Modern 2:Evolution of Wavepackets
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Evolution of Wavepackets
The Schroedinger Equation is first order in time:
This means that the wavefunction at any time is completely determined by its value at an earlier time.
Consider a generic first order (in time) differential equation written in operator form:
Where H is a differential operator (such as ) that does not depend on time. A solution of this equation is given by:
The operator e − Ht can be thought of as an evolution operator since it evolves the wavefunction forward in time. NB, there is nothing special about t = 0. We can give the initial conditions at any time then by shifting the time axis, this becomes t = 0. This would be an extremely handy tool to be able to apply to the Schroedinger equation but we have to figure out how to deal with the exponential of a differential operator.
The simplest approach is to Fourier transform the spatial part of the Schroedinger equation:
In case you're shakey on this argument...
Suppose we have a first order equation:
We know that
Hence differentiating this with respect to t and x and using Equation 1 we have:
Thus
The only way this can be true in general is for the quantity in the square brackets to be zero. This leaves, in effect, the Fourier Transform of Equation 1:
So, in practice we can replace space derivatives with multiplaction by i k in the wavenumber (momentum) domain. For vectors this generalizes in a completely natural way. Gradients become multiplication by , and the Laplacian .
Now back to Schroedinger... In the wavenumber domain we have:
which gives the nice evolution equation:
Hence the evolution operator can be written via:
The exponent of this expression can be written as well in terms of momentum or energy (for a free particle) .
A slight digression...
Another approach to computing the exponential of an operator (e.g. a matrix) is to make a Taylor series expansion of the exponential. We have
ex = 1 + x + x2 + ...
So it's at least plausible that we could define the exponential of an operator as
eA = 1 + A + A2 + ...
where an expression such as A2 really means to apply the operator A twice to some object in its domain of definition; something like A(A(f)).