The Wave-Function Continuity

Each particle, either in free movement or in bound state under a potential that do not depend explicitly on time may be associated with a wave-function Y which contains, in principle all, information of the system moreover it is a continuously function of coordinate (including for classical turning points) 

In following, the cmcial quantum effect of tunneling may be explained only through employing the wave-function s shape and continuity constraints then, an eminent application at nuclear level unfolds an important certification of the quantum mechanics reliability. 

Having learned the basic types of wave-function either associated with a quantum particle evolving either within a classically allowed (E V) or 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

Due to the fact that on region I we have both incident and reflected wave (as in classical picture), in region II the classically forbidden behavior may include stationary waves back and forth inside the barrier, wile the region III being without wave source at infinity hosts only the transmitted (advanced) wave coming from tunneling region II. 

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Typically, one can assess the nature of the state (ground vs. excited) after convergence of the wave function. Continuing with our example, let us say that we have optimized the A state. We can then take that wave function, alter it so that the occupation number of the highest occupied a orbital is zero instead of one, and the occupation of the lowest formerly... 

Schrodinger equation, for which solutions can be obtained to the left of the barrier, inside the barrier where the kinetic energy is less than zero, and again to the right of the barrier where the kinetic energy is positive. By making the wave functions continuous at the interfaces, the transmission probability for any potential function can be solved, although not necessarily in a closed analytical form (Steinfeld et al., 1989 Nikitin, 1974). 

It is easy to realize that if there is another allowed region to the right of the barrier, the wave function continues there with the value of rp in the barrier. In the quantum mechanical description, a particle may thus penetrate a forbidden region. 

The success of this approximation, which allows the calculation of (s for molecules without knowledge of the wave functions, continues to be demonstrated as more accurate calculations of o (including electron correlation) become available. 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

In Ih e quail tiiin mechanical description of dipole moment, the charge is a continuous distribution that is a I linction of r. and the dipole moment man average over the wave function of the dipole moment operator, p ... 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

Clearly, the above procedure can be continued (in principle) as many times as required. Thus, if the wave function includes n = —4 3 paths, we have simply to dehne the function I 4((t)) = —+ 8ti), and then map onto the (j) = 0 16ti cover space, which will unwind the function completely. In general, if there are h homotopy classes of Feynman paths that contribute to the Kernel, then one can unwind ihG by computing the unsymmetrised wave function ih in the 0 2hn cover space. The symmetry group of the latter will be a direct product of the symmetry group in the single space and the group... 

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... 

The wave function for the particle is obtained by joining the three parts ipi, tpii, and fill such that the resulting wave function f(x) and its first derivative f x) are continuous. Thus, the following boundary conditions apply... 

The only solution for Equation (3.5) when V = 0 is ip = 0. So that if ip is to be singlevalued and continuous, it must be zero at the walls, that is, at x = 0 and x = I. Thus the potential energy walls impose what are called boundary conditions on the form of the wave function. Figure 3.3 shows (a) the particle-in-a-box potential, (b) a wave function, that satisfies the boundary conditions and, (c) one that does not. We see that only certain wave func-... 

The second term in region II represents particles moving to the left, and since this is zero by definition, D — 0. The wave function must be continuous at x — 0 and this requires that ipj(0) = tpri(0), and also that the first derivatives... 

The boundary conditions for continuity are that the wave functions and first derivatives should match at x = 0, a. These conditions determine the value of the constants relative to A. At x = 0 ... 

The first step beyond the statistical model was due to Hartree who derived a wave function for each electron in the average field of the nucleus and all other electrons. This field is continually updated by replacing the initial one-electron wave functions by improved functions as they become available. At each pass the wave functions are optimized by the variation method, until self-consistency is achieved. The angle-dependence of the resulting wave functions are assumed to be the same as for hydrogenic functions and only the radial function (u) needs to be calculated. 

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