JHansen
- 8
- 1
- TL;DR
- Want to show that ##S(-p) =S^\dagger (p)##.
So let's say that we have som unitary matrix, ##S##. Let that unitary matrix be the scattering matrix in quantum mechanics or the "S-matrix".
Now we all know that it can be defined in the following way:
$$\psi(x) = Ae^{ipx} + Be^{-ipx}, x<<0$$ and $$ \psi(x) = Ce^{ipx} + De^{-ipx}$$.
Now, A and D cmpts. are the ongoing waves and B & C the outgoing ones. So we can define the S-matrix by.
$$
\begin{pmatrix}
C\\
B
\end{pmatrix} =
\begin{pmatrix}
S_{11}& S_{12}\\
S_{21}& S_{22}
\end{pmatrix}
\begin{pmatrix}
A\\
D
\end{pmatrix}
$$.
Now, of course, we can show that the matrix is unitary via the probability current density (or time-reversal symmetry I think?). Anyway,how would I actually show that ##S(-p) = S^\dagger (p)## ? hmmHere are my thoughts. We notice that letting p -> -p in the wave functions is the same thing as letting i-> -i, i.e. taking the complex conjugate. So what remains to show is that ##S^* = S^\dagger##, or that ##S^* S = 1## as well. And this can be achieved via time-reversal symmetry. But maybe this restricts our potential to be real?I would like something more rigorous to be more certain.
Now we all know that it can be defined in the following way:
$$\psi(x) = Ae^{ipx} + Be^{-ipx}, x<<0$$ and $$ \psi(x) = Ce^{ipx} + De^{-ipx}$$.
Now, A and D cmpts. are the ongoing waves and B & C the outgoing ones. So we can define the S-matrix by.
$$
\begin{pmatrix}
C\\
B
\end{pmatrix} =
\begin{pmatrix}
S_{11}& S_{12}\\
S_{21}& S_{22}
\end{pmatrix}
\begin{pmatrix}
A\\
D
\end{pmatrix}
$$.
Now, of course, we can show that the matrix is unitary via the probability current density (or time-reversal symmetry I think?). Anyway,how would I actually show that ##S(-p) = S^\dagger (p)## ? hmmHere are my thoughts. We notice that letting p -> -p in the wave functions is the same thing as letting i-> -i, i.e. taking the complex conjugate. So what remains to show is that ##S^* = S^\dagger##, or that ##S^* S = 1## as well. And this can be achieved via time-reversal symmetry. But maybe this restricts our potential to be real?I would like something more rigorous to be more certain.