Raising and Lowering Operators

[jjohnson]

In Griffith and Shankar, the raising and lowering operator seem to appear from nowhere, but it seems really elegant.

Where do they come from, is it some corollary from certain mathematical structure?

Can anyone give a brief explanation?

 

[vanhees71]

In which context? There are lowering and raising operators for the harmonic oscillator, angular-momentum eigenstates (closely related by the way due to the SU(N) symmetry of the N-dimensional symmetric harmonic oscillator) and for free-particle modes in QFT.

 

[jjohnson]

Thank you for replying.
So for harmonic oscillator, all dimensions are separable (H = Hx+Hy+Hz), does it depend on dimensionality? Even for one-dimensional harmonic oscillator, we can use raising and lowering operators.

Thanks