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In the B\"uttiker-Landauer perturbation approach to electron tunnelling,through a time-modulated rectilinear potential barrier, the Tien-Gordonidentity was invoked, together with its infinite energy spectrum. Here, anexact treatment is presented which is based on the temporal wave-functionmatching procedure, that led to a finite energy spectrum. In seeking thecondition governing the time evolution of the tunnelling process, the Eulerformula provided the crucial ingredient for time quantization, whichdiscretised the continuous time in the oscillating barrier potential and energyharmonic equations. As a result, a finite system of inelastic scatteringchannels was created. When an electron entered the elastic channel, it wasscattered, instantaneously, into finite neighbouring energy-level scatteringchannels, by absorption (emission) of photon energy from (to) the oscillatingfield, during the transit period across the dynamic barrier. The absorption andemission times of barrier traversal, T_+ and T_-, respectively, were derivedfor the low and high frequency regimes of the barrier oscillations.Calculations revealed that in the low (high) frequency situation T_+= T_- (T_+<T_-).

Quantization of Time in Dynamic Barrier Tunnelling