Diffusion Monte Carlo
Diffusion Monte Carlo (DMC) is a quantum Monte Carlo method that uses a Green's function to solve the Schrödinger equation. DMC is potentially numerically exact, meaning that it can find the exact ground state energy within a given error for any quantum system. When actually attempting the calculation, one finds that for bosons, the algorithm scales as a polynomial with the system size, but for fermions, DMC scales exponentially with the system size. This makes exact large-scale DMC simulations for fermions impossible; however, DMC employing a clever approximation known as the fixed-node approximation can still yield very accurate results. What follows is an explanation of the basic algorithm, how it works, why fermions cause a problem, and how the fixed-node approximation resolves this problem.
To motivate the algorithm, let's look at the Schrödinger equation for a particle in some potential in one dimension:
We can condense the notation a bit by writing it in terms of an operator equation, with
So then we have
where we have to keep in mind that H is an operator, not a simple number or function. There are special functions, called eigenfunctions, for which , where E is a number. These functions are special because no matter where we evaluate the action of the H operator on the wave function, we always get the same number E. These functions are called stationary states, because the time derivative at any point x is always the same, so the amplitude of the wave function never changes in time. Since the overall phase of a wave function is not measurable, the system does not change in time.
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