Andrew Zimmerman Jones

String theory, often called the “theory of everything,” is a relatively young science that includes such unusual concepts as superstrings, branes, and extra dimensions. Scientists are hopeful that string theory will unlock one of the biggest mysteries of the universe, namely how gravity and quantum physics fit together.

This Cheat Sheet is intended to supplement Quantum Physics For Dummies, 3rd edition, by Andrew Zimmerman Jones. It begins by reviewing some useful operators used in quantum mechanics calculations. Then it covers a useful method for solving the Schrödinger equation for the quantum wave function, and then how you can use that wave function to calculate probabilities in quantum physics.

General relativity was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account non-inertial frames of reference — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time.

The theory of the space-time continuum already existed, but under general relativity Einstein was able to describe gravity as the bending of space-time geometry. Einstein defined a set of field equations, which represented the way that gravity behaved in response to matter in space-time. These field equations could be used to represent the geometry of space-time that was at the heart of the theory of general relativity.

In 1905, Albert Einstein published the theory of special relativity, which explains how to interpret motion between different inertial frames of reference — that is, places that are moving at constant speeds relative to each other. Einstein explained that when two objects are moving at a constant speed as the relative motion between the two objects, instead of appealing to the ether as an absolute frame of reference that defined what was going on.

Albert Einstein was influenced by the concept of an unchanging universe. His general theory of relativity predicted a dynamic universe — one that changed substantially over time — so he introduced a term, called the cosmological constant, into the theory to make the universe static and eternal. This term represented a form of repulsive gravity that exactly balanced out the attractive pull of gravity, and it would prove to be a mistake when, several years later, astronomer Edwin Hubble discovered that the universe was expanding.

String theory depicts strings of energy that vibrate, but the strings are so tiny that you never perceive the vibrations directly, only their consequences. To understand these vibrations, you have to understand a classical type of wave called a standing wave — a wave that doesn’t appear to be moving. In a standing wave, certain points, called nodes, don’t appear to move at all.

Since string theory is related to the theory of quantum gravity, a good place to begin is by giving an overview of the scientific understanding of gravity, which is defined by Einstein’s theory of general relativity. Albert Einstein would revolutionize the way physicists saw gravity. Instead of gravity as a force acting between objects, Einstein instead envisioned a universe in which each object’s mass caused a slight bending of space (actually space-time) around it.