Clasical and quantum harmonic oscillator - correspondence principle

[exponent137]

At classical harmonic oscillator, total energy is proportional to square of frequency, but at quantum harmonic oscillator, total energy is proportional to frequency.
Are those two frequencies the same?
How it is with transition from quantum harmonic oscillator to classical harmonic oscillator? How omega^1 and omega^2 agree?

One example of this is http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc6.html
but where is the frequency?

Maybe I found answer:
http://en.wikipedia.org/wiki/Correspondence_principle
Is this enough?

 

[exponent137]

The next question is:
How more simply to imagine formula W = (n+1/2)ω ?
W is energy ω is frequency and n is integer quantum number.
The common calculation for quantum oscillator is too long for such short result.

 

[exponent137]

Does relativistic generalization of quantum harmonic oscillator exist?
Maybe:
http://www.quantumsciencephilippines.com/1811/lowest-order-relativistic-energy-correction-of-1-d-harmonic-oscillator/