- #1

John Finn

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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.

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

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

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