Semiclassical Theory and the Koopman-van Hove Equation

Abstract

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton-Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton-Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys-Wentzel-Kramers-Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein-Brillouin-Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

Figures (6)

• Figure 1: (Left) In phase space, each harmonic oscillator JWKB eigenfunction is supported on a Lagrangian submanifold. For the classical allowed region $m\omega x^2\leq 2J$ (shaded blue), the submanifold is a circle with $p\in \mathbb R$. For the classically forbidden region $m\omega x^2\geq 2J$ (shaded red), the submanifold is defined by hyperbolae with $-ip\in \mathbb R$. The wavefunction only has support on the decaying branches (solid red), not the growing branches (dashed black). (Right) Harmonic oscillator JWKB eigenfunctions as a function of angle $\theta=\tan^{-1}({p/m\omega x})$ for $n=0,1,2,3$. Here, no amplitude factor is applied.
• Figure 2: The KvH harmonic oscillator eigenfunctions in phase space using Bohr-Sommerfeld quantization are continuous: $n=0$ plotted with finite radius (upper left), $n=1$ (upper right), $n=2$ (lower left), $n=3$ (lower right). For each eigenfunction, the complex phase is plotted as a ribbon the rotates around the ring that defines the Lagrangian submanifold; the phase is also indicated by the hue of the ribbon.
• Figure 3: The KvH harmonic oscillator eigenfunctions in phase space using EBK quantization with the Keller-Maslov correction: $n=0$ plotted with finite radius (upper left), $n=1$ (upper right), $n=2$ (lower left), $n=3$ (lower right). For each eigenfunction, the complex phase is plotted as a ribbon the rotates around the ring that defines the Lagrangian submanifold; the phase is also indicated by the hue of the ribbon. Note the discontinuous jump in phase at classical turning points.
• Figure 4: A uniform superposition of KvH harmonic oscillator eigenfunctions in phase space including $n=0,1,2,3$: (left) Bohr-Sommerfeld quantization, (right) EBK quantization. The $n=0$ Bohr-Sommerfeld eigenfunction is illustrated on a circle of finite radius.
• Figure 5: The real part of the phase space harmonic oscillator eigenfunctions with JWKB matching conditions and $\left| p \right|^{1/2}$ amplitude factor: $n=0$ (upper left), $n=1$ (upper right), $n=2$ (lower left), $n=3$ (lower right). For each eigenfunction, the circular submanifold for $(x/x_0)^2\leq 2J/\hbar$ is plotted vs. $p$ and the (half) hyperbolic submanifold for $(x/x_0)^2> 2J/\hbar$ is plotted vs. $-ip$. For each plot, the coordinates $x/x_0$ and $p/p_0$ are scaled to $(2J/\hbar)^{1/2}$ to focus on the region of interest.