On the quantization of linear maps

✍ Scribed by A Lakshminarayan; N.L Balazs

• Publisher: Elsevier Science
• Year: 1991
• Tongue: English
• Weight: 784 KB
• Volume: 212
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(91)90115-o

✦ Synopsis

The quantum mechanics of some maps have been studied as means to understand the implications of "chaos" in quantum systems. Maps exhibiting highly complex classical dynamics can also be quantized. Linear maps that are classically unstable are simple systems that can be exactly solved, and here we do the same for their quantum version, namely find the eigenfunctions of the unitary operator that propagates states in one discrete time step. Here we discuss maps in a two-dimensional (non-compact) phase space where the instability is a hyperbolic fixed point that is either ordinary or "reflecting." The essential technique uses a basis that separates the time evolution of the dynamical variables. This also provides us a convenient basis to study the transition in the eigenfunctions when the classical map's fixed point changes from a stable elliptic case to a hyperbolic one.