Orbitals, shapes and wave function relation

[ado sar]

Are the shapes of orbitals like s which is sphere due to wave function or due to square of the wave function ? many textbook refer that atomic orbital is the space where an electron can be found. But how is this possible when from wave function we dont get any information for the probability density ? do the textbooks refer to the ψ^2 when they refer to space ?

Edited November 4, 2018 by ado sar

[studiot]

Hello ado and welcome to SF.

The orbitals are derived from the square of the wave function.

To understand why we use the square and not the wave function, here is a short quote from Coulson

Quote

There is also a simple reason why the square of ψ and not ψ itself, must be associated with the probability density. When we solve the wave equation, we usually find some regions of space where ψ is positive, and others in which it is negative. But the probability must always be positive or zero; and we are therefore forced to aboandon ψ in favour of ψ² when trying to relate  the wave function and the density.

This is better than most texts which simple state that the probability density is proportional to the square.

If this is not enough, we can look at a simpler example to make it clear.
Please tell us if you are familiar with simple harmonic motion (the pendulum) and integration.

Here are some plots of ψ and ψ2 originally from Hertzberg.

Note that ψ generally has some negative and some positive parts, ψ² does not.