User:ReyHahn/Gutzwiller

In quantum chaos, the Gutzwiller trace formula is a semiclassical formula for the density of states, developed by Martin Gutzwiller in 1971. It applies for quantum systems where their classical analog hasisolated orbits, generally when quantizing classically chaotic systems.

Formula

The density of states $g(E)$ (for a given total energy $E$) can be written as

$g(E)=g_{0}(E)+{\frac {1}{\hbar \pi }}\sum _{p}T_{p}\sum _{r}{\frac {1}{\sqrt {|{\text{det}}(I-({\tilde {M}}_{p})^{r})|}}}\cos \left({\frac {rS_{p}}{\hbar }}-rm_{p}{\frac {\pi }{2}}\right)$,

where the first sum is over a family of classical primitive periodic orbit, and the second one is over the particular periodic orbits associated to the primitive ones, $g_{0}(E)$ is the smooth part, $\hbar$ is reduced Planck constant, $T_{p}$ is the period of the primitive orbit, $S_{p}$is the action of the primitive orbit, $m_{p}(E)$ is the topological number of the primitive orbit, $I$ is the identity matrix and ${\tilde {M}}_{p}$ is the monodromy matrix for a transversal section to the primitve orbit of the constant energy shell.

The topological number $m_{p}(E)$, sometimes called the Maslov index,

The monodromy matrix can be written by writing the analog classical Hamiltonian of the system in terms of canonical positions and momenta $\{q_{i},p_{i}\}$, given a surface of section transverse to the orbit within the constant energy $E=H(q_{i},p_{i})$ shell.

References

• Peter (1992). cas9.