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physics pp. 553–564 Probabilistic interpretation of resonant states

Naomichi Hatano, Tatsuro Kawamoto, Joshua Feinberg

Published Jun 17, 2015• 0 citations
OpenAlex

Abstract

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.

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Abstract. We provide probabilistic interpretation of resonant states. This we do 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 behaviors 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 particle number.

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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.
Abstract. We provide probabilistic interpretation of resonant states. This we do 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 behaviors 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 particle number.
Resonance states spatially diverge and are thus not square integrable. Instead, their norm is defined by the biorthogonal scalar product of left and right states. We replace the corresponding volume integral over infinite space by a convenient boundary integral in piecewise-homogeneous systems and apply this procedure to diverse physical settings. For a quantum particle in any number of dimensions we treat hard-wall and piecewise-constant potentials. For electromagnetic systems with piecewise-homogeneous material properties, we consider three-dimensional and effectively two-dimensional cavities.
In his review of The Interpretation of Quantum Mechanics by Roland Omnes (February p54), Alistair Rae states that "the key problem in the interpretation of quantum physics" for many people is the necessity of adding "by hand" our "interpretation of the squared modulus of the wave function as a probability distribution". As he goes on to explain, the problem is not simply that this step – asserting the equivalence of a squared modulus to a "classical probability distribution" – must be added from outside the theory proper; the real challenge is that the question of when it is appropriate to take the step seems to have no universally valid answer. Rae poses as "a real question to be...hopefully answered one day" the following: "under what circumstances does the squared wavefunction collapse into one of the alternative possibilities?"
Abstract The widely cited definition of quantization in terms of square‐integrable wave functions does not apply to continuum wave functions, to such phenomena as metastable states, or many‐body resonances. A better philosophical foundation for quantum mechanics separates the probabilistic aspects based on square integrable Hilbert space functions from the dynamical aspects based upon the solutions of Shroedinger's (or Dirac's) equation. A Hilbert space may have a non‐Hilbert space basis, which may be described by Stieltjes integrals and a spectrum measure. This viewpoint is expounded by reference to a very detailed analysis of a simple model, through which a precise definition of a Bohr–Feshbach resonance can be given. We propose a definition of a “metastable state,” showing that it is consistent with accepted usage, and that it overcomes a series of objections which have been catalogued by Simon. Its rate of decay is given by the Fourier–Stieltjes transform of the spectral density function; it is moreover the longest‐lived initially localized state which can be formed from a small span of energy eigenfunctions near its mean energy.
Abstract The article reviews the theory of open quantum system from a perspective of the non-Hermiticity that emerges from the environment with an infinite number of degrees of freedom. The non-Hermiticity produces resonant states with complex eigenvalues, resulting in peak structures in scattering amplitudes and transport coefficients. After introducing the definition of resonant states with complex eigenvalues, we answer typical questions regarding the non-Hermiticity of open quantum systems. What is the physical meaning of the complex eigenmomenta and eigenenergies? How and why do the resonant states break the time-reversal symmetry that the system observes? Can we make the probabilistic interpretation of the resonant states with diverging wave functions? What is the physical meaning of the divergence of the wave functions? We also present an alternative way of finding resonant states, namely the Feshbach formalism, in which we eliminate the infinite number of the environmental degrees of freedom. In this formalism, we attribute the non-Hermiticity to the introduction of the retarded and advanced Green’s functions.
The wave function in Quantum Mechanics has properties that render its physical interpretation unclear. On the one hand, its modulus squared is interpreted as a probability density, but, unlike conventional probabilities, it interacts with the material world. In addition, it is thought to have encapsulated nonlocal correlation and, according to modern thought, parallel information in a manner impossible according to classical physics. In this paper, an extended version of Stochastic Electrodynamics is presented which offers conventional models for these otherwise spooky features.