Harmonic oscillator (quantum vs classical)

[mcheung4]

(I am referring to section 3.1 in Burkhardt's "Foundations of Quantum Physics", if you happen to have the book.)

In that book it's pointed out that the apparent contradiction between the pdf's of the QM ground state solution to the harmoinc oscillator with its classical conterpart (at the same energy = ℏω/2) is due to the comparison setup itself; we should compare the corresponding states rather than the corresponding energy. so QM's groud state with E = ℏω/2 and the classical E = 0. But why? shouldn't energy defines the system uniquely so when we make comparison we use the same energy?

 

[optophotophys]

The author's intention is to emphasize that we should compare between the "corresponding" states. However, it's difficult to find the "corresponding"states among the completely different model, namely classical and quantum mechanics. The easiest solution is the ground states. It's a plausible idea that the ground states should be corresponding.

As for the energy difference, we should be careful about how to define the the origin of the energy. Classically, one can define the origin of the energy by the energy of the rest. However, in the quantum mechanics, there is no "rest state" because of the uncertainty principle. One conventional definition is the energy of nothing (this is different from the energy of the quantum mechanical vacuum). Another conventional definition is the energy of the ground state. So if we take that attitude, there is no energy difference. Your question is closely related to the problem of the zero point energy, which many physicist have (even now?) struggled to deal with.

Many people also think that the corresponding state to the classical harmonic oscillator should be the coherent state of the quantum mechanical oscillator.