Wavefunctions nodes

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 1.34 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. 

FIGURE 3.26 When two 1s-orbitals overlap in the same region of space in such a way that their wavefunctions have opposite signs, the wavefunctions (red and orange lines) interfere destructively and give rise to a region of diminished amplitude and a node between the two nuclei (blue line). 

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. 

The wavefunction of an electron associated with an atomic nucleus. The orbital is typically depicted as a three-dimensional electron density cloud. If an electron s azimuthal quantum number (/) is zero, then the atomic orbital is called an s orbital and the electron density graph is spherically symmetric. If I is one, there are three spatially distinct orbitals, all referred to as p orbitals, having a dumb-bell shape with a node in the center where the probability of finding the electron is extremely small. (Note For relativistic considerations, the probability of an electron residing at the node cannot be zero.) Electrons having a quantum number I equal to two are associated with d orbitals. 

According to the Wigner theorem (see Appendix A), because of time-reversal. symmetry, the functions i ground-state wavefunction does not have a node, we can always make the... 

Figure 2.6 Nodal properties of standing waves. A one-dimensional oscillation (wave) constrained within a space of length L can have amplitudes (wavefunctions) of discrete wavelengths only. The open circles are the nodes where the amplitude is always zero...

These concepts of the classical physics of standing waves have important implications in photophysics, in particular for the understanding of orbital symmetries and laser light emission. In the case of a standing wave the propagation velocity does not exist, and the important relationship defines the wavelengths as a function of the distance between the boundaries and the number of nodes in the wavefunction... 

In eqn. 2.7 the number n is a quantum number5. It is in fact related to the number of nodes in the wavefunction and must in this case be a positive integer (n = 1, 2, etc.). This would apply to a wave which follows one direction only. Since real space is three-dimensional a standing wave must be defined by three quantum numbers. The motions of electrons around nuclei are essentially circular, so that the use of polar coordinates is preferable and the three quantum numbers are ... 

While it is obvious that the principal quantum number n must be a positive integer (it is impossible to have less than zero nodes in a wavefunction), the values of the other quantum numbers can be negative when they are defined in polar coordinates. The allowed values of the various quantum numbers are... 

The wavefunctions of delocalized 7T orbitals of linear conjugated molecules can be drawn following the number of nodes (open circles)... 

A node is a point where the wavefunction passes through zero. 

The classic way of determining the energies of hydrogenic levels in a field is to solve the zero field problem in parabolic coordinates and calculate the effect of the field using perturbation theory. The zero field parabolic wavefunctions obtained by solving Eqs. (6.8a) and (6.8b) have, in addition to the quantum numbers n and m, the parabolic quantum numbers n, and n2, which are nonnegative integers.1 and n2 are the numbers of nodes in the iq and u2 wavefunctions and are related to n and m by... 

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