It seems like symmetries usually lead to ladder operators. For example in a central potential problems the conservation of angular momentum leads to angular momentum ladder operators being used in the energy eigenstates of the problem. If I am not wrong similar ladder operators can be constructed through the conservation of the Runge-Lenz vector for eigenstates of an inverse square Hamiltonian. Therefore, what are the conservation laws/symmetries that lead to the ladder operators in the harmonic oscillator?
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Ladder operators are a consequence of the root and weight structure of Lie algebras, such as so(3), etc. In the case of the Harmonic oscillator, the relevant algebra is the Lie algebra of the oscillator group, with the celebrated four generators, of which the 3-generator Heisenberg algebra is a subalgebra.
Further see this.
rob
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One could construct a ladder operator for any spectrum (for simplicity assuming it non-degenerate): $$ H|n\rangle=\epsilon_n|n\rangle \longrightarrow \hat{a}=\sum_{n=0}^{+\infty}|n-1\rangle\langle n| $$ Admittedly, this is a bit a tongue-in-cheek answer - the ladder operators for HO are special, but one needs to specify what one considers special about them to inquire about symmetry.
E.g., that the Hamiltonian be written in form $H=\hbar\omega a^\dagger a$ is a consequence of the equidistant spectrum - one can inquire about the symmetry that leads HO to having such a spectrum, but it is not per se a question about ladder operators.
One often uses second quantization to perform similar trick on any system in terms of fermionic operators: $H=\sum_n \epsilon_na_n^\dagger a_n$, although this implies constraining the total number of particles to one. Less ambiguously (but less popularly) the same thing is done in terms of Hubbard operators: $$X_{i,j}=|i\rangle\langle j|\longrightarrow H=\sum_n\epsilon_ nX_{0,n}^\dagger X_{0,n},\\ \hat{a}=\sum_{n=1}^{+\infty}X_{n-1,n}$$
Roger V.
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