Ground state wave function from Euclidean path integral

✍ Scribed by G.C Rossi; M Testa

• Publisher: Elsevier Science
• Year: 1983
• Tongue: English
• Weight: 960 KB
• Volume: 148
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(83)90333-0

✦ Synopsis

In this article we show how to compute from Euclidean path integral the wave function of the ground state of a quantum mechanical system. The method is sufficiently general to encompass the case of degenerate classical minima, as well as the case of multidimensional systems. In the one dimensional case we find in the forbidden region the WKB approximation.

1. INTR~DUCTI~N

Euclidean functional integrals have become powerful tools to study perturbative and nonperturbative properties of quantum systems. The continuation to Minkowskian time is guaranteed by the Osterwalder-Schrader theorem 11, 21.

In this article we want to show how it is possible to compute in quantum mechanics the ground state wave function directly from the Euclidean path integral.

The observation which makes it possible is that for systems with nondegenerate minima of the potential the mean value of the observable 6(x(O) -x) is the modulus squared of the ground state wave function at the point x, i.e., j Wr) es'*%W -xl = I@ u/n*(v) 4.v -x) w&) = I vn(x)12.

(1.1)

In Section 2 we will discuss the case of nondegenerate classical minima and show how, starting from (l.l), it is possible to reproduce perturbation theory and loop expansion for the wave function.

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