Supporting random wave models: a quantum mechanical approach

We show how two-point correlation functions derived within nonisotropic random wave models are in fact quantum results that are obtained in the appropriate limit in terms of the exact Green function of the quantum system. Since no statistical model is required for this derivation, this shows that taking the wave functions as Gaussian processes is the only assumption of those random wave models. We also show how for clean systems the two-point correlation function defined through an energy average defines a Gaussian theory which substantially reduces the spurious contributions coming from the normalisation problem. PACS numbers: 05.45.Mt,05.40-a Since Berry’s seminal paper in 1977 [1], the so-called Random Wave Model (RWM) has become by far the most popular and successful tool to describe the statistical properties of wave functions of classically chaotic systems which in this approach are modelled by a random superposition of plane waves. Its applications range from the realm of optics [2], passing by the general problem of wave mechanics in disordered media [3] to important issues in mesoscopic systems [4]. Owing to this robustness this approach has been regarded as the indicator of wave signatures of classical chaotic dynamics [5]. The reasons for this success can be traced back to two fundamental points. First, it can be formally shown that such a random wave function is a stationary random process [6] (roughly speaking a function taking random values at each point); second, such random process is Gaussian, namely, it is uniquely characterised by a two-point correlation function which expresses fundamental symmetries, like the isotropy of free space. The fact that the process is Gaussian represents a considerable advantage in an operational sense since it provides us with a set of rules to cope with averages over complicated expressions in the way Wick’s theorem and its variants do. At the same time the generality of the random wave two-point correlation makes the theory a remarkably good approximation when the effect of the boundaries can be neglected, like in the case of bulk properties. When applied to real quantum systems, however, there remain limitations related to the above mentioned ingredients. Concerning the Gaussian assumption, a formal proof showing that a chaotic wavefunction is indeed a Gaussian process is still lacking. Even more, as noted in [7] the Gaussian distribution explicitly contradicts the normalisation condition for the wavefunction. In practical terms this means that, when dealing with statistics beyond the two-point correlation function, the Letter to the Editor 2 Gaussian distribution produces spurious non physical contributions, and attempts to construct a RWM respecting the normalisation constraint lead to severe mathematical difficulties [8]. Still this assumption is supported by many arguments based on Random Matrix Theory [4], quantum ergodicity [5], information theory [8], and Berry’s original semiclassical picture [1]. Impressive numerical results also support the conjecture at the level of one-point statistics [9], and evidence for higher order statistics is given in [4, 5]. Hence, it is appealing to look for a RWM which minimises the spurious contributions due to the normalisation problem while keeping the wavefunction distribution still Gaussian. About the isotropic character of the theory, constructing a random superposition of waves satisfying both the Schrödinger equation and boundary conditions turns out to be at least as difficult as solving the full quantum mechanical problem by means of standard techniques. To our knowledge the attempts in the direction of a non-isotropic RWM can only deal with highly idealised boundaries such as an infinite straight wall [10], a linear potential barrier [11], and the edge between two infinite lines enclosing an angle of a rational multiple of π [12]. Also in [13] a variation of the RWM to include finite size effects is presented. The fact that these approximations already produce non-trivial deviations form the isotropic case is an indication of the importance of the inclusion of arbitrary boundaries. Our aim in this communication is twofold: first, we shall show that the mentioned results for the two-point correlation function defining the non-isotropic and finitesize RWM can be derived from quantum mechanical expressions, namely, they are independent of any statistical assumption about the wavefunction. Second, we shall show how for a statistical description of wavefunctions using an energy ensemble average, the spurious contributions coming form the normalisation problem are of order O(1/N) with N the number of members of the ensemble, making their effect negligible for high energies. Isotropic and non-isotropic random wave models. We consider solutions of the Schrödinger equation (