Monday, January 12, 2009

The physics behind the name

As promised, I want to explain the name of this blog. Tonight I am being a complete dork (as if I were ever anything else) and writing about physics while watching Star Wars on TV. Ahh, this is the life!

Let's start with "Simple Harmonics." First of all, it's musical! But mathematically, which is what I'll focus on here, a simple harmonic is the same thing as a sine or cosine wave. These are important in many areas of physics. Those of you who took intro mechanics learned about simple harmonic motion. This could, for example, be the up-and-down bouncing motion of a ball hanging from the end of a spring. When you graph the motion of the ball, with time on the horizontal axis and the vertical position of the ball on the vertical axis, you get a sine wave. I spent way too much time making an animation of this using Mathematica. I know I could have gotten one from another website, but I wanted to do it myself. I finally got it to look right, and I was so proud of it, but now it won't upload! So I'll keep trying to figure that out.

Simple harmonics are also important in electromagnetism. All types of light waves can be expressed as some combination of sine and cosine waves.

And finally, they show up in my favorite area of physics, quantum mechanics. Quantum mechanics deals with phenomena that other areas of physics can't explain. You probably learned at one point that light can act like both a wave and a particle. This is true not only for light but for everything else too! Quantum mechanics deals with this wave nature of other particles (such as electrons). It also explains the strange fact that the energies of these particles are only observed at certain quantities - a particle's energy jumps from one discrete value to another instead of increasing/decreasing smoothly through all values. The vehicle for this explanation comes in the form of the Schroedinger Equation, which is to quantum mechanics what Newton's F=ma is to classical mechanics. It is a way of predicting the behavior of a particle. But unlike Newton's law, which predicts behavior with certainty, the Schroedinger Equation deals with the probability that a particle will be at a certain position at a certain time.

Simple harmonics are the solutions to a basic quantum mechanics problem called the infinite square well. Imagine a particle, let's say an electron, that can only move along a line (i.e. in only one dimension) and that is in between two infinitely thick walls so that it can never escape. According to classical mechanics, it would just bounce back and forth between the walls. It would be able to move at any constant speed (which means it could have any amount of energy), and it would spend an equal amount of time at all points between the walls. But according to quantum mechanics, the particle does not have an equal probability of being at all locations, nor can it have any amount of energy. Solving the Schroedinger equation for this situation gives a set of simple harmonics, where the horizontal axis is the distance between the two walls. The vertical axis corresponds to the probability that the particle will be at each position between the walls. (Technically, the vertical axis of the square of the simple harmonic is the probability.) I called it a "set of simple harmonics" because there are many ways a sine wave can fit between the two walls. If the wavelength is shorter, more peaks of the wave will fit between the walls. These different configurations correspond to the allowed discrete energies of the particle. Below you will see the simple harmonics corresponding to the five lowest energies of a particle confined between walls at x=0 and x=10.



As I mentioned above, probabilities are actually calculated from the squares of these graphs. The squares of the above graphs look like this:





Let's look specifically at the second graph above (in bold). It's telling us that a particle confined between x=0 and x=10 and in the second lowest allowed energy will most likely be found at x=2.5 or x=7.5. It also says that the particle will never be found at x=5! If it can be found on either side of x=5, how does it not pass through that point? The answer: I have no idea! In general, it is because particles are not just particles; they are particles that also act like waves. But I've never been able to find a more specific explanation. I'm not sure if there is one yet. Quantum mechanics is very strange. It exhibits phenomena that we know to be true but that we still don't understand.

Hopefully this has given you an idea of what quantum mechanics is and why simple harmonics are important to the basic calculations of the infinite square well. If you think this all seems crazy and counterintuitive... good! It is certainly very different from what we experience and what we know about the motion of everyday objects. We know it's there, but we don't see it because it happens on such a small scale.

Well that should do it for simple harmonics. Now on to "transmitted reflections."

Transmitted reflections is a contradiction. In optics, they are opposites. The light that doesn't get reflected from a surface gets transmitted through the surface. But the phrase also reminds me of something called Frustrated Total Internal Reflection. Regular total internal reflection occurs when light traveling inside a substance completely reflects off the inside of the substance. This is, for example, responsible for the sparkliness of a diamond ring. Diamonds are cut so that the light that goes into the top of the diamond doesn't come out the bottom - it reflects within the diamond and comes back out the top, making the diamond look brighter. Now imagine this happening in a simple block of glass. If a beam of light traveling through the glass hits the side of the glass at the correct angle, it will completely reflect, and no light will leave the glass. However, something does leave the glass - it's called an evanescent wave. It decays very quickly (exponentially, in fact), and it doesn't carry any light or energy. Normally it has no effect on anything. But something interesting happens if you put a second glass block close to the first one, with just a small gap of air in between. Because the light doesn't escape the first block into the air, it seems like no light should make it to the second block. But the evanescent wave is still present in the air gap between the blocks, and the interaction between this wave and the second block disturbs (or frustrates) the electric field in such a way that some of the light passes from the first block through the air gap to the second block. This is Frustrated Total Internal Reflection, and it's probably the closest thing to a "transmitted reflection" because the light transmitted to the second block should have been a reflection and would have been if not for this bizarre interaction.

I was determined to find a physics-related name for my blog that could also be otherwise meaningful. Think of "simple harmonics" as the many layers of my thoughts and experiences, the ups and downs of my life, the analytical nature of my mind, and the uncertainties that can result from even a basic problem or event. Think of "transmitted reflections" as my written interpretations of all of these. If not for this blog, they would remain my personal reflections and never be transmitted.

I will welcome any questions about the physics in the post!!

4 comments:

  1. What a relief to find an analytical approach to wavefunction science with close mathematics. The Schrodinger wavefunction psi may also be solved by expansion to an additive series differential within time and space boundaries. This CRQT function network is Clough Relative Quantum Topodynamics, and develops the GT integral atomic model by combination of quantum wave equations with relativistic math to generate the 3D animated topological data map of an atom.
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