Brief Overview
We have created a suite of open source computer programs that numerically calculate, time evolve, and visualize arbitrary initial quantum-mechanical bound states. The suite of programs is based on the ability to expand an arbitrary wave function in terms of basis vectors in a reduced Hilbert space (the superposition principle). The approach is extremely stable, fast, and accurate at depicting the long-term time dependence of complicated bound states.
In the reduced Hilbert space approach, more generally called the spectral method, one limits the basis set of the Hilbert space, which is typically an infinite dimensional space, to some finite dimension, N. As a practical matter in solving quantum-mechanical problems one rarely constructs a complete Hilbert space. Indeed this is often impractical or impossible. Because of finite computer resources, computationally one cannot create an infinite number of basis vectors to represent a complete Hilbert space, but for many problems it suffices to have a reduced Hilbert space. To see how this reduction will affect (or won't affect) the end result, we first construct two projection operators,
where the sum of the two projection operators is the complete projection operator, P. If we consider an arbitrary initial state, |Ψ>, we define
which describes the part of the original wave function that is in the reduced Hilbert space and the part that is not. The left-hand side of the Schrödinger equation, can be written as
since PN + PD = P= I and PN |ΨD> = PD |ΨN>= 0. We can now construct the complete Schrodinger equation for |ΨD> and |ΨN> by operating PN and PD above:
If in our computations (by virtue of the state, |Ψ>, selected), |ΨD> is vanishingly small, we are left with
which when integrated yields
As long as we can compute a precise subset of energy eigenvectors, the {ψn(x)}, [ H, PN ] = 0 and the above equation simplifies to
where
and hence,
The construction of a linear superposition in position space using the reduced
Hilbert space approach, therefore, proceeds like the usual case for a linear
superposition of energy eigenstates as long as we satisfy two conditions: (1) we
have a Hilbert space that is large enough that a finite subset of basis states
is sufficient to properly represent the initial state and (2) we have accurate
energy eigenfunctions, ψ(x), and hence the energy eigenvalues,
E, corresponding to a given potential energy function, V(x).
Energy eigenfunctions and eigenvalues can be calculated using a numerical
algorithm, but one can also use analytic expressions.
© Mario Belloni and Wolfgang Christian (2007).