Ah sorry I didn't read your first post properly, got the a's confused - thought you wanted to find A not a...
Unfortunately its not true that A = a; instead I think you want to say that the expected quantum value for energy (the average value, or observed value) should equal the observed energy of the classical harmonic oscillator, i.e.
Eav=<ψ0∣H^∣ψ0∗>The classical harmonic oscillator has a constant total energy, which is given by
Etot=2mwc2a2hence I think this total energy can be equated to the expected energy of the quantum system. Therefore I think that the equation
2mwc2a2=<ψ0∣H^∣ψ0∗> should hold. From this I think you should be able to determine a and hence solve the problem... I hope!