Relfection and transmission coefficients for wave function in $\delta$-potential Let's assume we have some one-dimensional Delta-potential $V(x)=V_0 \delta(x)$. Then I have found numerous problems where the approach for a wave function is $$\varphi(x)=\begin{cases}e^{ikx}+re^{-ikx},\ & x<0\\te^{ik'x},\ &x>0\end{cases}$$ I have two questions about this: 


*

*The Schrödinger equation for this wave function outside of $x=0$ yields $\frac{\hbar^2 k'^2}{2m}=E=\frac{\hbar^2k^2}{2m}$. This means $k'=k$. Is this correct? Can we say that in general the wave vector $k$ must be the same if the wave propagates in the same potential (which outside of $x=0$ is just $V=0$)? And if not, why do we then have a legit approach where the reflected part and the incoming part of the WV in the area $x<0$ have the same wave vector?

*By definition, for the transmission coefficient $T$ we have $T=\mid\frac{\varphi(\infty)}{\varphi(-\infty)}|^2=|t|^2$ which confuses me. Isn't $t$ already the coeffient of transmission? What else is $t$ if not the coeffiecient? And if it is the coefficient, what did I get wrong about the definition of $T$?
 A: *

*You have understood this aspect correctly. The bottom line is: $\psi(x)$ is claimed to satisfy the time independent Schrodinger equation, so if in doubt, plug it in and check that it does!

*Transmission here is defined to be the ratio of two physically observable rates, namely $T = R$(transmit) $/ R$(incident) where $R$(incident) is the rate at which right-moving particles would be detected before the barrier if a detector were placed there, and $R$(transmit) is the rate at which right-moving particles would be detected after the barrier if a detector were placed there. These rates are proportional to the modulus-squared of the quantum amplitude associated with each plane wave, not the quantum amplitude itself, and they also involve a factor $k$ or $k'$ to account for the faster motion (higher flux) when the wavevector is high. To be precise,
$$
T = \frac{ |t|^2}{ |1|^2 } \frac{k'}{k}
$$
where I include the $|1|^2$ term to keep the logic clear (your incident wave has amplitude $1$) and the ratio of wavevectors obviously evaluates to $1$ when $k'=k$, but more generally this will not always happen. To understand this really fully you need to learn about the probability current or flux which is given by
$$
{\bf j} = \frac{\hbar}{2 mi}(\Psi^* \nabla \Psi - \Psi \nabla \Psi^*)
$$
(this expression can be related to the continuity equation which expresses the conservation of the number of particles, or if you prefer, the conservation of probability).
