SemM
Gold Member
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Hi, in QM literature the inadmissible solutions to the Schrödinger eqn are often , if not always, quoted in the text as "inadmissible", because they are discontinuous, not-single valued, not square integrable and not infinitely differentiable. However in a discussion with Dr Du yesterday, various functions resulted as inadmissible, such as x, ##x^2## which the author later realized the reason for (not being infinity differentiable).
However ,as the discussion continued, the next suggestions, ##e^{−x}## and ##e^{−3x}## were easily identified by Dr Du to also be inadmissible. At first, and even second glance, I cannot see why these are inadmissible, because they are infinitely differentiable , they appear as being indeed single valued, however they are perhaps not square integrable.
In a textbook, the criteria say
"(...) where the domain ##\mathscr{D}(D) \subset L^2(-\infty, +\infty)## consists of all functions ##\psi \in L^2(-\infty, +\infty)## which are absolutely continuous on every compact interval on R, and such that ##D\psi \in L^2(-\infty,+\infty).##"
Here again, the functions are absolutely continuous in an infinite interval. But I wonder, how can one verify that without using mathematical software?
How can one "train" or which immediate features can one spot in a function to see rapidly, as Dr Du did, that a function does not satisfy the 4 criteria of QM?
As an example, any function including tan(x) would be assumed discontinuous, and any function without "i" in it, can be assumed to be not-square integrable. But what about functions that are more complex than so, how can one find if they are accepted as solutions to the Schrödinger eqn without excessive use of software?
Thanks
However ,as the discussion continued, the next suggestions, ##e^{−x}## and ##e^{−3x}## were easily identified by Dr Du to also be inadmissible. At first, and even second glance, I cannot see why these are inadmissible, because they are infinitely differentiable , they appear as being indeed single valued, however they are perhaps not square integrable.
In a textbook, the criteria say
"(...) where the domain ##\mathscr{D}(D) \subset L^2(-\infty, +\infty)## consists of all functions ##\psi \in L^2(-\infty, +\infty)## which are absolutely continuous on every compact interval on R, and such that ##D\psi \in L^2(-\infty,+\infty).##"
Here again, the functions are absolutely continuous in an infinite interval. But I wonder, how can one verify that without using mathematical software?
How can one "train" or which immediate features can one spot in a function to see rapidly, as Dr Du did, that a function does not satisfy the 4 criteria of QM?
As an example, any function including tan(x) would be assumed discontinuous, and any function without "i" in it, can be assumed to be not-square integrable. But what about functions that are more complex than so, how can one find if they are accepted as solutions to the Schrödinger eqn without excessive use of software?
Thanks