Question

What is a well-behaved wave function? Illustrate with diagrams.

Answer 1

Solution.

$$t = |\psi^2|$$

Well-behaved wave function has a number of features that distinguish it from others. For example,

1. Well-behaved wave function (t) is unambiguous. That is, each point in space has only one value.

2. Well-behaved wave function (t) is finite and continuous at all points in space. Also, the first and second derivatives are finite and continuous.

3. Well-behaved wave function (t) must have a finite integral over the entire space, that is, have a certain value.

An example of a well-behaved wave function is the following graph:

$$f(x) = cos(x)$$

Or, for example, the function

$$f(x) = x^2$$

is not a well-behaved wave function, since as,

$$x\to\infty$$

the function becomes infinite, which contradicts the second property.

Diagrams:

Well-behaved wave function

$$f(x) = cos(x)$$

Not well-behaved function

$$f(x) = x^2$$

Answer:

$$t = |\psi^3|$$

Well-behaved wave function has a number of features that distinguish it from others. For example,

1. Well-behaved wave function (t) is unambiguous. That is, each point in space has only one value.

2. Well-behaved wave function (t) is finite and continuous at all points in space. Also, the first and second derivatives are finite and continuous.

3. Well-behaved wave function (t) must have a finite integral over the entire space, that is, have a certain value.

An example of a well-behaved wave function is the following graph:

$$f(x) = cos(x)$$

Or, for example, the function

$$f(x) = x^2$$

is not a well-behaved wave function, since as,

$$x\to\infty$$

the function becomes infinite, which contradicts the second property.