Wavefunctions probability densities and

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. 

If we define the wavefunction in terms of the probability density, and it s the probability density that determines what we measure in the laboratory, then why do we need the wavefunction at all The wavefunction is a mathematical convenience that allows us not only to predict the properties of individual quantum states, but also to predict results that stem from combinations of quantum states. It is the wavefunction, not the probability density, that solves the Schrodinger equation, and—like a wave—the wavefunction can undergo constructive and destructive interference with other wavefunctions to accurately predict the resulting probability densities. 

Two wavefunctions that differ only in sign or phase have the same probability density, and so they are not distinct states. Thus, the allowed values of n are 1,2,3, and so on. Finally, the normalization condition determines the constant a, which turns out to be independent of n ... 

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 ... 

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. 

FIGURE 1.27 The two lowest energy wavefunctions (i <, orangei for a particle in a box and the corresponding probability densities (i] 2, blue). The probability densities are also shown by the density of shading of the bands beneath each wavefunction. 

All s-orbitals are independent of the angles 0 and c[>, so we say that they are spherically symmetrical (Fig. 1.31). The probability density of an electron at the point (r,0,ct>) when it is in a ls-orbital is found from the wavefunction for the ground state of the hydrogen atom. When we square the wavefunction (which was given earlier, but can also be constructed as RY from the entries for R and V in Tables 1.2a and 1.2b) we find that... 

A note on good practice Be careful to distinguish the radial distribution function from the wavefunction and its square, the probability density ... 

The radial parts of the wavefunctions for the hydrogen atom can be constructed from the general form of the associated Laguerre polynomials, as developed in Section 5.5.3. However, in applications in physics and chemistry it is often the probability density that is more important (see Section 5.4.1). This quantity in this case represents the probability of finding the electron in the appropriate three-dimensional volume element. 

For over a decade, the topological analysis of the ELF has been extensively used for the analysis of chemical bonding and chemical reactivity. Indeed, the Lewis pair concept can be interpreted using the Pauli Exclusion Principle which introduces an effective repulsion between same spin electrons in the wavefunction. Consequently, bonds and lone pairs correspond to area of space where the electron density generated by valence electrons is associated to a weak Pauli repulsion. Such a property was noticed by Becke and Edgecombe [28] who proposed an expression of ELF based on the laplacian of conditional probability of finding one electron of spin a at t2, knowing that another reference same spin electron is present at ri. Such a function... 

The term 11 (0) 2 is the square of the absolute value of the wavefunction for the unpaired electron, evaluated at the nucleus (r = 0). Now it should be recalled that only s orbitals have a finite probability density at the nucleus whereas, p, d, or higher orbitals have nodes at the nucleus. This hyperfine term is isotropic because the s wavefunctions are spherically symmetric, and the interaction is evaluated at a point in space. 

For the analysis of the wavefunctions we computed the expectation values of a number of operators, using formulas in the works already cited. The quantities (5(ri)) and ( (ri2)) give probability densities for pairwise particle coincidences (5(ri)5(ri2)) gives the same data for the triple coincidence. The quantity Vi is... 

In addition to operators corresponding to each physically measurable quantity, quantum mechanics describes the state of the system in terms of a wavefunction F that is a function of the coordinates qj and of tune t. The function l F/(qj,t)l2 = P P gives the probability density for observing the coordinates at the values qj at time t. For a many-particle system such as the H2O molecule, the wavefunction depends on many coordinates. For the H2O example, it depends on the x, y, and z (or r,0, and < )) coordinates of the ten... 

For example, if motion is constrained to take place within a rectangular region defined by 0 < x < L 0 < y < Ly, then the continuity property that all wavefunctions must obey (because of their interpretation as probability densities, which must be continuous) causes A(x) to vanish at 0 and at Lx. Likewise, B(y) must vanish at 0 and at Ly. To implement these constraints for A(x), one must linearly combine the above two solutions exp(ix(2mEx/h2)1 /2) and exp(-ix(2mEx/h2)l/2) to achieve a function that vanishes at x=0 ... 

In addition to initial conditions, solutions to the Schrodinger equation must obey certain other constraints in form. They must be continuous functions of all of their spatial coordinates and must be single valued these properties allow VP P to be interpreted as a probability density (i.e., the probability of finding a particle at some position can not be multivalued nor can it be jerky or discontinuous). The derivative of the wavefunction must also be continuous except at points where the potential function undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This condition relates to the fact that the momentum must be continuous except at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

Wavefunctions of electrons in atoms are called atomic orbitals. The name was chosen to suggest something less definite than an orbit of an electron around a nucleus and to take into account the wave nature of the electron. The mathematical expressions for atomic orbitals—which are obtained as solutions of the Schrodinger equation—are more complicated than the sine functions for the particle in a box, but their essential features are quite simple. Moreover, we must never lose sight of their interpretation, that the square of a wavefunction tells us the probability density of an electron at each point. To visualize this probability density, we can think of a cloud centered on the nucleus. The density of the cloud at each point represents the probability of finding an electron there. Denser regions of the cloud therefore represent locations where the electron is more likely to be found. 

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