Tetra Quark

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Don’t drink and derive

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Quantum scattering in one dimension

In this blog post, we explore quantum scattering in one-dimensional systems, focusing on the case of a rectangular potential barrier. We start by examining the Schrödinger equation in one dimension and then move towards a more efficient solution for the scattering problem. Rather than following the conventional approach of solving from left to right and imposing continuity conditions at the boundaries, we introduce a faster method by assigning coefficients directly from the transmitted wave and building in the boundary conditions. We cover both scenarios where the particle’s energy is below and above the potential barrier, and we detail how to calculate the transmission and reflection coefficients in each case. To simplify the calculations, we introduce dimensionless parameters, which allow us to rewrite the transmission coefficient in a more intuitive form. This also helps us identify resonance wavelengths, which occur when integer multiples of half-wavelengths fit within the potential barrier.