近年來控制驅動場技術的進展，提升了量子領域中系統控制的可能性，比如說分子動力學或是化學反應的控制等等。其中Rabitz 和Schirmer 提出的量子最佳化控制理論，乃針對正定可觀測物理量算符的期望值之最佳化控制，提出一種稱之為糾纏回授的演算法，並以此方法設計其外在控制場，來達成所要控制的目標。
另一方面，由於多體量子系統的控制分析往往是複雜的，一般而言，通常以簡化的型式來描述量子系統及其控制的過程，在本篇論文中，我們試著使用投影算符之方法來研究多維度的量子系統之控制行為，利用P和Q兩個投影算符，將量子系統的狀態分為主導量子狀態之行為與誤差修正項兩個部份，並結合上述Schirmer 的最佳化控制和演算法，降低誤差修正項對系統的影響，最後針對一個氟化氫分子，探討某一特定能階躍遷轉換為最大化的研究。

Recent advances in driven system have opened up new possibilities for control of phenomena in the quantum regime, such as molecular dynamics or chemical reaction dynamics. Rabitz and Schirmer presented an entangled feedback algorithm for quantum optimal control of the physical expectation value of a positive-definite observable operator.They designed to find the control field by this method and show that
achieving the control objective.
On the other hand, the analysis of control of the many-body quantum system is complex. For the reason it is described to have a reduced formulation of the control process generally. In this thesis we investigate the processes of multi-dimension of quantum control system by using projection operator method. Then we expect that the projection operator P onto the subspace which composed of strongly coupled states dominates the system dynamics, and let Q onto the remaining states which regard as the correction from the influence of weakly coupled states. We hope to decay or eliminate the correction by the above
algorithm method. Finally, the way is then applied to the problem of maximizing the population of objected level for a HF (hydrogen fluoride) molecule with Morse oscillator potential model.

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