Energy-Classical Frequency in Quantum Oscillator

  In a previous note (1) we examined the idea of a classical frequency such as that of an oscillator appearing in quantum mechanical results i.e. solutions for the quantum oscillator. In this note, we examine the problem differently. We begin with the quantum oscillator time-independent Schrodinger equation and note that  the LHS becomes an equation in the variable y only where  yy = sqrt(km) xx with E → E/w. This is already known in the literature, but it suggests  E/w must be independent of k and m. For example, if the LHS only has the variable y with no k or m present and y is integrated from -infinite to infinite, E/w should also have no k, m dependence. At this point, however, w is not really a frequency in the quantum sense. It is a frequency in the classical analysis so  one should compare the quantum and classical cases. One may consider the classical conservation of energy equation and integrate over x which may be converted into an integral over t. “wt” then allows one to integrate over a cycle, but E/w appears on the RHS as in the quantum case. This seems to show that quantum mechanics picks out special classical energies for which the classical cycle integral produces a value which is not dependent on k or m.