From my understanding of quantum mechanics, I know that it is difficult to support an understandable answer to this question.
In all physics textbooks it is written that the Psi wave function and its first derivative must be continuous.
Meanwhile, in these same books you find many situations where the Psi solution is continuous and its derivative is not, the simplest example is the particle in a one-dimensional box [1].
My own explanation is that Psi is a complex mathematical function that only has physical meaning as the product of Psi. Psi* represents the probability density of finding a particle and therefore:
Psi.Psi* and its derivative must be continuous while the derivative of Psi alone may or may not.
Let's take the simple example of real Psi "Psi = Psi *" with fixed impenetrable limits where Psi = 0
It is to be expected that the “probability function” Psi^2 and its derivative are continuous.
It follows that the derivative of Psi is continuous if Psi is not equal to zero.
Which is probably missing in the literature.
1-Quora Questions/Answers, 2021.
1. Continuous does not exclude sudden changes in first derivative
2. Usually you only find such jumps in d psi/ dx when using non physical infinate
Potentials.
Imagine a simple "moutain-like" or "saw tooth" function, linearly increasing and suddenly decreasing. This is a perfectly continuous function but its
derivative at the top is indeterminate.
The sad reality is that Schodinger's PDE itself is not complete and sometimes misleading.
There are many answers to this question, essentially containing a considerable amount of black magic.
1- One explanation is that Psi is a complex mathematical function that only has physical meaning as a product of Psi. Psi* represents the probability density of finding a particle and therefore:
Psi.Psi* and its derivative must be continuous while the derivative of Psi alone may or may not.
2-When the wave function has a boundary condition involving infinite potential, there is a discontinuity in the derivative of Psi because the wave function must immediately disappear at the boundary which means you will never find the particle in the potential, and that she may not do it. smoothly, usually creating a CORNER.
3-For a finite potential, the particle can be found INSIDE it, although this is unlikely - so the wave function is NOT required to IMMEDIATELY fall to zero and instead transitions smoothly to an exponential.
The question arises:
Did E. Schrödinger deeply understand quantum mechanics?
I believe that E. Schrödinger understood quantum mechanics deeply.
Only, he did not have the power to impose his own ideas against the giants N. Bohr, Heisenberg, Einstein, etc.
Check out his brilliant quotes:
i-Entanglement is not one but rather the characteristic feature of the quantum mechanical.
ii-He wrote about the probabilistic interpretation of quantum mechanics saying:
"I don't like it and I'm sorry I ever had anything to do with it."
Today, after 100 years, we are more reliable about what is true and what is false:
1- The three giants Einstein, Bohr, Schrödinger were all wrong in confining entanglement to quantum physics phenomena when it applies to classical physics in the same way as in QM (Sound waves in audio rooms for example).
2-Most of the great scientists of the time and the iron guards who followed believe that entanglement transfers instantly (spooky action at a distance) while the statistical theory of Cairo techniques predicts that the entanglement signal has a maximum speed equal to the speed of light C.
3-Quantum entanglement comes before Schrödinger PDE and not after.
4-Sentimentally I feel discomfort every time I hear or read the sentence:
Separation of variables
in time-dependent PDE.
To me this looks like separating the head from the body of a living subject (entanglement).
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