量子能量的物理來源，和機率流體中的位能場以及渦流場分佈的詳細結構的平衡狀況，還有雙原子所組成的分子( Diatomic molecule )在固定狀況下的量子位能變化，這些與物理當中最基本的量子能量之間的平衡與相互關係，種種重要課題在過去幾世紀以來，一直未能被人們所解決。這些特別的問題包含了：量子能量是否從量子位能而來？如果是的話，那其物理來源又是什麼？以及其在零點最基態的量子能量?而它是如何參與包括位能場、渦流場與量子位能之間彼此所達成的平衡關係。最近，邱輝煌老師成功地證明出在各種範圍內量子數中的原子結構關係，其與外圍所環繞的電子所達成的平衡關係；並且發展出一套量子模式平衡關係( Quantum Modal Balance )的方法，找出原子當中量子能量與其物理結構的基本來源。這套量子模式平衡關係的分析可有效提供在固定狀態中，似氫原子或者單電子與單位電荷中，彼此相互之間能量平衡的關係式。這套理論亦可用來解決多電子的原子，分子，和人工合成的原子或分子關係問題。
本研究所使用的理論方法便是運用邱輝煌老師所證明出的量子模式平衡關係的理論，來計算以及探討在雙原子所組成的分子中，各種能量的平衡機制關係。我們發現在位能場（即振動模式）的平衡裡，可以證明出跟量子能量中部分的動能以及在機率流體中，由於擴散所散生出的dilatation能量是相互達到平衡的。而在雙原子的運動中，包含了三種運動自由度：線性移動、兩原子之間相互的旋轉運動及相互振動。在此研究中，我們只考慮兩原子之間的相互振動問題，而對於另兩種相對運動則暫不考慮。
從此研究的探討我們可以發現,在所有的量子系統中,量子能量最基本的物理來源便是由於機率流體因擴散效應所產生的動能.並且藉著這兩種能量之間的儲存,釋放以及相互變換,所以才會造成各種量子現象的發生.
而機率流體中的量子能量是從量子位能中的動能( kinetic energy ) 而來,此能量的產生係為部分的徑向方向的流體擴散速率,所造成的能量變化.而渦流場的動能是由於 方向角的漲縮能( Dilatation energy )供給而達到動力平衡.而位能 與其部分徑向方向的擴散動能及漲縮能,和部分 方向角的帳縮能達到平衡.
最後,這種量子模式平衡理論 ( Quantum Modal Balance Theory ), 能夠有效擴大到各層次的問題範圍, 包含了帶電的原子, 以及雙原子到多原子所組成的分子, 更能計算出各種量子能量之間的平衡關係, 包含量子能量的來源, 以及各種不同量子狀態下的能量變化.

The physical origin of the quantized energy and the mechanism of detailed balance among the Potential energy field, vortex flow of the probability fluid, and the quantum potential of a diatomic molecule in a stationary state have been the outstanding unsolved problems of basic interest over the past century. Specific questions are: Does quantized energy originate in part of the quantum potential, and if so, what is the source of the quantized energy, what is origin and mechanism of the zero-point energy i.e. , and sensible quantized energy, and how does it take part in the global dynamic balance, involving Potential energy, vortex and quantum potential? Recently, professor H.H Chiu succeeded in identification of the mechanisms of the balance of the electron in atomic structure at all the range of quantum numbers, in hydrogen-like atom, and the physical source of the quantized energy and the structure of an atom by the method of the quantum modal balance. The theory can be generalized to deal with many-electron atoms, diatomic molecules, and artificial atoms and diatomic molecule.
This study also use the Quantum Modal Balance to calculate and make a discussion for the modal energy balance of a diatomic molecule. It was found that the zero-point energy and sensible quantized energy of the potential energy field (vibrational mode) balances with a part of the dilatation, , and a part of the kinetic energy, , both created by the diffusion of the fluid in radial and polar direction. The kinetic energy of the vortex-induced flow balances with a part of the azimuthal quantum dilatation energy. The central potential of the two atoms have three degree of freedom motions, including the linear translation, rotation and vibration. Here, we only concerned with the vibration of the two attracting atoms, and we do not consider the linear translation and rotation any further. The areas of the future research on quantum energy as a new type of energy of atoms and diatomic molecules are also discussed.

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