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The Schrödinger wave equation,
can be written
which bears remarkable resemblance to the linear ordinary differential equation
with solution
In fact, using the theory of linear operators, it can be shown that the general solution to the Schrödinger wave equation is
where exponentiation of operators is defined using power series. Now remember that
where the kinetic energy,
, is given by
and the potential energy, often depends only on position, written . We can write
It is tempting to write
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
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 .
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,
This requires only a pointwise multiplication. The next step is to apply the middle operator,
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
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
This sequence can be summarized as
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.
The error in the pseudo-spectral method is overwhelmingly due to discretization error.
Needed: a more in-depth error analysis
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