Quantum Mechanics
Introduction
Whenever I read through the theory of quantum mechanics I am reminded of a priest trying to explain the existence of God. They go through greats lengths to obfuscate and confuse, thinking to themselves that they if they can't be complete and thorough, at least they can make it sound like they are attempting to cover all possible weaknesses in their arguments. What I will try to do here is to first explain what problem quantum mechanics was developed to solve, and from this, what answers quantum mechanics provides that cannot be provided using another theory.
Questions that I would like answered:
- Why is energy absorption quantified?
- Why is electromagnetic energy quantified?
- Why are there electron and nuclear orbitals rather than a random distribution of particles?
Mathematics is a useful tool that approximates reality. The key word is 'approximates'. Reality is always more complicated than math can handle. Any real world system under study always has too many factors involved to draw any absolute conclusions, so the observer has to ignore a bunch of the factors, claiming that their contribution is too small to worry about. This is a common theme in all science, unfortunate but necessary if any useful conclusions are to be made.
States
To get quantum mechanics one has to have a good understanding of the idea of states and there is no better way to understand something than to see it in action. So here is a two-state system that can be constructed easily. A video showing it in action will be posted at some point:
Materials:
- 3 bar magnets with North/South on the sides rather than the ends
- 1 swivel
- 1 container to hold one of the magnets
- Calipers
Construction:
- Fix one of the magnets to a hard surface with super-glue.
- Place another magnet into a container such as a centrifuge tube.
- fix the third magnet to a surface.
- Glue a swivel to the center of the centrifuge tube and the center of the fixed magnet.
Two states in action: At equilibrium the container will always line up long end with the fixed magnet. It can be rotated with energy input, but as soon as you let go, it goes back to long-way alignment again. Thus there are two states, and only two states, both equivalent in that the container can twist 180 degrees to align the other direction, but there is no equilibrated half-way state. Any attempt to rotate requires energy, but no change in state happens unless the correct amount is added to cause a complete half-rotation.
For the calipers, the width will correspond to the photon size. A large width will be a high energy photon, while a small width is a low energy photon.
The energy needed to change state is thus just enough to rotate the container barely past 90 degrees, the size of the caliper is barely big enough so that both sides of the caliper touch the container. Once past 90 degrees, the magnetic field of this system will then push it (or pull it, depending on your point-of-view) to align the other direction.
Discussion:
There is no continuum of states here, where energy is added a little at a time until the 'lowest energy' state is achieved. Instead there are just two states. The size of photon (amount of energy) needed to rotate a container to 90 degrees can be called a quantum. Any energy more or less than the quantum does not cause a change in state.
Two non-equivalent states:
- Glue the third magnet near one end of the first fixed magnet.
Discussion:
This is now a two-state system where one of the states is lower in energy than the other because the magnetic strength at one end of the system is different.
Math
Now that we have a model of a two-state system along with a way to provide photons that will change the states, it would be nice to be able to describe the system logically and ultimately make predictions about it. This is where a tool like mathematics comes in. Unfortunately, regular math can't describe a system with states. The most complex regular math uses continua, like sine waves or curves, with differentials and integrals, infinitesimals and summations.
Most equations in math are derived, meaning that they can be derived from basic principles: given a thought problem and fundamental concepts, an equation solving the problem can be logically developed. One could either go through the derivation, which can be very long and convoluted, or accept that others have proven that a particular equation is true and instead focus on the uses and limitations of it. In other words, when an equation is presented and the author asks "This is true, do you believe me?". Answer yes.
Examples are the sine equation, which involves tracking the position of a point on the circumference of a circle, and the wave equation, which involves tracking the position of a vibrating line fixed at both ends. There are many texts that provide these derivations and, for the moment, will not be presented in this wiki. The resulting equations are very useful, especially in quantum mechanics, so a brief survey of the basic ones will be presented.
A big problem in all science, but especially mathematics, is notation. Writing F(x) could mean "the value of the function F at x", or "x is a member of the group F", or even "the constant F multiplied by the variable x". Each meaning has dramatic implications for the use of math as a tool for a particular problem. This inconsistency is one reason why I dislike mathematicians. I will try to avoid this problem by listing notation meanings for each mathematical section, but be wary.
