Quantization of Kramers' Rate Theory

With use of the Wigner representation, the Kramers equation for a Brownian particle in a potential field of external force is extended to the quantum mechanical case. As a simple application of the equation, the rate of jump over a potential barrier is estimated to the lowest order approximation in \(\). The rate increases due to the quantum mechanical effect by the factor \(1{+}(βω)²ω²/6(η²{+}ω²\)), where ω, η and β are the angular frequency of the particle in the potential well, viscosity and the inverse of temperature, respectively.