倒易对称与统一的经典量子振子及势阱中粒子与谐振子的相容性
Reciprocal Symmetry and Unified Classico-Quantum Oscillator And Consistency between a Particle in a Potential Well and a Harmonic Oscillator
摘要：
函数exp（iwt）描述振荡运动。振荡器的能量与w的平方成正比。exp（IWT）是微分方程的解。我们用相应的有限时间差分方程代替了这个微分方程，其中d是最小的时间跨度。然后，我们对方程进行了对称化，使得它在d变为-d时保持不变。这个对称方程有成对的解。角速度w被修改为w '或w "。w '包含带整数的部分。w "包含半整数部分。这对应于量子力学振荡器能级。F = a。exp（iwt）描述-a和+a之间的振荡。如果我们使w = 0，F描述了-a和+a之间的自由振荡。因此，互易对称振子一方面统一了量子和经典谐振子，另一方面统一了谐振子和势阱中的自由粒子。不涉及量子力学假设。
英文摘要：
The function exp(iwt) describes an oscillating motion. Energy of the oscillator is proportional to the square of w. exp(iwt) is the solution of a differential equation. We have replaced this differential equation by the corresponding finite-time difference equation with d as the smallest span of time. We have, then, symmetrized the equation so that it remains invariant under the change d going to -d. This symmetric equation has solutions in pairs. The angular speed w is modified to w' or w". w' contains a part with an integer. w" contains a part with a half-integer. This corresponds to quantum mechanical oscillator energy levels. F= a.exp(iwt) describes oscillation between -a and +a. If we make w=0, F describes free oscillation between -a and +a. Reciprocal symmetric oscillator, thus, unifies quantum and classical harmonic oscillators on one hand, and a harmonic oscillator and a free particle in a potential well on the other hand. No quantum mechanical postulates are involved.
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