Bound states Boundary conditions

The fulfillmenf of Eq. (53) generates a complex energy if fhe inward matrix incorporates outgoing boundary conditions (with a real coordinate) or bound state boundary conditions (with a complex rotated coordinate). 

The additional problem present in the bound state case, at energies below the dissociation energy of the complex, is that of locating energies which are eigenvalues of the coupled equations, where a solution may be found that satisfies bound state boundary conditions. There are several procedures available for doing this, "" but the stablest are the log-derivative methods.In the many-channel case, the log-derivative matrix Y(R) is defined by ... 

The basis for the determination of an upper bound on the apparent Young s modulus is the principle of minimum potential energy which can be stated as Let the displacements be specified over the surface of the body except where the corresponding traction is 2ero. Let e, Tjy, be any compatible state of strain that satisfies the specified displacement boundary conditions, l.e., an admissible-strain tieldr Let U be the strain energy of the strain state TetcTby use of the stress-strain relations... 

This chapter has focused on reactive systems, in which the nuclear wave function satisfies scattering boundary conditions, applied at the asymptotic limits of reagent and product channels. It turns out that these boundary conditions are what make it possible to unwind the nuclear wave function from around the Cl, and that it is impossible to unwind a bound-state wave function. 

Of course, the distinction between reactive- and bound-state wave functions becomes blurred when one considers very long-lived reactive resonances, of the sort considered in Section IV.B, which contain Feynman paths that loop many times around the CL Such a resonance, which will have a very narrow energy width, will behave almost like a bound-state wave function when mapped onto the double space, since e will be almost equal to Fo - The effect of the GP boundary condition would be therefore simply to shift the energies and permitted nodal structures of the resonances, as in a bound-state function. For short-lived resonances, however, Te and To will differ, since they will describe the different decay dynamics produced by the even and odd n Feynman paths separating them will therefore reveal how this dynamics is changed by the GP. The same is true for resonances which are long lived, but which are trapped in a region of space that does not encircle the Cl, so that the decay dynamics involves just a few Feynman loops around the CL... 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

The bound-state energies and eigenfunctions can be obtained by solving the Schrodinger equation with boundary conditions that the radial wave function vanishes at both ends... 

If the kinetic balance condition (5) is fulfilled then the spectrum of the L6vy-Leblond (and Schrodinger) equation is bounded from below. Then, in each case there exists the lowest value of E referred to as the ground state. In effect, this equation may be solved using the variational principle without any restrictions. On the contrary, the spectrum of the Dirac equation is unbounded from below. It contains the negative ( positronic ) continuum. Therefore the variational principle applied unconditionally would lead to the so called variational collapse [2,3,7]. The variational collapse maybe avoided by properly selecting the trial functions so that they fulfil the boundary conditions specific for the bound-state solutions [1]. 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

To find bound state solutions, W < 0, for the H atom we apply the r = 0 and r = oo boundary conditions. Specifically we require that xp be finite as r — 0 and that xf>—> 0 as r— °°. We can see from Eqs. (2.12) that only the/functions are allowed due to the r — 0 boundary condition. As < we require that ip — 0, and, as indicated by the asymptotic form of the / function, this requirement is equivalent to requiring that sin nv be zero or that v be an integer. Combining the angular function of Eq. (2.7) with the / radial function yields the bound H wavefunction... 

For W > 0, the Rydberg electron is no longer in a bound state but in the continuum. In developing a wavefunction for the continuum we realize that the r — oo boundary condition has been removed only the r — 0 boundary condition remains. For H, the r=0 boundary condition excludes the g function so... 

The nuclear wavefunctions are continuum, i.e., scattering, wavefunctions which asymptotically behave like free waves rather than decaying to zero like the bound-state wavefunctions, scattering wavefunctions fulfil distinct boundary conditions in the limit R — oo. 

In addition to the direct methods, in which one calculates first the continuum wavefunctions and subsequently the overlap integrals with the bound-state wavefunction, there are also indirect methods, which encompass the separate computation of the continuum wavefunctions the artificial channel method (Shapiro 1972 Shapiro and Bersohn 1982 Balint-Kurti and Shapiro 1985) and the driven equations method (Band, Freed, and Kouri 1981 Heather and Light 1983a,b). Kulander and Light (1980) applied another method, in which the overlap of the bound-state wavefunction with the continuum wavefunction is directly propagated. The desired photodissociation amplitudes are finally obtained by applying the correct boundary conditions for R —> oo. 

The theory outlined above can be used to calculate the exact bound-state energies and wavefunctions for any triatomic molecule and for any value J of the total angular momentum quantum number. We can solve the set of coupled equations (11.7) subject to the boundary conditions Xjfi (R Jp) —> 0 in the limits R —> 0 and R — oo (Shapiro and Balint-Kurti 1979). Alternatively we may expand the radial wavefunctions in a suitable set of one-dimensional oscillator wavefunctions ipm(R),... 

The rationale behind this approach is the variational principle. This principle states that for an arbitrary, well-behaved function of the coordinates of the system (e.g., the coordinates of all electrons in case of the electronic Schrodinger equation) that is in accord with its boundary conditions (e.g., molecular dimension, time-independent state, etc.), the expectation value of its energy is an upper bound to the respective energy of the true (but possibly unkown) wavefunction. As such, the variational principle provides a simple and powerful criterion for evaluating the quality of trial wavefunctions the lower the energetic expectation value, the better the associated wavefunction. 

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