Tunnelling between tori in phase space

Abstract

This paper calculates the energy splitting Δ E due to tunnelling between a pair of quantum states which correspond to classical motion on tori in phase space. The calculation assumes that the states are separated in coordinate space. However, the result of the calculation, ΔE=Aħ 3 2 e -2 h ̵ , has canonically invariant expressions for the tunnelling actions S and prefactor A, and it is conjectured that the results apply even when the tori overlap in coordinate space. This makes the results relevant to a very general problem connected with the Einstein-Brillouin-Keller quantization scheme. There is a surprising consequence of this calculation. When the system is separable, it is clear that Δ E vanishes for most pairs of states. This does not occur because the prefactor A vanishes, but because the tunnelling action S is infinite. As a parameter is varied to make the system separable, S diverges logarithmically. The results of this paper only apply to quasi-integrable systems, where S is finite;p in the case of any exactly integrable system S is infinite and Δ E vanishes in a more singular manner than e -const. h ̵ as ħ→0.