Quantum Mechanical Tunneling in a Resonant Tunneling Diode I am currently trying to simulate a Resonant Tunneling Diode (RTD) on a somewhat basic level, so the question is rather general.
Quantum-mechanically its structure is shown well in this figure: Two barriers creating a well in between them. Using the stationary Schrödinger Equation for an electron and interpreting the potential barriers as the conduction band, the transmission and reflection coefficients can be obtained for an electron travelling from the left from the right side. Using the argument of continuation Reflection $R(E)$ & Transmission $T(E)$ should satisfy $R+T=1$ (Electron Energy $E$). In my numerical simulations this can also be observed.
However, if a Voltage $U$ is applied across the Device the Schrödinger Equation becomes an Airy-Equation i.e. the potential is shifted:
$U_{(da)}>0$)">
In principal, this also works in the numerical simulation. If, however, the correct value for the lower potential (right hand side) is used, it can be observed that $R+T<1$, even down to $R+T\approx 0.7$! It seems the higher $V_{\mathrm{l}}$ is, the closer becomes the test sum $R+T$ to $1$.
The Professor told me there's a physical reason for this, but I just can't figure it out. Am I missing s/th obvious here?
 A: I going to take a guess that $T$ and $R$ are being calculated incorrectly. If calculated correctly (for a lossless system), it must be that $T+R=1$ because probability current is a conserved quantity.
The most common way to incorrectly calculate $T$ and $R$ is to assume that they depend only on the amplitude (or square of amplitude) of traveling waves leaving the system. For example, in transmission over a step function, one might think that the $T = B_{\rightarrow}/A_{\rightarrow}$ (or something similar with a square root) --- i.e. the amplitude of the transmitted wave divided by the amplitude of the incident wave.
However, the transmission does not only depend on the amplitude of the wavefunction, it also depends on how fast the wave is moving. This is just like a normal current: the amount of current depends on both the amount of stuff moving (the amplitude) but also on how fast it is moving.
So, the correct formula for the transmission is $T = \frac{\left|\vec{j}_{trans}\right|}{\left|\vec{j}_{inc}\right|}$, where $\vec{j}_{inc}$ is the incident probability current and $\vec{j}_{trans}$ is the transmitted probability current.
You see this in the solution to the step function (second link), where it notes that $B_{\rightarrow}=\sqrt{T \frac{k_1}{k_2}}$, where $k_{1,2}$ are the wavenumbers (proportional to momentum and thus "speed") for the incoming and outgoing wave. The $\frac{k_1}{k_2}$ term is telling you that the waves' "speed" matters.
So my guess is that your problem will go away if you use the proper definitions for transmission and reflection.
