Probabilistic interpretation of resonant states

Naomichi Hatano, Tatsuro Kawamoto, Joshua Feinberg

Research output: Contribution to journalArticlepeer-review


We provide probabilistic interpretation of resonant states. We do this by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviours of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particles.

Original languageEnglish
Pages (from-to)553-564
Number of pages12
JournalPramana - Journal of Physics
Issue number3
StatePublished - Sep 2009

Bibliographical note

Funding Information:
The work of NH is supported by Grant-in-Aid for Scientific Research No. 17340115 from the Ministry of Education, Culture, Sports, Science and Technology as well as by Core Research for Evolutional Science and Technology (CREST) of Japan Science and Technology Agency. The work of JF was supported in part by the Israel Science Foundation (ISF).


  • Normalization
  • Probabilistic interpretation
  • Resonance

ASJC Scopus subject areas

  • General Physics and Astronomy


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