Definition of "Quantizing" Could anyone explain to me what "quantize" means in the following context?

Quantize the 1-D harmonic oscillator for which 
  $$H~=~{p^2\over 2m}+{1\over 2} m\omega^2 x^2.$$

I understand that the energy levels of a Q.H.O. are quantized. But here it is used as a verb... Aren't the energy levels naturally quantized?
 A: Quantization is a process of constructing a quantum (field) theory out of a classical (field) theory. 
In your case you have a classical Hamiltonian (which governs the dynamics of your classical system, in your case the harmonic oscillator), the underlying classical theory is Hamiltonian mechanics. If you say to quantize $H$ one means to find the corresponding observable in the corresponding quantum theory (in your case usual non relativistic quantum mechanics). In your case this will be the Hamiltonian of the system which in particular determines the dynamics of the corresponding quantum system. 
There are several approaches to quantization. The simplest one is called canonical quantization, where just the position variable $q$ (or $x$ in your notation) is replaced by the canonical position operator $\hat{Q}$ and the momentum $p$ by the momentum operator $\hat{P}$.
In your case the quantized version of $H$, i.e. the quantum Hamiltonion would be:
$$
\hat{H} = \frac{\hat{P}^2}{2m} + \frac{1}{2} m \omega^2 \hat{Q}^2
$$
In my opinion it is not fully understood what quantization really means, it's often just a more or less handwaving rule to construct a quantum theory out of a classical one. However there are more sophisticated approaches to quantization which try to make things conceptually clearer (but are mathematically very advanced).
One may ask, why people quantize theories and do not construct quantum theories without referring to the classical one. 
I think there are several reasons for this:


*

*Historically it was very successfull to start with a classical picture to get e good theory of the quantum world

*It gives us something which makes it easier to interpret the quantum observables

*If a quantum theory emerged through quantization of a classical one in a well defined way one has reason to hope that it would be easier to understand the classical limit (i.e. how the classical world emerges out of the quantum one)


Some people would say that it is better to formulate a quantum theory without constructing it out of a classical one (for example in axiomatic quantum field theory). However I think it is accepted that quantization at least is a very usefull heuristics to discover a quantum theory or less ambitious to see how certain observables (for example your quantum hamiltonian) will look like.
A: I think of 'quantize' as a verb that refers to converting the classical to the quantum picture. The Hamiltonian you wrote down is, after all, the classical one, which is not 'quantized'. Once you solve it by (1) writing down annihilation/creation operators, (2) finding that only particular wavefunctions are allowed, (3) numerically evaluated the energy levels with a computer, (4) used other quantum kung-fu, you've 'quantized' it. Argueably, you've quantized it once you've written down the appropriate quantum mechanical operators.
