I know I've seen this point about roots discussed somewhere but I can't for the life of me remember where. I'm hoping someone can point me the right direction.(adsbygoogle = window.adsbygoogle || []).push({});

Here's the situation:-

The standard derivation of the quantum HO starts with the classic Hamiltonian in the form H = p^{2}+ q^{2}(ignoring constants)

Then the raising and lowering operators are introduced with complex values such that the QM hamiltonian becomes something of the form (p + iq)(p-iq), which expands into p^{2}+ q^{2}plus the commutator [q.p] as an imaginary part and the rest follows.

I found myself questioning this. Beyond the fact that 'it works' (which is fair enough). When you consider that the commutator in question evaluates to ih/2, what we have actually done is to simply add h/2 to the original classical Hamiltonian without any clear justification, disguising the fact by a subtle ordering of the steps.

I know this can be justified mathematically in other ways. I just need to know where I can find it.

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# Harmonic oscillator Hamiltonian.

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