The tunnelling rate in real time in the semiclassical limit has been calculated for arbitrary energy levels in the cubic potential. For the ground state it agrees well with the result found by the instanton method.
The periodic instanton method is used to study the decay rates of metastable ground state and excited states of the cubic potential. The imaginary part of the energy is calculated through the standard procedure in the path-integral scheme. A formula of the decay rate valid for the entire region of energy is obtained. This formula provides a linkage between classical thermal activation at high temperatures and purely quantum tunneling at zero temperature. It is shown that in the low energy limit this more general result reduces exactly to the vacuum result. The temperature dependence of the decay rate agrees with earlier works in the literature.
We adapt the semiclassical technique, as used in the context of instanton transitions in quantum field theory, to the description of tunneling transmissions at finite energies through potential barriers by complex quantum mechanical systems. Even for systems initially in their ground state, not generally describable in semiclassical terms, the transmission probability has a semiclassical (exponential) form. The calculation of the tunneling exponent uses analytic continuation of degrees of freedom into a complex phase space as well as analytic continuation of the classical equations of motion into the complex time plane. We test this semiclassical technique by comparing its results with those of a computational investigation of the full quantum mechanical system, finding excellent agreement.