As example let's consider very simple task: the solution of one-dimensional stationary Schrodinger equation with step-function potential: $V(x) = 0$ if $x0$ (see this, for example, https://en.wikipedia.org/wiki/Solution_of_Schr%C3%B6dinger_equation_for_a_step_potential ).
The method of solving is very clear. We have to find solutions for $x0$ independently and after that apply the continuety for wave-function and it's first derivative in the point $x=0$.
My interest is this point $x=0$. The value of potential in this point is undefined (could vary from 0 to $V_0$). This means that Schrodinger equation is undefined in this point, so we cannot find the second derivative of the wave function, and wave function itself. Is it correct logic? Do you have some sources for the explanation of such type problems?
The way one usually thinks of this - and I am not sure how mathematically rigorous this is - as two Schrödinger equations on the half lines R>0 and R
Dear Andrew James Bruce,
Thank you, for your reply. I want to understand better how to overcome this inconsistency mathematically or even could we find some examples where this problem crutially occurs and it has to be resolved in some mathematical way.
To begin with, an obvious example of an unbounded second derivative of the wave function is a quartic potential ~ x^4. I've never heard about any problems or paradoxes with it.
On the other hand, the Schroedinger equation is only the end result in a series of approximations: QED -> the Dirac equation -> the Pauli equation -> the Schroedinger equation. That the Dirac equation would not accept potentials varying in a range exceeding 2mc^2 is a well-known fact (sorry can't think about a reference; ask your lecturer). From the top of my head, the Schroedinger equation is free of paradoxes, but one can never be sure. There can well be Bryzgalov's paradox still hidden there. Good luck!
Sorry, have just realised that my example of quartic potential is erroneous: the second derivative of the wave function is potential times the wave function. The rest remains.
The math answer to the question (and related ones) can be extracted from the (remarkable) book by V S Vladimirov "Equations of mathematical physics" (originally published in Russian) or Gel'fand-Shilov one on theory of distributions. From physics point of view
there is no interface of zero size in Nature. You have to introduce a finite-size interface as a regularization making potential smooth and continuous around x=0 and then take a limit to zero. In the distant past you might address the question to AS Schwarz, who was MIFI professor, and immediately got the answer.
Dear Alexander Turbiner,
Thank you very much. Your reply gives the solution what I have to do to overcome this inconsistency. More or less I understood, since I have already been familiar with these references. What about some examples where the mathematical problem like this arrives in physical meaning? Or do you know some physical examples where this (or any other) regularization tecnique is essentially needed?
At finite volume (or finite number of particles) a phase transition does not occur but a sharp, but smooth interphase. Taking infinite limit of volume or number of particles you get sharp interphase of zero size.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Quantum Physics
- Quantum Mechanics