Why is this the definition of the scattering matrix and reflection/transmission coefficients?

Question

I've been having trouble with the choice commonly used to name the elements of the scattering matrix in quantum mechanics. Let's say we are dealing with a simple potential barrier in 1D, such that if we fix the energy the only two possible states far from it are those with right moving ($k>0$) and left moving ($-k>0$). Given an incoming state, the scattering matrix gives the outgoing state:

$|\psi_{out}\rangle = S|\psi_{in}\rangle$

where we can restrict ourselves to states of the form:

$ |\psi_{in}\rangle = a|k\rangle + |-k\rangle $

If the incoming state is just $|k\rangle$, then I might define the reflection $r_k$ and the transmission $t_k$ such that $|r_k|^2$ is the probability that the system has been reflected (that the final state is found in $|-k\rangle$) and $|t_k|^2$ is the probability to be transmitted (that the final state is found in $|k\rangle$):

$ |\langle -k|S|k\rangle|^2 = |r_k|^2 \qquad |\langle k|S|k\rangle|^2 = |t_k|^2$

Similarly for $r_{-k}$ (I'm not assuming yet time-reversal, so these coefficients may be different from the one above for now):

$ |\langle k|S|-k\rangle|^2 = |r_{-k}|^2 \qquad |\langle -k|S|-k\rangle|^2 = |t_{-k}|^2$

This implies that, up to a phase:

$ \begin{cases} S|k\rangle = r_k|-k\rangle + t_k|k\rangle\\ S|-k\rangle = r_{-k}|k\rangle + t_{-k}|-k\rangle \end{cases} $

(I know this is not the way to define it, but I just want to place the elements of $S$ in the right place, not compute their values)

Therefore, in matrix representation:

$ |k\rangle = \left(\begin{matrix} 1\\ 0 \end{matrix}\right) \qquad |-k\rangle = \left(\begin{matrix} 0\\ 1 \end{matrix}\right) \qquad S = \left(\begin{matrix} t_k & r_{-k} \\ r_k & t_{-k} \end{matrix}\right) $

However I believe the standard representation for the S-matrix (and the one used by my professor) is:

$ S = \left(\begin{matrix} r_k & t_{-k} \\ t_k & r_{-k} \end{matrix}\right) $

with the same representation for $|-k\rangle$ and $|k\rangle$. What is wrong with my reasoning? Why is it different? Was it a computation error? Or am I looking at reflection and transmission from the wrong side? I can't figure it out.

Answer 1

I'm not sure if this is what OP is asking but there exist at least 2 conventions for the 1D S-matrix:

1. In the convention of e.g. Wikipedia & Griffiths, the $2\times 1$ column is ordered according to left and right asymptotic regions, so that the free particle $V=0$ has $S=\sigma_x$.

2. In the convention of e.g. Gasiorowicz (see problem 4.1), the $2\times 1$ column is ordered according to left and right movers, so that the free particle $V=0$ has $S={\bf 1}_{2\times 2}$.

For more information, see e.g. this Phys.SE post and my Phys.SE answer here.