Quantum mechanical model probability distribution

List the most important ideas of the quantum mechanical model of the atom. Include in your discussion the terms or names wave function, orbital, Heisenberg uncertainty principle, de Broglie, Schrodinger, and probability distribution. 

SECTION 6.5 In the quantum mechanical model of the hydrogen atom, the behavior of the electron is described by mathematical functions called wave functions, denoted with the Greek letter 0. Each allowed wave function has a precisely known energy, but the location of the electron cannot be determined exactly rather, the probability of it being at a particular point in space is given by the probabWty density, 0. The electron density distribution is a map of the probability of finding the electron at all points in space. 

The quantum-mechanical model of the atom replaced the Bohr model in the early twentieth century. In the quantum-mechanical model Bohr orbits are replaced with quantum-mechanical orbitals. Orbitals are different from orbits in that they represent, not specific paths that electrons follow, but probability maps that show a statistical distribution of where the electron is likely to be found. The idea of an orbital is not easy to visualize. Quantum mechanics revolutionized physics and chemistry because in the quantums-mechanical model, electrons do not behave like particles flying through space. We cannot, in general, describe their exact paths. An orbital is a probability map that shows where the electron is likely to be found when the atom is probed it does not represent the exact path that an electron takes as it travels through space. 

The maximum in the radial distribution function, 52.9 pm, turns out to be the very same radius that Bohr had predicted for the innermost orbit of the hydrogen atom. However, there is a significant conceptnal difference between the two radii. In the Bohr model, every time you probe the atom (in its lowest energy state), yon would find the electron at a radius of 52.9 pm. In the quantum-mechanical model, yon would generally find the electron at various radii, with 52.9 pm having the greatest probability. 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

As an example. Fig. 18 shows the diabatic electronic population probability for Model I. The quantum-mechanical results (thick line) are reproduced well by the QCL calculations, which have assumed a localization time of to = 20 fs. The results obtained for the standard QCL (thin full line) and the energy-conserving QCL (dotted line) are of similar quality, thus indicating that the phase-space distribution p]](x, p) at to = 20 fs is similar for the two schemes. Also shown in Fig. 18 are the results obtained for a standard surface-hopping calculation (dashed line), which largely fail to match the beating of the quantum reference. 

Fig. 9. Radial probability density distribution, derived from quantum-mechanical predictions. Ibr rubidium, along with Bohr planetary model for the same atom...

Particle-in-a-box models and the qnantnm harmonic oscillator illustrate a number of important features of quantum mechanics. The energy level structure depends on the natnre of the potential in the particle in a box, E n, whereas for the harmonic oscillator, E n. The probability distributions in both cases are different than for the classical analogs. The most probable location for the particle-in-a-box model in its gronnd state is the center of the box, rather than uniform over the box as predicted by classical mechanics. The most probable position for the quantum harmonic oscillator in the ground state is at its equilibrium position, whereas the classical harmonic oscillator is most likely to be fonnd at the two classical turning points. Normalization ensures that the probabilities of finding the particle or the oscillator at all positions add np to one. Finally, for large values of n, the probability distribution looks mnch more classical, in accordance with the correspondence principle. 

In classical mechanics It Is assumed that at each Instant of time a particle is at a definite position x. Review of experiments, however, reveals that each of many measurements of position of Identical particles in identical conditions does not yield the same result. In addition, and more importantly, the result of each measurement is unpredictable. Similar remarks can be made about measurement results of properties, such as energy and momentum, of any system. Close scrutiny of the experimental evidence has ruled out the possibility that the unpredictability of microscopic measurement results are due to either inaccuracies in the prescription of initial conditions or errors in measurement. As a result, it has been concluded that this unpredictability reflects objective characteristics inherent to the nature of matter, and that it can be described only by quantum theory. In this theory, measurement results are predicted probabilistically, namely, with ranges of values and a probability distribution over each range. In constrast to statistics, each set of probabilities of quantum mechanics is associated with a state of matter, including a state of a single particle, and not with a model that describes ignorance or faulty experimentation. 

To distinguish the quantum mechanical description of an atom from Bohr s model, we speak of an atomic orbital, rather than an orbit. An atomic orbital can be thought of as the wave function of an electron in an atom. When we say that an electron is in a certain orbital, we mean that the distribution of the electron density or the probability of locating the electron in space is described by the square of the wave function associated with that orbital. An atomic orbital, therefore, has a characteristic energy, as well as a characteristic distribution of electron density. 

The development of quantum mechanics enabled chemists to describe electron energies and locations outside the nucleus more accurately than was possible with the planetary model for the atom. The meanings and implications of quantum numbers, photons, electromagnetic radiation, and radial probability distributions are central to describing the atom in terms of quantum mechanics. Other central ideas include the aufbau principle and the uncertainty principle. 

It is only possible to understand how two electrons can be bound to one proton by considering the electron wave functions. In quantum mechanics, the electrons cannot be modeled as pointlike particles orbiting the nucleus, but must be pictured as fuzzy distributions of probability. In H, the electrons are in close enough proximity that their probability distributions, or wave functions, overlap. This overlap induces a positive correlation that allows the bound state of the ion. This means that the electrons do not have simple individual independent wave functions, but share a different and more complicated wave function. 

In the usual quantum-mechanical implementation of the continuum solvation model, the electronic wave function and electronic probability density of the solute molecule M are allowed to change on going firom the gas phase to the solution phase, so as to achieve self-consistency between the M charge distribution and the solvent s reaction field. Any treatment in which such self-consistency is achieved is called a self-consistent reaction-field (SCRF) model. Many versions of SCRF models exist. These differ in how they choose the size and shape of the cavity that contains the solute molecule M and in how they calculate t nf... 

Many of the models used for studying molecules with large numbers of atoms are said to be classical because they rely on assumptions that allow the use of Newton s equations for the motion of the nuclei. Coming to this subject for the first time, one may wonder why we don t use quantum mechanics, since that is a more modern and comprehensive theory of the behavior of matter. A system of N atoms can be described in a probabilistic way using the Schrodinger equation. For illustration, consider a water molecule which has 10 electrons and 3 nuclei, each of which is identified with a point in three dimensions, thus the basic object describing the probability distribution for an isolated water molecule is a complexvalued function of 39 variables. Denote a point in this space by the coordinates... 

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