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# Quantum Numerical Control for Particles at Matter Surface

In this work, time-depended Schrodinger equation described particles at matter (crystal,
catalysis, metal) surface could be considered as propose of numerical control of quantum system. Accessing existing physical experimental results on the motion of particles (molecules,
atoms) at surface, based on variational method of quantum control theory in Hilbert space,
using density function theory (DFT), time-depended Schrodinger equation to proceed the
investigation of computational approach. To do quantum calculation at surface, physically,
first needs a concept as control goal: such as breaking a chemical bond as target; reducing
energy of high intensity shaped laser pulse. Particles at surface is a kind of constrain control
for spatial variable. Optimal control is to find and characterize the quantum optima for
minimizing or maximizing the cost functional. Control methods contain selecting chemical
reagent, designing chemical reaction, making control scope for a quantized system: time
varying Schrodinger equation.
Precisely, for general quadratic cost function, in two or three dimensional cases, a semi discrete (time continuous, spatial discrete) algorithm consisting of finite element method
and conjugate gradient method, would be utilized for solving a numerical solution of state
system, and obtaining quantum optimal control from a initial guess of control input. It is quite curious: what is the difference of control particles occurred at surface than control
free particles? whether one can develop a suit of theory or methodology for quantum surface
control? It is certainly expected to connect theoretical control, to numerical or computational
control, and to experimental control as carrying out quantum system control of particles on
the surface. It is desired that quantum control theory (QCT) for quantum dot at surface
would be evidenced in visualization method, and attained confidential verification in the
guidance of real-time computer-aided experiments in the viewpoint of chemistry and physics.