Creation and annihilation operator in quantum mechanics?

[saravanan13]

What is exactly the creation and annihilation operator in quantum mechanics?
What its physical significance?

 

[Bob_for_short]

Take a harmonic oscillator Hamiltonian like H(p,x) = p² + x². Then make the variable changes: a = p + ix and a⁺ = p - ix. Now your Hamiltonian becomes H(a,a⁺) = a⁺a or so. So a and a⁺ are just some other variables.

 

[haushofer]

In QM this can be used, for instance for the QM harmonic oscillator, to construct eigenfunctions of the Hamiltonian by acting with these creation operators on the vacuum |0>.

In quantum field theory you normally promote the field itself to be an operator in order to treat space and time as labels of these fields; after all, in special relativity they are on equal foot so you shouldn't treat one as an operator (x) and the other as a label (t) parametrizing motion.