The wave function

Since an electron has wave properties, it is described as a wave function, or 4 C JX ), the latter meaning that t/ is a function of coordinates x,y, and The wave function can take on positive, negative, or imaginary values. The probability of finding an electron in any volume element in space is proportional to the square of the absolute value of the wave function, integrated over that volume of space. This is the physical significance of the wave function. Measurements we make of electronic charge density, then, should be related to not Expressed as an equation, we have 

By way of further explanation, it should be noted that the probability of finding an electron in any volume element must be real and positive, and always satisfies this requirement. 

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The wave function T i oo ( = 11 / = 0, w = 0) corresponds to a spherical electronic distribution around the nucleus and is an example of an s orbital. Solutions of other wave functions may be described in terms of p and d orbitals, atomic radii Half the closest distance of approach of atoms in the structure of the elements. This is easily defined for regular structures, e.g. close-packed metals, but is less easy to define in elements with irregular structures, e.g. As. The values may differ between allo-tropes (e.g. C-C 1 -54 A in diamond and 1 -42 A in planes of graphite). Atomic radii are very different from ionic and covalent radii. 

Only certain energy values ( ) will lead to solutions of this equation. The corresponding values of the wave functions are called Eigen functions or characteristic wave functions. 

The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

While the presence of sign changes in the adiabatic eigenstates at a conical intersection was well known in the early Jahn-Teller literature, much of the discussion centered on solutions of the coupled equations arising from non-adiabatic coupling between the two or mom nuclear components of the wave function in a spectroscopic context. Mead and Truhlar [10] were the first to... 

We have used the above analysis scheme for all single- and two-surface calculations. Thus, when the wave function is represented in polar coordinates, we have mapped the wave function, it a i(, 0 r, t) in each... 

In hyperspherical coordinates, the wave function changes sign when <]) is increased by 2k. Thus, the cotTect phase beatment of the (]) coordinate can be obtained using a special technique [44 8] when the kinetic energy operators are evaluated The wave function/((])) is multiplied with exp(—i(j)/2), and after the forward EFT [69] the coefficients are multiplied with slightly different frequencies. Finally, after the backward FFT, the wave function is multiplied with exp(r[Pg.60]

The projection on the final channel is done in the following manner. We let the trajectory decide on the channel—just as in an ordinary classical trajectory program. Once the channel is detemrined we project the wave function (in the DVR representation) on the appropriate wave function for that channel... 

A somewhat different viewpoint motivates this chapter, which stiesses the added meaning that the complex nature of the wave function lends to our understanding. Though it is only recently that this aspect has come to the forefront, the essential point was affimied already in 1972 by Wigner [5] in his famous essay on the role of mathematics in physics. We quote from this here at some length ... 

What is addressed by these sources is the ontology of quantal description. Wave functions (and other related quantities, like Green functions or density matrices), far from being mere compendia or short-hand listings of observational data, obtained in the domain of real numbers, possess an actuality of tbeir own. From a knowledge of the wave functions for real values of the variables and by relying on their analytical behavior for complex values, new properties come to the open, in a way that one can perhaps view, echoing the quotations above, as miraculous. ... 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

Section IB presents results that the analytic properties of the wave function as a function of time t imply and summarizes previous publications of the authors and of their collaborators [29-38]. While the earlier quote from Wigner has prepared us to expect some general insight from the analytic behavior of the wave function, the equations in this secbon yield the specific result that, due to the analytic properties of the logarithm of wave function amplitudes, certain forms of phase changes lead immediately to the logical necessity of enlarging... 

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

In addition, it can occasionally be useful to regard some physical parameter appearing in the theoi y as a complex quantity and the wave function to possess analytic properties with regard to them. This formal procedure might even include fundamental constants like e, h, and so on. 

Note [240] that the phase in Eq. (13) is gauge independent. Based on the above mentioned heuristic conjecture (but fully justified, to our mind, in the light of our rigorous results), Resta noted that Within a finite system two alternative descriptions [in temis of the squared modulus of the wave function, or in temis of its phase] are equivalent [247]. 

A time-varying wave function is also obtained with a time-independent Hamiltonian by placing the system initially into a superposition of energy eigenstates ( n)), or forming a wavepacket. Frequently, a coordinate representation is used for the wave function, which then may be written as... 

Next, for the log term (which normalizes the wave function), we have to choose, as in Eq. (15), suitable functions P t) that will correct the behavior of that term along the large semicircles. Among the multiplicity of choices, the following are the most rewarding (since they completely cancel the log term) ... 

This establishes the functional form of the phase for real (physical) times. The phase of the solution given in [261,262] indeed has this functional form. The fractions/] and/2 cannot be determined from our Eqs. (17) and (18). However, by compaiing with the wave functions in [261,262], we get the following values for them ... 

The connection holds separately for the coefficient of each state component in the wave function and is not a property of the total wave function (as is, e.g., the dynamical phase [9]). 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

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