Wave function complex phase

We consider two metallic free-electron systems, with atomically flat surfaces separated by vacuum over a distance Ax (Figure 20). In fact, the model system is an extension of the metal surface considered in Section 4.5. The complex potential energy barrier at a metal surface, discussed in Section 4.5 is simplified here to a rectangular barrier. We look for the quantum-mechanical probability that an electron in phase A is also present in phase B. This probability is given by the ratio of squared amplitudes, and A, of the free-electron wave function in phase B and A, respectively. It is quantified by the transmission coefficient ... 

The complex phase which is fundamental to gauge theory is commonly defined in terms of symmetry groups without consideration of its physical meaning, which emerges most clearly in its characterization of the quantum wave functions. Whereas phase relationships between point particles are hard to imagine, they appear naturally in wave structures. With respect to electrons and other chemical entities a wave model in terms of complex wave functions is therefore the most satisfactory physical model. The complex phase represents the fundamental attribute of non-classical systems and the major difference between classical particles and quantum waves. Simulation of chemical systems based on real basis sets is essentially classical. It is therefore wrong, although fashionable, to refer to such simulations as quantum chemistry. 

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]. 

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. 

Making use of the polar representation of a complex number, the nuclear wave function can be written as a product of a real amplitude, A, and a real phase, S,... 

Equality between the 1, 2 wave function and the modulus of the 2, 1 wave function, v /(j2, i), shows that they have the same curve shape in space after exchange as they did before, which is necessary if their probable locations are to be the same. The phase factor orients one wave function relative to the other in the complex plane, but Eq. (9-17) is simplified by one more condition that is always true for particle exchange. When exchange is canied out twice on the same particle pair, the operation must produce the original configuration of particles... 

In the vector potential approach [6], the (real) electronic wave function (4>) is multiplied by a complex phase factor/(4>), defined such that... 

By considering only elastic scattering events, the interaction of the specimen with the electron beam can be described through a complex transmission function (object wave-function) 0(f) which represents the ratio between the outgoing and the incoming electron wave-functions f = (x, y) is a two-dimensional vector lying on a plane perpendicular to the optic axis z which is parallel, and in the same direction, to the electron beam. In the standard phase object approximation ... 

The product of a function and its complex conjugate is always real and is positive everywhere. Accordingly, the wave function itself may be a real or a complex function. At any point x or at any time t, the wave function may be positive or negative. In order that F(x, t)p represents a unique probability density for every point in space and at all times, the wave function must be continuous, single-valued, and finite. Since F(x, /) satisfies a differential equation that is second-order in x, its first derivative is also continuous. The wave function may be multiplied by a phase factor e , where a is real, without changing its physical significance since... 

A theory for nonequilibrium quantum statistical mechanics can be developed using a time-dependent, Hermitian, Hamiltonian operator Hit). In the quantum case it is the wave functions [/ that are the microstates analogous to a point in phase space. The complex conjugate / plays the role of the conjugate point in phase space, since, according to Schrodinger, it has equal and opposite time derivative to v /. 

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