Pseudo-spectral method
The pseudo-spectral method is a method in computational physics used for direct simulation of a particle with an arbitrary wavefunction interacting with an arbitrary potential.
Background
The Schrödinger wave equation,
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can be written
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which bears remarkable resemblance to the linear ordinary differential equation
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with solution
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In fact, using the theory of linear operators, it can be shown that the general solution to the Schrödinger wave equation is
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where exponentiation of operators is defined using power series. Now remember that
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where the kinetic energy,
, is given by
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and the potential energy,  often depends only on position, written  . We can write
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It is tempting to write
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so that we may treat each factor separately. However, this is only true if the operators and commute, which is not true in general. Luckily, it turns out that
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is a good approximation for small values of  . This is known as the symmetric decomposition. The heart of the pseudo-spectral method is using this approximation iteratively to calculate the wavefunction  for arbitrary values of .
The method
For simplicity, we will consider the one-dimensional case. The method is readily extended to multiple dimensions.
Given  , we wish to find  where  is small. The first step is to calculate an intermediate value  by applying the rightmost operator in the symmetric decomposition,
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This requires only a pointwise multiplication. The next step is to apply the middle operator,
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This is an infeasible calculation to make in configuration space. Fortunately, in momentum space, the calculation is greatly simplified. If  is the momentum space representation of , then
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which also requires only a pointwise multiplication. Numerically, is obtained from using the Fast Fourier transform (FFT) and  is obtained from  using the inverse FFT.
The final calculation is
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This sequence can be summarized as
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Analysis of algorithm
If the wavefunction is approximated by its value at  distinct points, each iteration requires 3 pointwise multiplications, one FFT, and one inverse FFT. The pointwise multiplications each require  effort, and the FFT and inverse FFT each require  effort. The total computational effort is therefore determined largely by the FFT steps, so it is imperative to use an efficient (and accurate) implementation of the FFT. Fortunately, many are freely available.
Error analysis
The error in the pseudo-spectral method is overwhelmingly due to discretization error.
Needed: a more in-depth error analysis
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