Classical fields are usually constructed using a collection of classical harmonic oscillators, e.g. masses connected to springs. The energy of a classical harmonic oscillator is proportional to the amplitude squared. QFT uses quantized versions of those same classical fields. But, in the quantum domain energy is proportional to frequency, not amplitude squared. Is there any mathematical connection between this classical definition of energy and the quantum definition (frequency)? Or is the classical definition of energy simply discarded and arbitrarily replaced with the quantum definition? As always, thanks in advance.