When solving the Schrodinger equation in 2D polar coordinates, one has to deal with various Bessel functions. In the most simple example, the infinite circular potential well, the solutions to the radial differential equation are the Bessel functions of first $[J_m(kr)]$ and second $[Y_m(kr)]$ kind. One usually discards the $Y_m(kr)$ functions on account of their asymptotic behavior at $r = 0$, $$Y_m(kr) \sim (kr)^{-m}$$ and so they are not square integrable functions. However, in the case of zero angular momentum, $m=0$, the Neumann function of zeroth order, $$Y_0(kr) \sim \ln (kr),$$ although infinite at the origin, is square integrable! So why do we have to discard it as well? What are the boundary conditions that have to be satisfied by a radial wave function at the origin?
Subscribe to:
Post Comments (Atom)
-
Why can't we use fissions products for electricity production ? As far has I know fissions products from current nuclear power plants cr...
-
How can we know the order of a Feynman diagram just from the pictorial representation? Is it the number of vertices divided by 2? For exampl...
-
I have searched for equations regarding craters and I came across two of them. The first one is from this NOAO website in the level two sec...
-
As the title says. It is common sense that sharp things cut, but how do they work at the atomical level? Answer For organic matter, such a...
-
This image from NASA illustrates drag coefficients for several shapes: It is generally accepted that some variation of the teardrop/airfoil...
-
Problem Statement: Imagine a spherical ball is dropped from a height $h$, into a liquid. What is the maximum average height of the displaced...
-
In most books (like Cardy's) relations between critical exponents and scaling dimensions are given, for example $$ \alpha = 2-d/y_t, \;\...
No comments:
Post a Comment