Wavefunction probability density

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. 

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 ... 

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 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. 

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. 

The probability density for a particle at a location is proportional to the square of the wavefunction at that point the wavefunction is found by solving the Schrodinger equation for the particle. When the equation is solved subject to the appropriate boundary conditions, it is found that the particle can possess only certain discrete energies. 

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... 

SOLUTION The probability density is independent of angle when 1=0. We calculate the following ratio of the squares of the wavefunction at the two points... 

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

FIGURE 1.34 The radial wavefunctions of the first three s-orbitals of a hydrogen atom. Note that the number of radial nodes increases (as n 1), as does the average distance of the electron from the nucleus (compare with Fig. 1.32). Because the probability density is given by ip3, all s-orbitals correspond to a nonzero probability density at the nucleus. 

Born interpretation The interpretation of the square of the wavefunction, i j, of a particle as the probability density for finding the particle in a region of space. 

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. 

The conclusion is then that the wavefunction representing a system composed of indistinguishable particles must be either symmetric or antisymmetric under the permutation operation. On purely physical grounds, this result is apparent, as the probability density must be independent of the permutation of indistinguishable particles or 1 (1,2) 2 = (2,1) 2. 

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. 

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