Raising and lowering operators

[John Finn]

Suppose you take a Schroedinger-like equation $$-\psi''+F(x)\psi=0$$. (E.g. F(x)=V(x)-E, and not worried about factors of 2 etc.) This is positive definite if $$\int \left( \psi'^2+F(x)\psi^2 \right)dx>0$$. Is so, you can write this as the product $$(d/dx+g(x))(-d/dx+g(x))\psi=0$$, i.e. as $$a^{\dagger}a=0$$ for $$a=-d/dx+g(x)$$ [its adjoint is $$a^{\dagger}=d/dx+g(x)$$] if $$g(x)$$ satisfies a Riccati equation, $$dg/dx+g^2=F$$. So raising and lowering operators hold for any potential.

Is this true? Is it useful? Is it well known?

MENTOR NOTE: Post edited changing single $ to double $ for latex/mathjax expansion.

 

[div_grad]

That you can solve second-order differential equations (of the Schrödinger type) using "raising" and "lowering" operators has been well known for many years now, and the general method for factorizing the equations into products of such operators can be found in this nice paper:
L. Infeld and T. E. Hull, "The Factorization Method", Rev. Mod. Phys. 23, 21 (1951)

If you prefer more accessible explanation of the method before jumping straight into this paper, I suggest that you check out the section in R. Shankar's textbook "Principles of Quantum Mechanics (second edition)" titled "The free particle in spherical coordinates" on page 346. The general techniques and many specific examples (as regards to your question about the validity for "any potentials") can be found in the paper by Infeld and Hull linked above.