Energy eigenfunctions probability density

The term scar was introduced by Heller in his seminal paper (Heller, 1984), to describe the localization of quantum probability density of certain individual eigenfunctions of classical chaotic systems along unstable periodic orbits (PO), and he constructed a theory of scars based on wave packet propagation (Heller, 1991). Another important contribution to this theory is due to Bogomolny (Bogomolny, 1988), who derived an explicit expression for the smoothed probability density over small ranges of space and energy... 

These are the classical analogues of quantum scattering resonances except that these latter ones are associated with the wave eigenfunctions of the energy operator, although the eigenstates of the LiouviUian operator are probability densities or density matrices in quanmm mechanics. Nevertheless, the mathematical method to determine the Pollicott-Ruelle resonances is similar, and they can be obtained as poles of the resolvent of the LiouviUian operator... 

Figure 1.1 The one-dimensional potential given in Eq. (1) is depicted for two different choices of a (a) a = 0 and (b) a = 0.05. The following parameters are the same for both potentials V o = 1, /S = 1. The potential in (a) supports a single bound state at the energy Eo = 0.5[a.u.], and the corresponding probability density is drawn, where the energy of the state serves as a baseline. The potential in (b) supports no bound states but only continuum states. Still we draw the probability density of the continuum eigenfunction close in energy to Eq = 0.5[a.u.].

To extract information from the wavefunction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfunctions. An extensive discussion of these important elements of the formalism of quantum mechanics is precluded by space limitations. For further 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 information about the system. The third postulate relates to certain properties of these operators ... 

Figure 4-4 The volume-weighted probability density for the lowest-energy eigenfunction of the hydrogenlike ion. The most probable value of r occurs at rmp-...

Each eigenfunction ( ) thus is determined by a particular value of an integer quantum number (n), and the energies ( ) increase quadratically with this number. Figure 2.2 shows the first five eigenfunctions, their energies, and the corresponding probability density functions Outside the box, the wavefunctions must be... 

Fig. 2.2 Eigenfunctions (A) and probability densities (B) of a particle in a one-dimensional rectangular box of length / and infinitely high walls. The dashed lines indicate the energies of the first five eigenstates ( = 1,2,... 5), and the eigenfunctions and probability densities (solid curves) are displaced vertically to align them with the corresponding energies...

Figure 16.2 shows the probability density for the first four energy eigenfunctions of a particle in a box. Each graph is placed at a height proportional to the energy eigenvalue corresponding to that wave function. These probability densities are very different from the predictions of classical mechanics. If the state of a classical particle in a box is known, the probability at a given time would be nonzero at only one point, as in Figure 16.3a. Since the classical particle moves back and forth with constant...

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