it is not true that in-plane momentum needs to be conserved. As there is a scattering region, the momentum along the direction of transport is no longer a good quantum number.
what mean's "a good quantum number"?
They define the quantum number as a value that is used when describing the energy levels available to atoms and molecules. An electron in an atom or ion has four quantum numbers to describe its state and yield solutions to the Schrödinger wave equation for the hydrogen atom.
There are four quantum numbers:
n - principal quantum number - describes the energy level. ℓ - azimuthal or angular momentum quantum number - describes the subshell. mℓ or m - magnetic quantum number - describes the orbital of the subshell. ms or s - spin quantum number - describes the spin.
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