I’m doing some work in semiconductors and I routinely read about “the Schrodinger wave equation and Poisson equation need to be solved self-consistently”.
I’ve always just glossed over that phrase but I recently realized that I don’t really understand what Self-consistent means. Does it mean that PE and SWE need to be solved simultaneously? Does it mean each one must converge? Does it mean that one is dependent on the other and you must iterate to solve both?
Silly question but I really don’t think I understand this.
Integrating the Poisson equation gives the electric potential for a given charge distribution.
Integrating the Schrodinger equation gives the electron wave function (and thus the charge distribution) for a given potential.
So the Poisson equation and Schrodinger equation are coupled, and you have to numerically solve the system of equations by some sort of iterative method. In other words, what you said:
Elaborating a bit:
Generally “self-consistent solution” means some kind of iterative fixed point algorithm. That is, you start with equations like:
x = f(x,y)
y = g(x,y)
Then you guess some value (x,y), plug it into the equations to get some new value for (x,y), and keep repeating the procedure until you find an (x,y) which when input into the equations just yields itself. In other words, you’ve found a fixed point, or a self-consistent solution.
Of course, whether that algorithm succeeds depends on your initial guess, and there are some other ways of numerically solving such a system of equations that in a given case may be more efficient or less sensitive to the initial guess. But that’s the general idea.
Actually for the sort of problem you’re talking about x and y will be functions rather than numbers – i.e., the electric potential and charge density are functions of position. Thus f and g are functionals, rather than functions. But I think the method of solution is essentially the same. For instance, you can break the domain of your functions into discrete pieces, and assume the function is roughly constant on each of those pieces. Then the function is defined by a list of numbers, namely the average value of the function on each of those pieces.
Of course, this kind of approach will only yield accurate results if each piece of the domain is sufficiently small – but on the other hand smaller pieces means more total pieces, and thus more numerical effort. I imagine one would start with a fairly coarse-grained solution to generate your initial guesses, and then progressively move to finer grains, each time using the result of the previous calculation as your new initial guess. This is somewhat outside my area of experience, however, so there may be much more efficient methods that I don’t know about.
I just have to add, only on the SDMB would there be a “silly question” about self-consistent solutions to the Poisson equation and the Schrodinger wave equation.
