Here are the list of quantum-mechanical potentials. My question is how did scientists physically model these potentials, what were the parameters they consider before mathematically constructing a potential? Few of these potentials were devised when technology was not available to observe the behaviour and properties of quantum systems, so were these potentials mere ansatzes?

For example, while solving the Kronig-Penny model, we use a periodic square potential and the Bloch function (to account for the periodicity) to obtain solutions. For solving the Gamov's alpha particle tunnelling model, we use the barrier potential. These approximations are easy to deal with mathematically. If, for instance, I want to model my own potential without observation of experimental evidence, then how do I make an ansatz about the potential I want to construct?


closed as too broad by Kyle Kanos, heather, Jon Custer, Wolpertinger, AccidentalFourierTransform Nov 28 '16 at 17:42

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    $\begingroup$ It's very strange to me to think of a single list of quantum potentials. Schrodinger's equation works for whatever potential you want, and the choice of what mathematical function to use for the potential depends on entirely on what physical system you want to model. I guess there is a set of very common potentials, and they are common exactly because they model common physical systems. That said, I don't really understand what you're asking. You want to know how to pick a potential based on experimental evidence, or something else? $\endgroup$ – DanielSank Nov 26 '16 at 18:00
  • $\begingroup$ @DanielSank I have edited the question to make it more clear. What I'm asking is more to do with how the potentials were mathematically constructured. How did they make the ansatz to choose a specific potential that would be suitable for a system a priori to having an experimental evidence? $\endgroup$ – Naveen Balaji Nov 26 '16 at 18:23
  • $\begingroup$ The Kronig-Penny MODEL IS THE periodic square potential. It is not like you pick it. The same for Gamov's MODEL, which effectively is the tunneling barrier. You should reformulate this part to start from an actual physical system. $\endgroup$ – user1583209 Nov 26 '16 at 18:29
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    $\begingroup$ Asking how models are generically made seems too broad. $\endgroup$ – Qmechanic Nov 26 '16 at 18:33
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    $\begingroup$ the periodic square potential model leads to an analytically and simply solvable problem, moreover the solution does have all the essential properties that would be expected from the metallic behavior implying that if you massaged the corners or the middle of the square it would not matter much in practice because the periodicity is more important than the finer details of the well. $\endgroup$ – hyportnex Nov 26 '16 at 22:13

Difficult to generalise but you could try these steps:

  1. Find out the real potential (basically adding all potentials in the system).
  2. If necessary/possible, simplify this potential. (e.g. you can probably ignore the moon's gravity if you are interested in the electrons in some metal on your desk).
  3. Plot the potential
  4. Try to approximate this potential with simple segments. Typically constant potentials or steps or delta functions or some from your linked list work best. Also periodic potentials are easier to deal with than potentials which are different everywhere. Start with something as simple as possible.
  5. Calculate observable quantities and see if they can make sense.
  6. If nothing interesting pops up in 5, repeat steps 4 and 5 making your model more complicated (closer to the real potential). Usually you don't want to get too complicated though as this could make the interpretation (What is going on?) more difficult.
  7. Find an experimental physicist to do measurements and compare the data.

Also in many cases, experienced physicists would already have some effect in mind that could happen. You are not going out studying a random system looking for whatever.

  • $\begingroup$ Thanks a lot for answering! Just another question, how did they do this to find out the potentials the first time around. How would they be sure of their constructed potential as they did not have the tech to check if were right back in the day. $\endgroup$ – Naveen Balaji Nov 27 '16 at 4:30
  • $\begingroup$ What do you mean by "first time around"? For example, if there are charged particles, Coulomb's law has been known since the 18th century. $\endgroup$ – user1583209 Nov 27 '16 at 20:30

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