I've been trying to solidify my understanding of quantum theory and linear algebra for almost a month now and I've finally gotten it! I was reading Quantum Physics by Gasiorowicz this morning when I read this line: "...we drew attention to the fact that the complete set of eigenfunctions was similar to a set of unit vectors in a vector space, and the expansion theorem was similar to the expansion of an arbitrary vector in terms of unit vectors that span the vector space."
I almost shat a brick. Seriously.
Finally! Once again, months of my physics education clicked in an instant. This statement by itself isn't very complicated, but it's contents are fundamental and deeply profound when things like the uncertainty principle are examined. The uncertainty principle (one version of it, anyway) states that you can't simultaneously know with absolute precision both the position and momentum of a particle. The first reaction to these experimental woes was that somehow the measurement techniques were flawed. When examined from this mathematical perspective, the uncertainty is inherently a part of the system! Now, with just this alone not much can be said. The uncertainty principle could just be a consequence of the model we've constructed, however, there are many other factors involved that have a much more solid physics basis.
All in all, when things like this click in my head I feel as if I've unlocked some new door to the universe. I couldn't focus on much of anything for about half an hour... so profound!
Lab work is going well. I got my Radiological Worker II training, which means I can now work in High Radiation and High Contamination areas. I don't expect to have to do that often, but if I do I know how to put on the crazy radiation suit to do it.
Part of our helium-jet system has a recirculation cart that controls the flow of helium, provides pressure measurements at key points, and also recirculates the helium creating a closed system. Since this cart was shipped in from Canada and hasn't been used in 15 years, we need to test it for leaks. This involves pumping the system to a vacuum and using a leak detector.
The leak detector is essentially a mass spectrometer tuned specifically for helium. You use the detector to pump your system down and use a helium gas cylinder to spray tiny amounts of helium gas over parts you think may be leaking. If any helium gets through, it gets sucked into the detector and it makes this awful beeping noise. I found a pretty huge leak, which is good. However, I can't isolate it from the rest of the system. This is a problem because with such a big leak, I can't get the vacuum to the level I need it at. Because of this, the background noise of the detector can't get low enough for me to detect smaller, albeit problematic, leaks. I'm probably going to have to wait for JJ to get back from India before I can go any further.
I'm also working on a calculation concerning our Argon flow. We have our system set up such that Argon flows out of the flowmeter at a specific rate through about a meter of tubing and into the ion source chamber. Now, the whole reason that things flow is due to pressure differences. Mass flows from areas of high pressure to low pressure. This means that if my Argon is coming out of my flowmeter at a specific rate, but travels through some tubing, there is an additional pressure differential between the flowmeter and the ion source chamber. There is a formula that relates the pressure drop between to reservoirs and the associated flow rate. What I need to figure out is how our output flow rate will be affected by traveling through this tubing via the pressure differential. My problem has been with converting between different types of flow.
The flow that I know isn't actually a mass flow. It's in unites of Joule/second, but it's written as pressure*volume/second. What I need for the formula is volumetric flow rate: volume/second. So I've got to look at a lot of factors when converting between the two, like the relationship between density and pressure/temperature. Since I'm not expecting a huge pressure drop between the flowmeter and the chamber (since our flow rate is so small), I'm going to assume the Argon is at the pressure of the chamber and go from there. Now I just need to find a pressure/density relationship.
I'll be Atlanta inbound about 4 hours from now. Another chapter of my fraternity is visiting, so I'm stoked to see those guys. There's also a party at CoLab on Saturday that I'm really looking forward to. All in all, an exciting weekend awaits!
Friday, February 19, 2010
Tuesday, February 16, 2010
First shot at Group Theory
Ok, so I've been tossing this around ye olde noggin for a few days now. I've been reading about elementary group theory lately and I've been fascinated by it's implications. I've never really cared too much about abstract mathematics since this. I think the reason why I'm loving it so far is that it isn't buried in obscure jargon (not yet, anyway) and it's core principles are based solely on logic. The very basics of the theory aren't buried in a ton of numbers and odd greek letters.
After reading about a few simple number groups, and a few more abstract logical ones, I've tried to come up with a group concerning the logic of something I see in every day life. I chose how I think I could model the process involved in governing the final temperature of the water as it hits you. I, most appropriately, got this idea while I was in the shower. I'm still trying to work through the details but I think I might have something.
First, how I think the temperature is governed. It's not that big of a deal if this part is way off... what's important is the logic it follows. The idea is if such a system existed, then I would have, in essence, modeled a real-life system in an alarmingly mathematical way.
You have two shower knobs, hot and cold. A few notes about the knobs. There is some number of "ticks" you can make before the knob has gone from fully closed to fully open, i.e. a discreet system. The knob may feel continous, but at the smallest level it could be billions of tiny little steps. Imagine a really loooooong staircase with tiny steps. Anyway, each knob has a certain number of turns. Each knob can only send a single temperature of water to the final spiget. By turning the knob the intensity of the water is increased or decreased. Positive numbers denote hot water intensities and negative numbers denote cold water intensities. If we had a pair of knobs with 3 settings, they would be enumerated like this: -3, -2, -1, 1, 2, 3.
That's just how the water works. I now need to decide my set. My set consist of all possible final temperatures, illustrated as the set of all possible pairs of hot and cold temperatures. I won't list them all, but some possible pairs would be (-2, -2), (-1, 1), (1, -4), etc. The operation would "turning the knob", akin to adding the components of two pairs (-1, 2) + (-1, 1) = (1, 3).
To illustrate the group operation, if I turn the hot knob a lot and the cold knob just a little, the final temperature will be pretty hot, but not quite the hottest it can be. Now, I need to check my 4 rules and this is where I'm a bit iffy.
I'll start with the ones I think I've figured out.
1) Closure under the group operation.
Any combination of knob turns will reside within the realm of possible temperatures (since there are finite turns of each knob).
2) Associativity:
The order in which you turn the knobs doesn't matter. Any hot cold followed by a cold turn is the same were it reversed. -1 + 2 = 2 + -1 = 1
3) Existance of the identity element:
If I simply "do not turn", then the temperature remains unchanged.
4) Existence of the inverse:
Any combination of each knob that's at the same intensity yields the same exact temperature (felt as warm). 3 + -3 = 0.
However, this is wrong. Here's the zinger! I've modeled this system, under all the logic, based on a set of numbers (-3, -2, -1, 1, 2, 3) with the operation of simple addition. If you examine this from a purely mathematical standpoint (dealing with numbers only), the first rule is broken. If you add a positive and negative number that are the same intensity, you get zero. -3 + -3 = 0. The number 0 is NOT part of the original set of numbers, so this proposed group breaks closure and is therefore not a group. What's awesome is that the logical way that I've defined my group (with shower turns, and knobs) also fails BECAUSE they are the same exact group! These two concepts, on the most general level, are exactly the same in that their group tables are identical! The reason they both fail as groups is because there are more possible elements than should be allowed! -3 + -3 = 0; if you never turn the knobs, the temperature is undefined. These two fake-groups are homomorphisms! (I think?)
There are logical fallacies in the other rules, but I won't go into them.
I think I've exhausted my mental power for now. I shall continue to try to find my group.
EDIT: When trying to visualize the "group table", imagine the multiplication tables you saw in elementary schools.
After reading about a few simple number groups, and a few more abstract logical ones, I've tried to come up with a group concerning the logic of something I see in every day life. I chose how I think I could model the process involved in governing the final temperature of the water as it hits you. I, most appropriately, got this idea while I was in the shower. I'm still trying to work through the details but I think I might have something.
First, how I think the temperature is governed. It's not that big of a deal if this part is way off... what's important is the logic it follows. The idea is if such a system existed, then I would have, in essence, modeled a real-life system in an alarmingly mathematical way.
You have two shower knobs, hot and cold. A few notes about the knobs. There is some number of "ticks" you can make before the knob has gone from fully closed to fully open, i.e. a discreet system. The knob may feel continous, but at the smallest level it could be billions of tiny little steps. Imagine a really loooooong staircase with tiny steps. Anyway, each knob has a certain number of turns. Each knob can only send a single temperature of water to the final spiget. By turning the knob the intensity of the water is increased or decreased. Positive numbers denote hot water intensities and negative numbers denote cold water intensities. If we had a pair of knobs with 3 settings, they would be enumerated like this: -3, -2, -1, 1, 2, 3.
That's just how the water works. I now need to decide my set. My set consist of all possible final temperatures, illustrated as the set of all possible pairs of hot and cold temperatures. I won't list them all, but some possible pairs would be (-2, -2), (-1, 1), (1, -4), etc. The operation would "turning the knob", akin to adding the components of two pairs (-1, 2) + (-1, 1) = (1, 3).
To illustrate the group operation, if I turn the hot knob a lot and the cold knob just a little, the final temperature will be pretty hot, but not quite the hottest it can be. Now, I need to check my 4 rules and this is where I'm a bit iffy.
I'll start with the ones I think I've figured out.
1) Closure under the group operation.
Any combination of knob turns will reside within the realm of possible temperatures (since there are finite turns of each knob).
2) Associativity:
The order in which you turn the knobs doesn't matter. Any hot cold followed by a cold turn is the same were it reversed. -1 + 2 = 2 + -1 = 1
3) Existance of the identity element:
If I simply "do not turn", then the temperature remains unchanged.
4) Existence of the inverse:
Any combination of each knob that's at the same intensity yields the same exact temperature (felt as warm). 3 + -3 = 0.
However, this is wrong. Here's the zinger! I've modeled this system, under all the logic, based on a set of numbers (-3, -2, -1, 1, 2, 3) with the operation of simple addition. If you examine this from a purely mathematical standpoint (dealing with numbers only), the first rule is broken. If you add a positive and negative number that are the same intensity, you get zero. -3 + -3 = 0. The number 0 is NOT part of the original set of numbers, so this proposed group breaks closure and is therefore not a group. What's awesome is that the logical way that I've defined my group (with shower turns, and knobs) also fails BECAUSE they are the same exact group! These two concepts, on the most general level, are exactly the same in that their group tables are identical! The reason they both fail as groups is because there are more possible elements than should be allowed! -3 + -3 = 0; if you never turn the knobs, the temperature is undefined. These two fake-groups are homomorphisms! (I think?)
There are logical fallacies in the other rules, but I won't go into them.
I think I've exhausted my mental power for now. I shall continue to try to find my group.
EDIT: When trying to visualize the "group table", imagine the multiplication tables you saw in elementary schools.
Tuesday, February 9, 2010
Blah blah
These last two weeks have been rather boring. I've been mostly dealing with logistical stuff and trying to get this order from Matheson in. It's amazing that they make any money given the quality of their customer service. It's up in the air whether someone will ever answer the phone and god forbid I expect them to call or email me back. I wonder if Einstein's progress was ever halted because some bozo salesperson wouldn't call him back?
I've also been reading about something called outgassing. It turns out that when creating a vacuum, getting the molecules in the open air within your volume is the easiest part. The difficult part, then, is getting the molecules (like water vapor) that have absorbed into your surface. Water is a particular notorious substance. To better understand why, it's best to learn how evaporation works.
Evaporation is a process in which a substance in its liquid form transforms to it's gaseous form. Lots of factors effect the rate of evaporation off a surface including surrounding pressure, temperature, surface area, etc. For a water molecule to evaporate off of your head after a shower, it has to acquire enough kinetic energy (i.e., heat transfer) to overcome the tensile forces of the surrounding water molecules. This energy transfer mostly happens due to collisions with other molecules. This is why things dry faster when you heat them up; the extra heat you supply allows more intense collisions to happen which, in turn, allow more molecules to desorb (opposite of absorb) off of the surface. You can also lower the pressure of the surface's surroundings, which lowers the amount of exertion on the surface and lowers the required amount of kinetic energy to escape.
It's the lowering of the pressure of the surface's surroundings that we're interested in. Creating a vacuum is just that process: lowering the pressure in a sealed chamber. As soon as you get to even a modest vacuum, water starts to fly off of these surfaces at a fairly rapid rate. Thankfully, baking your surface at around 150 degrees C solves this problem wonderfully in that it greatly increases the rate of evaporation. Most reasonably sized surfaces can be made acceptably water free in an hour or two.
That is just one example of what is considered outgassing. Other gaseous molecules in the air, like Nitrogen or Argon, can absorb into metals. Baking is the most common method to accelerate outgassing, but there are other methods like electron-stimulation, ion-stimulation, photodesorption, etc. All of these methods use some special process to add kinetic energy to the molecules absorbed on the surface in question and allow them to evaporate off faster.
Star Trek Online is super fun. I'm finally being able to live out my fantasy of being a starship captain. I wish voice recognition software was a few decades ahead of its time so that I could shout "EMERGENCY POWER TO SHIELDS! FIRE A HIGH YIELD PLASMA TORPEDO! HARD TO PORT, HARD TO PORT!" instead of clicking it. Seriously, this game is like a dream come true. Sadly it's interfering with my Mass Effect 2 time (which is also super ballin). At least it will let me stretch the experience out a month or so.
I've also been reading about something called outgassing. It turns out that when creating a vacuum, getting the molecules in the open air within your volume is the easiest part. The difficult part, then, is getting the molecules (like water vapor) that have absorbed into your surface. Water is a particular notorious substance. To better understand why, it's best to learn how evaporation works.
Evaporation is a process in which a substance in its liquid form transforms to it's gaseous form. Lots of factors effect the rate of evaporation off a surface including surrounding pressure, temperature, surface area, etc. For a water molecule to evaporate off of your head after a shower, it has to acquire enough kinetic energy (i.e., heat transfer) to overcome the tensile forces of the surrounding water molecules. This energy transfer mostly happens due to collisions with other molecules. This is why things dry faster when you heat them up; the extra heat you supply allows more intense collisions to happen which, in turn, allow more molecules to desorb (opposite of absorb) off of the surface. You can also lower the pressure of the surface's surroundings, which lowers the amount of exertion on the surface and lowers the required amount of kinetic energy to escape.
It's the lowering of the pressure of the surface's surroundings that we're interested in. Creating a vacuum is just that process: lowering the pressure in a sealed chamber. As soon as you get to even a modest vacuum, water starts to fly off of these surfaces at a fairly rapid rate. Thankfully, baking your surface at around 150 degrees C solves this problem wonderfully in that it greatly increases the rate of evaporation. Most reasonably sized surfaces can be made acceptably water free in an hour or two.
That is just one example of what is considered outgassing. Other gaseous molecules in the air, like Nitrogen or Argon, can absorb into metals. Baking is the most common method to accelerate outgassing, but there are other methods like electron-stimulation, ion-stimulation, photodesorption, etc. All of these methods use some special process to add kinetic energy to the molecules absorbed on the surface in question and allow them to evaporate off faster.
Star Trek Online is super fun. I'm finally being able to live out my fantasy of being a starship captain. I wish voice recognition software was a few decades ahead of its time so that I could shout "EMERGENCY POWER TO SHIELDS! FIRE A HIGH YIELD PLASMA TORPEDO! HARD TO PORT, HARD TO PORT!" instead of clicking it. Seriously, this game is like a dream come true. Sadly it's interfering with my Mass Effect 2 time (which is also super ballin). At least it will let me stretch the experience out a month or so.
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