Quantum Analogs

Abstract

Many physical phenomena have mathematically equivalent derivations even when they describe completely different principles. Spherical harmonics are typically first encountered in an introductory upper-division quantum mechanics course while covering the Schrodinger equation in three dimensions, but they also exist in the Helmholtz equation and other branches of physics which incorporate the Laplacian operator in spherical coordinates. In this experiment, sound waves are emitted into a spherical cavity and the resonant frequencies are recorded. The wave amplitude at each resonant frequency is compared with its polar angle in the cavity and the shape is found to take on the form a spherical harmonic function. Eight total resonant frequencies are measured from 2.00000 kHz to 9.00000 kHz with an average error ranging from 1 Hz, up to 3Hz at higher frequencies. Of the eight, six are found to be spherical harmonic states. An analytic solution to the resonant states is given and analogous n, l, and m resonating numbers similar to the hydrogen atom’s quantum numbers are found. The theoretically predicted resonant frequencies are compared to the best fit measured states across a frequency spectrum of 110 Hz to 10000 Hz using the SpectrumSLC quantum analogs software and other supporting software. Six n=0 states are found near experimentally measured frequencies and come to agree with their theoretical value with an average error of 1.14% under the calculated value. Finally, degenerate resonance states are split apart by breaking the spherical symmetry inside the wave emitting chamber. Selected state functions corresponding to m=±1, m=±2, and m=±3 orbitals are measured. Their accuracy to their theoretical expressions exhibits some strong divergence, but the range still allows qualitative analysis for future experiments.

Link:
Paper

Application of the Radiating Orbital Equation

Please read my previous blog post, Deriving Kepler's Laws for Radiating Stars. In this blog post, we take the previously mentioned ideas and apply them to see if it would ever be necessary in calculation. Read about it here.

Deriving Kepler's Laws for Radiating Stars

In this next blog, I examine how Kepler's Laws (well, just the first one at least) changes when we look at a variable-mass system. Read about it here.

Eccentricity given mass, distance, and relative velocity

Suppose we have two stars; we know their masses, distance from each other, and relative velocity between them. We seek to predict the eccentricity of their orbit.

Read about it here.

Making Mistakes in Critical Thinking

I spent a great deal of time focusing on a simple problem in my recent midterm. I knew the correct answer, but sought a "more correct" answer. Read about it here.

Space Elevators: Exactly what it sounds like

There was a bit of math I had to write down, so please enjoy the PDF version here.

Star Wars - The Kessel Run in under twelve parsecs

"You've never heard of the Millennium Falcon? It's the ship that made the Kessel Run in less than twelve parsecs." - Han Solo, Star Wars: Episode IV A New Hope

That certainly sounds impressive, I think... What exactly does that mean though? Anyone reading this blog probably already knows that a parsec (pc) is a measurement of distance, not time or speed. A parsec (parallax arcsecond) is the distance an object would need to be to create a parallax angle of one arcsecond when the base between the two points of observation is an astronomical unit (average Earth to Sun distance) apart.

Most people with this much information will simply assume the George Lucas was just making up science fiction sentences and put no extra thought into that line, and I would be one to agree. But that is not why we are here. We are here because there is still an air of curiosity in that claim that is worth investigating, for no other reason that it is somewhat interesting.

Supporters of George Lucas will claim that this statement still has some validity when you consider relativistic effects. At speeds near the speed of light, the phenomenon of length contraction occurs, as does time dilation.

One of the key points of relativity is that, no matter which reference frame one is in, the speed of light is always the same and no physical object can exceed this speed. One of the consequences of this fact is that, if you are moving in respect to some object, that object will appear "thinner" or contracted if you looked at it. The relationship between these two properties is

where L_0 is the length of the object in its reference frame, L is the length of the object in your reference, v is the speed you are going, and c is speed of light. If we look, we see that v cannot exceed v, otherwise we would get an unreal number.

So why is this important? Couldn't Han Solo have just said the Millennium Falcon went at some speed v? Or made the Kessel Run in some time t? To answer these questions, let us look at an important fact, the change in speeds vs. the change in lengths. Assume the Millennium Falcon went at some speed v=0.9999c while making the Kessel Run in 12 pc. Solving for L_0, the proper length of the Kessel Run, we get 849 pc.

The significance of this is how much the speed changes versus how much the distance changes. If a regular ship could only make the run at v=0.9995c, then the relative length of the Kessel Run would be 26.83 pc. The difference in speeds is only 0.04%, whereas the difference in lengths is 124%! 

When it comes to relaying information to another human being, bigger changes grab more attention. Could this be the reason Han Solo toted the Falcon's speed with a measure of distance rather than time? I doubt George Lucas and his writers had originally thought so deeply about this line (despite them saying they did after the fact), but it's definitely an interesting way to say how fast something is when you move into the realm of relativity.