Probabilistic interpretation of resonant states

NAOMICHI HATANO^1,*, TATSURO KAWAMOTO² and JOSHUA FEINBERG^3,4

¹Institute of Industrial Science, University of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan

²Department of Physics, University of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan

³Department of Physics, University of Haifa at Oranim, Qiryat Tivon 36006, Israel

⁴Department of Physics, Technion, Haifa 32000, Israel

^*Corresponding author. E-mail: hatano@iis.u-tokyo.ac.jp

Abstract. 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.

Keywords. Resonance; probabilistic interpretation; normalization.

PACS No. 03.65.Nk