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Akano's Blog

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Back to School

Posted by Akano Toa of Electricity , in Math/Physics Sep 04 2012 · 58 views

So, today was the first official day of school at my grad school, but I didn't have any classes. Today was lab orientation for Monday and Tuesday lab sections (since we had yesterday off). Having nearly 50 students crammed into a room only able to seat 32 is rather entertaining.

Also, I have a talk to give on Friday on my research I did over the summer. This wouldn't be so much of an issue if I knew which part of the research to discuss, as I worked with two undergrads, and we have to split the topics of our research between us. However, one of the students worked in another lab over the summer as well, so she's probably not going to present on what the three of us did at all. Now the talk has to be divided in half.

Also, did I mention the talk was Friday? ._.

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Curiosity's on Mars

Posted by Akano Toa of Electricity , in Math/Physics Aug 06 2012 · 70 views

Science, Awesome, Space

Yay, science! 8D

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It's A Trap!

Posted by Akano Toa of Electricity , in Math/Physics Jun 25 2012 · 40 views

Optics, Physics, Laser

A magneto-optical trap. We just got it back up and running again after many days of realigning things (which is quite a pain, but it builds character ). The image is taken from a TV screen since the collection of atoms shown (the bright, white dot in the center) scatters light very dimly in the near-infrared, so our eyes can't see them, but security cameras can. The collection of atoms in the center is just above absolute zero (-273.15°C) by millionths of a degree. I don't remember off hand what the number of atoms is in the trap, but I'm assuming it's fairly large (~10⁶?).

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Summer: Week 1

Posted by Akano Toa of Electricity , in Math/Physics, Life May 24 2012 · 88 views

Already I'm nearly through my first week of summer research. That's kinda weird.

What I've gained from this experience thus far: aligning a laser beam so that it hits a fiber optic cord that's ~1-2 microns in diameter is tedious, painful, and annoying. However, magneto-optical traps and ultracold plasmas are awesome. I am learning a lot about optics and atomic physics despite it only being week one, and I'm sure this lab will be fun.

This weekend I hope to work on my Whirling Time Warper review (I have the pics, I just have to type and format everything) and maybe even make a new comic! Hurray for no homework and more free time!

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Quantum Mechanics

Posted by Akano Toa of Electricity , in Math/Physics May 08 2012 · 49 views

You were a worthy adversary, but in the end, it was I who came out superior.

Got an "A" in the course.

8 Comments

Take That, Quantum

Posted by Akano Toa of Electricity , in Math/Physics May 03 2012 · 51 views

quantum mechanics, exam, final and 1 more...

Finished 4/5 problems on my quantum final today AND almost half of the fifth one. I'd say I thoroughly owned that test to the best of my ability. 8D

Also, kudos to BZP's staff for keeping their cool and being able to bring the forums back. You guys are awesome.

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The Harmonic Oscillator

Posted by Akano Toa of Electricity , in Math/Physics Apr 12 2012 · 24 views

science, simple, harmonic and 3 more...

My Classical Mechanics professor quoted someone in class the other day: "The maturation of a physics student involves solving the harmonic oscillator over and over again throughout his/her career." (or something to that effect)

So, what is the harmonic oscillator? Otherwise known as the simple harmonic oscillator, it is the physical situation in which a particle is subject to a force whose strength is proportional to the displacement from equilibrium of said particle, known as Hooke's Law, or, in math terms,

F = -kx

where F is our force, x is our displacement, and k is some proportionality constant (often called the "spring constant"). That sounds swell and all, but to what situations does this apply? Well, for a simple example, consider a mass suspended on a spring. If you just let it sit in equilibrium, it doesn't really move since the spring is cancelling out the force of gravity. However, if you pull the mass slightly off of its equilibrium point and release it, the spring pulls the mass up, compresses, pushes the mass down, and repeats the process over and over. So long as there is no outside force or friction (a physicist's dream) this will continue oscillating into eternity, and the position of the mass can be mapped as a sine or cosine function.

What is the period of the oscillation? Well, it turns out that the square of the period is related to the mass and the spring constant, k in this fashion:

T² = 4π²m/k

This is usually written in terms of angular frequency, which is 2π/T. This gives us the equation

(2π/T)² = ω² = k/m

This problem is also a great example of a system where total energy, call it E, is conserved. At the peak of the oscillation (when the mass is instantaneously at rest), all energy is potential energy, since the particle is at rest and there is no energy of motion. At the middle of the oscillation (when the mass is at equilibrium and moving at its fastest) the potential energy is at a minimum (zero) and the all energy in the system is kinetic energy. Kinetic energy, denoted by T (and not to be confused with period) is equal to mv²/2, and the kinetic energy of the simple harmonic oscillator is kx²/2. Thus, the total energy can be written as

E = mv²/2 + kx²/2 = p²/2m + kx²/2

Where I've made the substitution p = mv. Advanced physics students will note that this is the Hamiltonian for the simple harmonic oscillator.

Well, this is great for masses on springs, but what about more natural phenomena? What does this apply to? Well, if you like music, simple harmonic oscillation is what air undergoes when you play a wind instrument. Or a string instrument. Or anything that makes some sort of vibration. What you're doing when you play an instrument (or sing) is forcing air, string(s), or electric charge (for electronic instruments) out of equilibrium. This causes the air, string(s), and current to oscillate, which creates a tone. Patch a bunch of these tones together in the form of chords, melodies, and harmonies, and you've created music. A simpler situation is blowing over a soda/pop bottle. When you blow air over the mouth of the bottle, you create an equilibrium pressure for the air above the mouth of the bottle. Air that is slightly off of this equilibrium will oscillate in and out of the bottle, producing a pure tone. Also, if you have two atoms that can bond, the bonds that are made can act as Hooke's Law potentials. This means that, if you vibrate these atoms at a specific frequency, they will start to oscillate. This can tell physicists and chemists about the bond-lengths of molecules and what those bonds are made up of. In fact, the quantum mechanical harmonic oscillator is a major topic of interest because the potential energy between particles can often be approximated as a Hooke's Law potential near minima, even if it's much more complex elsewhere.

Also, for small angles of oscillation, pendula act as simple harmonic oscillators, and these can be used to keep track of time since the period of a pendulum can be determined by the length of its support. Nowadays, currents sent through quartz crystals provide the oscillations for timekeeping more often than pendula, but when you see an old grandfather clock from the olden days, you'll know that the pendulum inside the body is what keeps its time.

Hopefully you can now see why we physicists solve this problem so many times on our journey to physics maturity.

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It's All Relative

Posted by Akano Toa of Electricity , in Math/Physics Mar 22 2012 · 39 views

math, physics, Einstein and 1 more...

Being a physics grad student has seen me be in quite the scientific mood lately, hasn't it? Well, unfortunately, I still don't have a new comic made (I'm sorry, everyone! ><), but I do have another idea for a blog entry. Last week, Pi day (March 14) marked Einstein's 133^rd birthday, and since my Classical Mechanics course is covering the Special Theory of Relativity, I thought I'd try to cover the basic ideas in blog form.

According to the laws of physics laid down by Sir Isaac Newton, all non-accelerating observers witness the same laws of physics. This included an idea of spontaneity, the idea that someone traveling on the highway at 60 mph would witness an event occur at the exact same time as someone who was just sitting on the side of the highway at rest. The transformation from a reference frame in motion to one at rest for Newtonian physics is known as a Galilean transformation, where x is shifted by -vt, or minus the velocity times time. Under such transformations, laws of physics (like Newton's second law, F = ma, remain invariant (don't change).

However, during the 19^th century, a man by the name James Clerk Maxwell formulated a handful of equations, known now as Maxwell's equations, that outline a theory known as electromagnetic theory. Of the many new insights this theory gleaned (among these the ability to generate electricity for power which every BZP member uses) one was that light is composed of oscillating electric and magnetic fields; light is an electromagnetic wave. By using his newly invented equations, Maxwell discovered what the speed of light was by formulating a wave equation. When his equations are used to describe electromagnetism, the speed of light is shown to be the same regardless of reference frame; in other words, someone traveling near the speed of light (as long as they weren't accelerating) would see light travel at the same speed as someone who was at rest. According to Newton's laws, this didn't make sense! If you're in your car on the highway and traveling at 60 mph while another car in the lane next to you is traveling at 65 mph, you don't see the other car moving at 65 mph; relative to you, the other car moves at 5 mph. The reason that light is different is because a different theory governs its physics.

This brought about a dilemma: is Maxwell's new electromagnetic theory wrong? Or does Newtonian mechanics need some slight revision? This is where Einstein comes in. He noticed the work of another physicist, Lorentz, who had worked on some new transformations that not only caused space to shift based on reference frames moving relative to each other, but also shifted time. Einstein realized that if light had the same speed in all non-accelerating reference frames, then objects moving faster experienced time differently than those that moved slower. This would come to be known as the Special Theory of Relativity.

How does this make sense? Well, if you have some speed that must remain constant no matter how fast one is traveling, you need time to shift in addition to shifting space to convert between both reference frames, since speed is the change in distance over the amount of time that displacement took place. If you have two reference frames with some relative speed between them, the only way to shift your coordinates from one to another and preserve the speed of light is if both frames experience their positions and times differently. This means that, if something moves fast enough, a journey will take less time in one frame than the other. Special relativity says that moving clocks progress more slowly than clocks at rest, so someone traveling in a rocket at a speed comparable to the speed of light will find that the journey took less time than someone who had been anticipating his arrival at rest. This also means that if someone left Earth in a rocket traveling near the speed of light and came back ten years later would not have aged ten years, but would be younger than someone who was his/her age before his journey took place. Weird, huh?

If you think this is crazy or impossible, there have been experiments done (and are still going) to try to confirm/reject the ideas of special relativity, and they all seem to support it. There's another relativity at play as well known as general relativity, which states that gravitational fields affect spacetime (the combination of space and time into one geometry). General relativity says that the higher up you are in a gravitational field, the faster clocks run (time speeds up). A proof of this theory is GPS; the satellites that help find your position by GPS are all higher up in Earth's gravitational field than we are, and thus their clocks run faster than those on Earth's surface. If general relativity weren't considered in the calculations to figure out where you are on Earth, your GPS would be off by miles.

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Complex Numbers

Posted by Akano Toa of Electricity , in Math/Physics Feb 19 2012 · 46 views

math, i, e, pi

Math is a truly wonderful topic, and since I'm procrastinating a little on my physics homework, I'm going to spend some time talking about the complex numbers.

Most of us are used to the real numbers. Real numbers consist of the whole numbers (0, 1, 2, 3, 4, ...), the negative numbers (-1, -2, -3, ...), the rational numbers (1/2, 2/3, 3/4, 22/7, ...), and the irrational numbers (numbers that cannot be represented by fractions of integers, such as the golden ratio, the square root of 2, or π). All of these can be written in decimal format, even though they may have infinite decimal places. But, when we use this number system, there are some numbers we can't write. For instance, what is the square root of -1? In math class, you may have been told that you can't take the square root of a negative number. That's only half true, as you can't take the square root of a negative number and write it as a real number. This is because the square root is not part of the set of real numbers.

This is where the complex numbers come in. Suppose I define a new number, let's call it i, where i² = -1. We've now "invented" a value for the square root of -1. Now, what are its properties? If I take i³, I get -i, since i³ = i*i². If I take i⁴, then I get i²*i² = +1. If I multiply this by i again, I get i. So the powers of i are cyclic through i, -1, -i, and 1.

This is interesting, but what is the magnitude of i, i.e. how far is i from zero? Well, the way we take the absolute value in the real number system is by squaring the number and taking the positive square root. This won't work for i, though, because we just get back i. Let's redefine the absolute value by taking what's called the complex conjugate of i and multiplying the two together, then taking the positive square root. The complex conjugate of i is obtained by taking the imaginary part of i and throwing a negative sign in front. Since i is purely imaginary (there are no real numbers that make up i), the complex conjugate is -i. Multiply them together, and you get that -i*i = -1*i² = 1, and the positive square root of 1 is simply 1. Therefore, the number i has a magnitude of 1. It is for this reason that i is known as the imaginary unit!

Now that we have defined this new unit, i, we can now create a new set of numbers called the complex numbers, which take the form z = a + bi, where a and b are real numbers. We can now take the square root of any real number, e.g. the square root of -4 can be written as ±2i, and we can make complex numbers with real and imaginary parts, like 3 + 4i.

How do we plot complex numbers? Well, complex numbers have a real part and an imaginary part, so the best way to do this is to create a graph where the abscissa (x-value) is the real part of the number and the ordinate (y-axis) is the imaginary part. This is known as the complex plane. For instance, 3 + 4i would have its coordinate be (3,4) in this coordinate system.

What is the magnitude of this complex number? Well, it would be the square root of itself multiplied by its complex conjugate, or the square root of (3 + 4i)(3 - 4i) = 9 + ~~12i~~ - ~~12i~~ +16 = 25. The positive square root of 25 is 5, so the magnitude of 3 + 4i is 5.

We can think of points on the complex plane being represented by a vector which points from the origin to the point in question. The magnitude of this vector is given by the absolute value of the point, which we can denote as r. The x-value of this vector is given by the magnitude multiplied by the cosine of the angle made by the vector with the positive part of the real axis. This angle we can denote as ϕ. The y-value of the vector is going to be the imaginary unit, i, multiplied by the magnitude of the vector times the sine of the angle ϕ. So, we get that our complex number, z, can be written as z = r*(cosϕ + isinϕ). The Swiss mathematician Leonhard Euler discovered a special identity relating to this equation, known now as Euler's Formula, that reads as follows:

e^iϕ = cosϕ + isinϕ

Where e is the base of the natural logarithm. So, we can then write our complex number as z = re^iϕ. What is the significance of this? Well, for one, you can derive one of the most beautiful equations in mathematics, known as Euler's Identity:

e^iπ + 1 = 0

This equation contains the most important constants in mathematics: e, Euler's number, the base of the natural logarithm; i, the imaginary unit which I've spent this whole time blabbing about; π, the irrational ratio of a circle's circumference to its diameter which appears all over the place in trigonometry; 1, the real unit and multiplicative identity; and 0, the additive identity.

So, what bearing does this have in real life? A lot. Imaginary and complex numbers are used in solving many differential equations that model real physical situations, such as waves propagating through a medium, wave functions in quantum mechanics, and fractals, which in and of themselves have a wide range of real life application, along with others that I haven't thought of.

Long and short of it: math is awesome.

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New Comic And Some Maths

Posted by Akano Toa of Electricity , in Math/Physics Oct 22 2011 · 73 views

comic, diverging, numbers, sum and 1 more...

Hi, all,

New comic (and topic) have been posted over in the comics forum! Check it out!

To address the maths in the new comic, the first sum is

1 + 2 + 4 + 8 + ... = Sum(2^n), where n = 0, 1, 2, 3, ... all the way up to infinity. Now, if we look at the partial sums of this series, we see that the sum of the first term is 1, the first two terms is 3, the first three terms is 7, first four terms is 15, etc. Clearly, the sum gets bigger the more terms you add on. However, let's say that we have

s = 1 + 2 + 4 + 8 + 16 + ...

We can rewrite this by using the distributive property,

s = 1 + 2 (1 + 2 + 4 + 8 + ...)

But, what's inside the parentheses is clearly the sum we had before, so we can rewrite this as

s = 1 + 2s

Subtracting s from both sides, we get

0 = 1 + s

Therefore, s = -1 = 1 + 2 + 4 + 8 + ... So, while the sum does not converge to a number in the traditional sense, it still has some other meaning that says that it is equivalent to -1. Similarly,

1 - 2 + 4 - 8 + ... = Sum((-2)^n) where n = 0, 1, 2, 3, ... all the way up to infinity, or

s = 1 - 2 + 4 - 8 + ...

We can again rewrite this as

s = 1 - 2(1 - 2 + 4 - 8 + ...)

What's inside the parentheses is, again, the sum, so

s = 1 - 2s, or 3s = 1. Therefore, s = 1/3 So, while the partial sums (1, -1, 3, -5, 11, ...) get larger in magnitude (with alternating +/- signs), the sum is still on some higher level equal to 1/3. Neat, huh?

If this doesn't make sense, that's okay, because it is confusing and highly mathematical. What I think is cool, though, is that you can still show these two statements using alternative mathematical methods, meaning that these values are consistent with different techniques. It's about as awesome to me as how classical physics comes out of quantum mechanics when you take the limit of quantum mechanics for a "large" (classical) system.

I love math and science. 8D

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About Me

Akano Toa of Electricity
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Name: Akano
Real Name: Forever Shrouded in Mystery
Age: 23
Gender: Male
Likes: Science, Math, LEGO, Bionicle, Ponies, Comics, Yellow, Voice Acting
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning
Misses the 90's. A lot.
Twitter: @akanotoe