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We consider a quantum particle with energy E incident upon a one-dimensional potential. We show that the probabilities of transmission and reflection are the same for incidence upon a general potential from either side (from ‘the left’ or ‘the right’). This equality holds true for any potential which goes to constant values as and is finite for all x. We present a remarkably simple proof that the probabilities are equal. The simplicity of our proof is the most important pedagogical result of our paper, and will be easily understood by undergraduate students in second to fourth year. We discuss several cases, including the step potential and the potential barrier.

This article examines resonant tunnelling of multi-atomic systems, extending the body of work on resonant tunnelling in this area beyond diatomic homo-nuclear systems. We consider diatomic molecules with distinct atoms having different masses, and linear triatomic molecules with indistinguishable atoms. The molecule is incident in the bound state upon a step potential with an energy increase of V0 for each atom. We calculate the probabilities of reflection pR and transmission pT in the bound state and the dependence on the energy of the molecule. We find that, as was the case for the homo-nuclear molecule, resonant transmission for diatomic molecules with distinguishable atoms occurs for arbitrarily weak binding, and pT is close to one for finite binding energy. We also find resonant transmission for linear triatomic molecules. We show results for the time-dependent Schrödinger equation which agree with resonant transmission. We also consider the transmission of an N-atom one-dimensional molecule.