Contributions in Mathematical Physics pp 161-176 | Cite as

# Infinite Divisibility in Euclidean Quantum Mechanics

Chapter

- 1 Citations
- 191 Downloads

## Abstract

An interesting and well studied class of quantum mechanical Hamiltonians consists of examples of the form
expressed in units where m = ,
and therefore

$$\mathcal{H} = - \tfrac{1}{2}{\partial ^2}/\partial {x^2} + V(x),$$

*ћ =*1^{l}, 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$$\mathcal{H} = - \tfrac{1}{2}[{\partial ^2}/\partial {x^2} - {\phi ''_o}(x)/{\phi _o}(x)],$$

*φ*_{ o }(*x*) — just as much as \( \mathcal{H} \) itself — can be said to determine all the properties of the system.## Preview

Unable to display preview. Download preview PDF.

## References

- [1]E. Lukas,
*Characteristic Functions*, 2nd Ed., (Hafner Publishing Company, New York, 1970).Google Scholar - [2]B. De Finetti,
*Theory of Probability, Vol. 2*, (Wiley & Sons, London, 1975).zbMATHGoogle Scholar - [3]B. Simon,
*Functional Integration and Quantum Physics*, (Academic Press, London, 1979).zbMATHGoogle Scholar - [4]I. M. Gel’fand and N. Ya. Vilenkin,
*Generalized Functions, Vol. IV: Applications of Harmonic Analysis*, (Academic Press, New York, 1964).zbMATHGoogle Scholar - [5]J. R. Klauder, “Isolation and Expulsion of Divergences in Quantum Field Theory”, Int. J. Mod. Phys. B
**10**, 1473–1483 (1996).MathSciNetCrossRefzbMATHGoogle Scholar - [6]Private communication.Google Scholar

## Copyright information

© Hindustan Book Agency 2007