- 07-176 Ulrich Mutze
- The direct midpoint method as a quantum mechanical integrator II
Jul 7, 07
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Abstract. A reversible integrator for the time-dependent
Schroedinger equation associated with an arbitrary (potentially time-dependent) Hamilton operator is defined.
This algorithm assumes the dynamical state of the system to be described
by a conventional quantum state vector and a velocity vector of the same
data structure and storage size.
The algorithm updates these two vectors by five additions of vectors,
three multiplications of vectors with real numbers,
and four actions of the Hamilton operator on a vector.
If the Hamilton operator is time-independent, additions of vectors reduce to three, and the actions of the Hamilton operator reduce to one action of its square.
In the first of a series of steps, the velocity has to be initialized by one action of the Hamilton operator on the initial state vector.
Further properties of this algorithm are derived only for finite dimensional state spaces and time-independent Hamilton operators.
Under these assumptions it is shown that the time step
evolution operator is symplectic
so that exact energy conservation holds.
Further, an explicit expression for the n-th power of the
time step evolution operator is derived.
This makes the system behavior completely transparent: There is a limiting time step, namely 2 divided by the norm of the Hamilton operator, so that for smaller time steps all trajectories remain bounded for all times, whereas for larger time steps there are always exponentially growing trajectories.
For time steps smaller than the limit there is approximate conservation of norm along each trajectory and the deviation from exact
conservation is controlled by explicit expressions proportional
to the square of the time step.