The cavity system

Consider a single particle whose canonical coordinates
are
moving inside a cavity of .
The Hamiltonian is

where is the confining potential. I have introduced a (unitless) deformation `field' , so the effect of changing the parameter is to distort the potential in space. We assume that inside the cavity. Outside the cavity the potential becomes very large. To be specific, one may assume that the walls exert a normal force , and we take the hard wall limit . With the above assumptions about it is clear that the deformation is completely specified by the scalar boundary deformation function , where is an outwards unit normal vector at the boundary point . This is shown in Fig. 3.1. The surface area of the cavity is . Its volume is , which is related to its typical length by . Quantum-mechanically, a second length scale appears, where is the wavenumber. The other parameters (particle velocity) and (particle mass), as well as , could be scaled away with the appropriate choice of units (this will only be done in numerical tests where we set ). The energy is . Sometimes I will use to denote corresponding to energy . Upon quantization the eigenenergies, which in general are -dependent, are .

My numerical calculations of the band profile in this and the following chapter, unless otherwise stated, will refer to the two-dimensional (2D) cavity illustrated in Fig. 3.2, which we call the generalized Sinai billiard. The shape has been chosen because it has `hard chaos': no mixed phase space, and absence of any marginally-stable orbits (see Section 3.4). In Fig. 3.2b we show three example deformations of this billiard.

The band profile calculations will generally be done classically (Appendix B), since this is easier by far than the quantum calculation (Appendix C). The two have been demonstrated equivalent in Section 2.3. In Section 3.3.2 I will verify this equivalence for special deformations.

- Form of correlation spectrum and timescales
- Conversion of time averages to averages over collision parameters