Infinite Divisibility in Euclidean Quantum Mechanics



An interesting and well studied class of quantum mechanical Hamiltonians consists of examples of the form
$$\mathcal{H} = - \tfrac{1}{2}{\partial ^2}/\partial {x^2} + V(x),$$
expressed in units where m = ћ = 1l, where the real potential V(x) is chosen so that the spectrum of \( \mathcal{H} \) is nonnegative and the real, normalizable, nowhere vanishing ground state φ o (x) has zero energy eigenvalue; such systems are referred to as “simple systems” in this article. In this case, the ground state itself determines the Hamiltonian completely since
$$\mathcal{H} = - \tfrac{1}{2}[{\partial ^2}/\partial {x^2} - {\phi ''_o}(x)/{\phi _o}(x)],$$
, and therefore φ o (x) — just as much as \( \mathcal{H} \) itself — can be said to determine all the properties of the system.


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© Hindustan Book Agency 2007

Authors and Affiliations

  1. 1.Department of Physics and Department of MathematicsUniversity of FloridaGainesvilleUSA

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