Wavefunction probability

Figure 4.11 Overlap of squared wavefunctions (probabilities of the presence of electrons) in the s orbitals of two atoms, A and B, separated by an internuclear distance d...

The wavefunctions, probability densities and shape of 2s atomic orbitals... 

The wavefunctions, probability densities [given by iA (x)], and energies for the first four energy levels for the particle in a one-dimensional box are plotted in Eigure 1.24. 

Before we compute the partition function, let s complete the expression for the wavefunction Probabilities must integrate to 1, Jg dx = 1. So... 

The analysis of this wavefunction in terms of contour maps of a weighted scattering wavefunction probability density defined by... 

Fig, 5. Contour maps of the scattering-wavefunction probability density at four values of s, in constant-s planes. The vertical axis is m, while the horizontal axis is n, the distance in the col-linear plane to the or Zp axis. The "clock in the upper right corner shows the intersection of the constant-s plane with the col-linear plane. The black dot on the n axis is the collinear reaction path. Cross-hatched regions are local density maxima. The contour... 

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

Typically, the ratio of this to the incident flux detennines the transition probability. This infonnation will be averaged over the energy range of the initial wavepacket, unless one wants to project out specific energies from the solution. This projection procedure is accomplished using the following expression for the energy resolved (tune-independent) wavefunction in tenns in tenns of its time-dependent counterpart ... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

Approximation Property We assume that the classical wavefunction 4> is an approximate 5-function, i.e., for all times t G [0, T] the probability density 4> t) = 4> q,t) is concentrated near a location q t) with width, i.e., position uncertainty, 6 t). Then, the quality of the TDSCF approximation can be characterized as follows ... 

This example relates to the well known Franck-Condon principal of spectroscopy in which squares of overlaps between the initial electronic state s vibrational wavefunction and the final electronic state s vibrational wavefunctions allow one to estimate the probabilities ofpopulating various final-state vibrational levels. 

When the measurement of Lz is made, the values 1 h. Oh, and -1 h will be observed with probabilities given by IC if, jCop, and jC. ip, respectively. For that sub-population found to have, for example, Lz equal to -1 h, the wavefunction then becomes... 

A basis set is the mathematical description of the orbitals within a system (which in turn combine to approximate the total electronic wavefunction) used to perform the theoretical calculation. Larger basis sets more accurately approximate the orbitals by imposing fewer restrictions on the locations of the electrons in space. In the true quantum mechanical picture, electrons have a finite probability of existing anywhere in space this limit corresponds to the infinite basis set expansion in the chart we looked at previously. 

In this equation, T is the wavefunction, in is the mass of the particle, h is Planck s constant, and V is the potential field in which the particle is moving. The product of P with its complex conjugate ( P P, often written as P) is interpreted as the probability distribution of the particle. 

This formulation is not just a mathematical trick to form an antisymmetric vravefunction. Quantum mechanics specifies that an electron s location is not deterministic but rather consists of a probability density in this sense, it can he anywhere. This determinant mixes all of the possible orbitals of all of the electrons in the molecular system to form the wavefunction. 

Even worse is the confusion regarding the wavefunction itself. The Born interpretation of quantum mechanics tells us that i/f (r)i/f(r) dr represents the probability of finding the particle with spatial coordinates r, described by the wavefunction V (r), in volume element dr. Probabilities are real numbers, and so the dimensions of i/f(r) must be of (length)" /. In the atomic system of units, we take the unit of wavefunction to be... 

Wavefunctions by themselves can be very beautiful objects, but they do not have any particular physical interpretation. Of more importance is the Bom interpretation of quantum mechanics, which relates the square of a wavefunction to the probability of finding a particle (in this case a particle of reduced mass /r vibrating about the centre of mass) in a certain differential region of space. This probability is given by the square of the wavefunction times dx and so we should concentrate on the square of the wavefunction rather than on the wavefunction itself. 

You are probably used to this idea from descriptive chemistry, where we build up the configurations for many-electron atoms in terms of atomic wavefunctions, and where we would write an electronic configuration for Ne as... 

Once an approximation to the wavefunction of a molecule has been found, it can be used to calculate the probable result of many physical measurements and hence to predict properties such as a molecular hexadecapole moment or the electric field gradient at a quadrupolar nucleus. For many workers in the field, this is the primary objective for performing quantum-mechanical calculations. But from... 

In order to calculate the total probability (which comes to 1), we have to integrate over both space dr and spin ds. In the case of the hydrogen molecule-ion, we would write LCAO wavefunctions... 

Because the square of any number is positive, we don t have to worry about i i having a negative sign in some regions of space (as a function such as sin x has) probability density is never negative. Wherever i , and hence i i2, is zero, the particle has zero probability density. A location where i]i passes through zero (not just reaching zero) is called a node of the wavefunction so we can say that a particle has zero probability density wherever the wavefunction has nodes. 

FIGURE 1.24 The Bom interpretation of the wavefunction. The probability density (the blue line) is given by the square of the wavefunction and depicted by the density of shading in the band beneath. Note that the probability density is zero at a node. A node is a point where the wavefunction (the orange line) passes through zero, not merely approaches zero. 

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