Effect of intrinsic degrees of freedom on the quantum tunneling of a collective variable

The effect of intrinsic degrees of freedom on tunneling through a potential barrier is discussed using a BKW-like approximation. In the present work intrinsic degrees of freedom are represented by a single harmonic oscillator. The theory leads to a formula for the effect of the coupling on the decay width Γ of a metastable state. When the frequency ω of the intrinsic degree of freedom is large, then Γ → Γ_ad where Γ_ad is the decay width calculated with the adiabatic barrier. An inequality Γ ≤ Γ_ad is proved for any form of the coupling hamitonian. Corrections are discussed and are shown to be of order $\text{1}\text{M}$ where M is the mass of the tunneling coordinate. An application to fission is considered. The generalization of our formula for Γ to the case of many intrinsic degrees of freedom is given explicitly.