Energy-Classical Frequency in Quantum Oscillator Part II

 In Part I we noted that the quantum oscillator result E= hbar w (n+.5) implies that for exp(-iEt) there are special times proportional to 1/E which in turn are proportional to 1/w which is proportional to the classical cycling time of the oscillator. In other words for oscillators with different w there are different 1/w times such that -Et (which is part of a free particle action A= -Et+px) is constant i.e. one cannot distinguish as in statistical mechanical equilibrium.

    In this note, we consider the free particle action in more detail. From A=-Et+px there are special times (1/E) and lengths 1/p proportional to wavelength such that different E and p values when multiplied by these times and length have the same value.  Furthermore A=0 for this values. We argue that it is this feature of px in exp(ipx) which is responsible for the energy of an oscillator being proportional to w, the classical angular frequency. Thus the idea of a special length (wavelength) such that different p’s when multiplied by 1/p produce the same value leads to a quantum energy such that 1/E is proportional to a physical cycling time in the classical problem. Thus different w’s lead to E/E being constant just as p/p is constant.