Multidimensional problems

The majority of interesting problems involve more than one coordinate and momentum. Immediately the Schrodinger equation becomes a partial differential equation and the solutions become more complicated. One of the simplest cases that illustrates a general method of solving the partial differential equation is the example of the particle in a three-dimensional box. 

Since the particles cannot exist in a region of infinite potential energy, we know that x]/ = 0 outside of and at the walls of the box. Since F = 0 in the interior of the box, we have the Schrodinger equation in the form 

We now assume that 1/ is a product of functions of the individual coordinates that is, 

We insert these expressions for the partial derivatives in the Schrodinger equation and divide through by ij/ this reduces the equation to the form 

Now suppose we keep x and y constant then the first terms in the equation, since they depend on x and on y respectively, remain constant. If we vary z, the third term would appear to vary since it depends on z. But in fact it cannot vary, since the addition of a varying third term to the two constant ones would make E vary and is a constant. Thus, we may write 

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The one-dimensional cases discussed above illustrate many of die qualitative features of quantum mechanics, and their relative simplicity makes them quite easy to study. Motion in more than one dimension and (especially) that of more than one particle is considerably more complicated, but many of the general features of these systems can be understood from simple considerations. Wliile one relatively connnon feature of multidimensional problems in quantum mechanics is degeneracy, it turns out that the ground state must be non-degenerate. To prove this, simply assume the opposite to be true, i.e. 

Now let us return to the Kolm variational theory that was introduced in section A3.11.2.8. Here we demonstrate how equation (A3.11.46) may be evaluated using basis set expansions and linear algebra. This discussion will be restricted to scattering in one dimension, but generalization to multidimensional problems is very similar. 

Note the meaning of this expression for each choice of the initial and final position a and a , calculate the classical path that takes you from x to x" m time t. Specifically, calculate tire momentum along the path and the final momentum, p", and find out how p" varies with the initial position. This would give, for a multidimensional problem, a matrix dp"-Jdx"- whose absolute detenninant needs to be inverted. 

If all the PES coordinates are split off in this way, the original multidimensional problem reduces to that of one-dimensional tunneling in the effective barrier (1.10) of a particle which is coupled to the heat bath. This problem is known as the dissipative tunneling problem, which has been intensively studied for the past 15 years, primarily in connection with tunneling phenomena in solid state physics [Caldeira and Leggett 1983]. Interaction with the heat bath leads to the friction force that acts on the particle moving in the one-dimensional potential (1.10), and, as a consequence, a> is replaced by the Kramers frequency [Kramers 1940] defined by... 

The identity of eqs. (2.6) (at T = 0) and (3.47) for the cubic parabola is also demonstrated in appendix A. Although at first glance the infinite determinants in (3.46) might look less attractive than the simple formulas (2.6) and (2.7), or the direct WKB solution by Schmid, it is the instanton approach that permits direct generalization to dissipative tunneling and to the multidimensional problem. 

Economical Difference Schemes for Multidimensional Problems in Mathematical Physics... 

In view of this, the economy requirement becomes rather urgent and extremely important in numerical solution of multidimensional problems arising time and again in mathematical physics. 

From here it seems clear that the admissible step in the explicit scheme is yet to be refined along with increasing the maximum value of the coefficient of heat conductivity. As a matter of fact, the last requirement is unreal for the problems with fastly and widely varying coefficients. Just for this reason explicit schemes are of little use not only for multidimensional problems, but also for one-dimensional ones (p = 1). On the other hand, the explicit schemes ofl er real advantages that the value y = on every new layer + t is found by the explicit formulas (3) with a finite... 

Reduction of a multidimensional problem to a chain of one-dimensioiial problems. The multiple equation we must solve is... 

Examples of reduction of multidimensional problems to chains of onedimensional ones. It is apparent from that discussion that some class of problems for which a solution of problem (6) or problem (11) coincides on the grid with the exact solution of the multidimensional problem (5) plays an important role. 

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