situations in which their individual wave functions do not overlap, the more usual case is where they have overlapping wave functions because of the high particle density in the system. In this latter case, it is necessary to invoke a many-particle wave function. To describe properly the properties of the many-particle wave function, it is necessary to understand the properties of the interactions that occur among and between these single particles. If the one-electron problem were our only interest, this more powerful development that is used for the interacting system would be worthless to us, for its complexity is not worth the extra effort. However, when we move to multi-electron problems, the power of the approach becomes apparent. In this chapter, we want to examine just the properties of these general wave functions for the case of electrons, which also possess their own spin angular momentum, and to introduce the interaction among the electrons.
Understanding the Atom and the Nucleus
Published in Robert E. Masterson, Nuclear Engineering Fundamentals, 2017
We will consider only the formulation dealing with the uncertainties in the measurement of the energy of a particle, which we will call ΔE, in the period of time we are able to measure it, which we will call Δt. The uncertainty principle states that the precision with which we can measure the characteristics of any given physical object (such as a nuclear particle) has limits to it. The precision with which we can measure the energy of any physical object ΔE in a time Δt will never be greater than that given by the following equation:
Introduction to Nanosensors
Published in Vinod Kumar Khanna, Nanosensors, 2021
Two fundamental differences exist between quantum and classical mechanics: The parameters of a quantum system, such as the position and momentum of an electron, are affected by the act of measuring them. This is expressed in the uncertainty principle, which states that the uncertainty in simultaneous measurement of the x-coordinate and the x-component of the momentum of the particle is greater than or equal to a minimum value defined by the equationΔxΔpx≥h4πwhere h is the Planck’s constant, a fundamental constant of quantum mechanics which relates the energy carried by a photon to its frequency.The parameters of a quantum system are not exact or precise values but are expressed as probability distributions. This happens because electrons act as particles and waves at the same time; this concept is called wave-particle duality. The wavelength (λ) associated with a particle of mass m moving with a velocity v isλ=hmv
Quantum models of cognition and decision
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
If P1P2 − P2P1 = 0, then operators commute (they are commutative). Then, order of events does not matter: P1P2 = P2P1. Such operators has common basis, and questions they represent are compatible. If P1P2 − P2P1 ≠ 0 then such operators do not commute (they are not commutative – they do not have a common basis). Then, P1P2 ≠ P2P1 and result of action on any state vector depends on order of events. It is reflected in quantum mechanics as the Heisenberg’s uncertainty principle: there are certain pairs of physical properties of a particle, known as complementary variables (e.g. momentum of a particle and its corresponding position) that cannot be simultaneously measured, because the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. If this is the case, order of measurements is crucial.
Nonclassical properties of a deformed atom-cavity field state
Published in Journal of Modern Optics, 2022
Naveen Kumar, Arpita Chatterjee
The uncertainty principle prevents a direct phase-space description of a quantum mechanical system. This fact leads to the creation of quasiprobability distributions, which are very useful in quantum mechanics because they provide a quantum-classical correspondence and make it easier to calculate quantum mechanical averages in a way that is similar to the calculation of classical phase-space averages [27]. The Q function is one such quasiprobability distribution, and its zeros are a sign of nonclassicality [28]. The Q function is calculated as follows: In Figure 6, we observe that the Q distribution shows the wavy and non-Gaussian nature and remains positive everywhere.
Dynamics of entropic uncertainty relation under quantum channels with memory
Published in Journal of Modern Optics, 2019
Guo-you Wang, You-neng Guo, Ke Zeng
The uncertainty principle is one of the most remarkable features of quantum mechanics initially proposed by Heisenberg (1) for two incompatible observables of position and momentum. It states that, one can not simultaneously predict the measurement outcomes of these two incompatible observables with certainty. Later this uncertainty principle is further formulated by Robertson (2) to generalize for arbitrary pairs of observables R and Q based on the standard deviations: , and are the standard deviations and is the commutator. Usually, the left hand side of this inequality is named uncertainty and the right hand of this inequality is called uncertainty bound. It is worthy noting that, this uncertainty bound is dependent of quantum state to be measured, which maybe lead to a trivial bound if the commutator has zero expectation value. To break through this drawback and to precisely capture its physical meanings, uncertainty relation has been developed in an information theoretical framework replacing the standard deviation with entropy by Deutsch (3). Afterwards, an improved version was given by Kraus and then proved by Maassen and Uffink (4). It states that, given arbitrary pairs of observables R and Q with eigenstates and , respectively, for any state ρ to be measured where quantifies the complementarity of R and Q, and or denotes the Shannon entropy of the probability distribution of the outcomes when R or Q is performed on a system ρ, respectively. Note that this uncertainty bound quantified by entropy provides a fixed lower bound compared with Heisenberg's uncertainty relation, since c does not depend on specific states to be measured.