I have a question concerning the numerical models that are frequently used for a scanning tunneling microscope in which three solutions of the Schrodinger equation using the potentials for the pre-barrier, barrier, and post-barrier regions are combined using the boundary conditions. This gives a piecewise solution with an incident and reflected wave in the pre-barrier region and only a transmitted wave in the post-barrier region. However, in a simple instrument the pre-barrier and post-barrier regions may be ideal metal electrodes at different potentials which are connected to a battery to provide the necessary step in the DC potential. Please consider this simple circuit with no amplifiers or other electronics added. How do the incident, reflected, and transmitted waves pass through or combine at an ideal battery?
I appreciate the comments that I received here, as well as in email, but please address my question: What is the significance of the reflected wave that is found when you have a piece-wise analytical solution in which a battery is connected directly to a tunneling junction? There is an analytical function for the pre-barrier, the barrier, and the post-barrier section. The wavefunction and its derivative are matched at each boundary to satisfy an incident wave and a reflected wave at one end of this model and a transmitted wave at the other end. Thus, if a battery is connected, how can you match the boundary conditions?
Hello Mark,
I think it a bit of quantum mechanical wizardry, but the piece-wise function accounts for all the energy in the circuit. So, the barrier forms the potential block for energy to overcome and causes reflected energy, mostly, hence, the overall formula must account for this energy, for it will be "subtracted" at a latter point. The transmitted wave is, of course, the portion of energy successfully tunnelling through, completing the circuit. So, after reading a current, you must subtract the background (incident plus reflected) waves to reveal the transmitted portion.
Now, "How do you match the boundary..." That is simply imposed by the experimenter. For example, you would adjust voltage and needle height to get a nice reading.
I have attached a paper that goes far more into detail, including the business of matching the wavefunctions.
- Badges
- Science topic
- Similar topics
- Electrochemistry
- Battery