Wentzel-Kremers-Brilouin Approximation for Dynamic Systems with Kinetic Coupling in Entangled State Representations

We study the Wentzel-Kramers-Brillouin (WKB) approaximation for dynamic systems with kinetic couplings in entangled state
representations. The result shows that the kinetic coupling will affects the position of classical turning points where the condition of using the WKB approximation breaks down. The modified WKB approximation formula is derived in the entangled state representation, for example, the common eigenvector of relative coordinate and total momentum of two particles. The corresponding Bohr-Sommerfeld quantization rule is also derived.