Answer to Question #86348 in Physical Chemistry for Chanchal

Question

What is well behaved function. Illustrate with diagram

Answer 1

Answer on Question #86348 Chemistry / Physical Chemistry

Functions which have derivatives of all orders at all points.

Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object—a function, a set, a space of one sort or another—is "well-behaved". The term has no fixed formal definition, and is dependent on context, mathematical interests, fashion, and taste. To ensure that an object is "well-behaved" mathematicians introduce further axioms to narrow down the domain of study. This has the benefit of making analysis easier, but cuts down on the generality of any conclusions reached. In mathematics, well-behaved is opposite to pathological function which means that its properties are considered atypically bad or counterintuitive.

 There are some conditions for well behaved function in quantum mechanics:

1. It must be continuous and single valued because probability can have one value at a particular place and time, and continuous.

2. Momentum considerations require partial derivatives  to be finite, continuous and single valued.

3. It must be normalizable which means that it must go to zero as X,Y,Z go to +/- infinity, in order that probability of finding the object over all space is finite constant.

Example of pathological function :  the Weierstrass function is continuous everywhere but differentiable nowhere.