A non-hermitean momentum operator for the particle in a box

Abstract

We construct a discrete non-hermitean momentum operator, which implements faithfully the non self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit, with the quarter wave as first non-trivial eigenstate. We show how to construct the corresponding hermitean Hamiltonian for the infinite well as concrete example to realize unitary dynamics. The resulting Hilbert space can be decomposed into a physical and unphysical subspace, which are mutually orthogonal. The physical subspace in the continuum limit reproduces that of the continuum theory and we give numerical evidence that the correct probability distributions for momentum and energy are recovered.

Figures (7)

• Figure 1: Sketch of the grid layout with genuine boundaries on which we formulate a momentum operator
• Figure 2: Example of two non-constant eigenvectors of $\mathds{P}$ in position space for $N_x=32$. Real-part (blue circles) and imaginary-part (mustard triangles) of the normalized eigenfunction to $l=N_x/2-1$ (top) and to $l=N_x/2-2$ (bottom). Note that on the boundary either the value, or the derivative of the functions is zero.
• Figure 3: Disappearance of boundary leakage under grid refinement by the three lowest lying non-constant eigenfunctions of the operator $\mathds{P}$.
• Figure 4: Examples of the two lowest lying physical (colored symbols) and unphysical (gray cross) energy eigenfunctions of the Hamilton operator $\mathds{H}$ built from our non-hermitean momentum operator $\mathds{P}$ on a $N_x=32$ grid. The continuum ground state (red) and first excited (blue) stationary state wavefunctions are provided as solid lines.
• Figure 5: Deviation $\Delta E^n$ of the lowest four $n=1,\ldots,4$ energy eigenvalues of $\mathds{H}$ from the continuum values of the infinite square well for different lattice spacings $\Delta x$.