# 10: Plane Polar and Spherical Coordinates

• • Contributed by Marcia Levitus

Objectives

• Understand the concept of area and volume elements in cartesian, polar and spherical coordinates.
• Be able to integrate functions expressed in polar or spherical coordinates.
• Understand how to normalize orbitals expressed in spherical coordinates, and perform calculations involving triple integrals.
• Understand the concept of probability distribution function.

• 10.1: Coordinate Systems
The simplest coordinate system consists of coordinate axes oriented perpendicularly to each other. These coordinates are known as cartesian coordinates or rectangular coordinates, and you are already familiar with their two-dimensional and three-dimensional representation.
• 10.2: Area and Volume Elements
In any coordinate system it is useful to define a differential area and a differential volume element.
• 10.3: A Refresher on Electronic Quantum Numbers
Each electron in an atom is described by four different quantum numbers.  The first three quantum numbers specify the particular orbital the electron occupies. Two electrons of opposite spin can occupy this orbital.
• 10.4: A Brief Introduction to Probability
We have talked about the fact that the wavefunction can be interpreted as a probability, but this is a good time to formalize some concepts and understand what we really mean by that. Let’s start by reviewing (or learning) a few concepts in probability theory.
• 10.5: Problems

Thumbnail: A globe showing the radial distance, polar angle and azimuth angle of a point P with respect to a unit sphere. In this image, r equals 4/6, θ equals 90°, and φ equals 30°. Image used with permission (CC BY-SA 4.0; SharkD).