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

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.

• So you're asking about various conventions? Commented Dec 8, 2021 at 17:49
• Yeah, pretty much. There might still be some mistake due to getting confused with matrix multiplication, but the main problem is what are the exact conventions. Commented Dec 8, 2021 at 17:50

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 $$S={\bf 1}_{2\times 2}$$.