Quantum Chaos
The quantum version of the billiards is readily studied in several ways. The classical Hamiltonian for the billiards, given above, is replaced by the stationary-state Schrödinger equation or, more precisely,
where is the Laplacian. The potential that is infinite outside the region but zero inside it translates to the Dirichlet boundary conditions:
As usual, the wavefunctions are taken to be orthonormal:
Curiously, the free-field Schrödinger equation is the same as the Helmholtz equation,
with
This implies that two and three-dimensional quantum billiards can be modelled by the classical resonance modes of a radar cavity of a given shape, thus opening a door to experimental verification. (The study of radar cavity modes must be limited to the transverse magnetic (TM) modes, as these are the ones obeying the Dirichlet boundary conditions).
The semi-classical limit corresponds to which can be seen to be equivalent to, the mass increasing so that it behaves classically.
As a general statement, one may say that whenever the classical equations of motion are integrable (e.g. rectangular or circular billiard tables), then the quantum-mechanical version of the billiards is completely solvable. When the classical system is chaotic, then the quantum system is generally not exactly solvable, and presents numerous difficulties in its quantization and evaluation. The general study of chaotic quantum systems is known as quantum chaos.
A particularly striking example of scarring on an elliptical table is given by the observation of the so-called quantum mirage.
Read more about this topic: Dynamical Billiards
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