8
Akano's Blog
Posted by
Akano
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Video Games
Aug 09 2016
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377 views
Zelda, Gannon banned, Link and 2 more...
Equation of the Day #17: The Rydberg Formula
Posted by
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Math/Physics
Aug 04 2016
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500 views
Hydrogen, Atoms, Molecules and 2 more...
Hydrogen is the simplest and most common neutral atom in the universe. It consists of two particles – a positively charged proton and a negatively charged electron. The equation that describes the hydrogen atom (or any oneelectron atom) in the nonrelativistic regime is the Schrödinger equation, specifically
where ħ is the reduced Planck constant, μ is the reduced mass of the electronnucleus system, Z is the number of positive charges in the nucleus that the electron is orbiting, e is the charge of a proton, τ is the circle constant, ε_{0} is the vacuum permittivity, and ψ is the wavefunction. Solving this equation (which is nontrivial and is usually done after a semester of Advanced Quantum Mechanics) yields a surprisingly simple formula for the energies of the atom,
where h is Planck's constant, c is the speed of light, m_{e} is the rest mass of the electron, and n is any integer larger than or equal to 1. The constant R_{∞} is known as the Rydberg constant, named after Swedish physicist Johannes Rydberg, the scientist who discovered a formula to predict the specific colors of light hydrogen (or any hydrogenlike atom) would absorb or emit. Indeed, the formula I gave, E_{n}/hc, is equivalent to the inverse wavelength, or spatial frequency, of light that it takes for the atom in its n^{th} energy state to free the electron of its atomic bond. Indeed, this was a puzzle in the early 20^{th} century. Why was it that hydrogen (and other atoms) only absorbed and emitted specific colors of light? White light, as Isaac Newton showed, is comprised of all visible colors of light, and when you split up that light using a prism or similar device, you get a continuous rainbow. This was not the case for light emitted or absorbed by atoms.
The equation above was first derived by Niels Bohr, who approached solving this problem not from using the Schrödinger equation, but from looking at the electron's angular momentum. If electrons could be considered wavelike, as quantum mechanics treats them, then he figured that the orbits of the electron must be such that an integer number of electron wavelengths fit along the orbit.
This condition requires that
The wavelength of the electron is inversely related to its momentum, p = mv, via Planck's constant, λ = h/p. The other relation we need is from the physics of circular motion, which says that the centripetal force on an object moving in a circular path of radius r is mv^{2}/r. Equating this to the Coulomb force holding the proton and electron together, we get
Plugging this into the quantization condition, along with some algebra, yields the energy equation.
What's incredible is that hydrogen's energy spectrum has a closedform solution, since most problems in physics can't be solved to produce such solutions, and while this equation only works exactly for oneelectron atoms, it can be modified to work for socalled Rydberg atoms and molecules, where a single electron is highly excited (large n) and orbits a positive core, which need not be a nucleus, but a nonpointlike structure. In my lab, we consider two types of Rydberg molecules.
The example on the left is an electronic Rydberg molecule, while the one on the right is called an ionpair Rydberg state, where a negative ion acts as a "heavy electron" coorbiting a positive ion. To model the energies of these kinds of states, we use a modified energy equation.
where I.P. represents the ionization energy of the electron, and the new quantity δ is known as the quantum defect. It's a number that, for electronic Rydberg states, has a magnitude that's usually less than 1, while for ionpair states can be quite large (around –60 or so in some cases); it in some sense contains information of how the core ion, e.g. H_{2}^{+}, is oriented, how the electron is spread over space, how its polarized, and so on. It's a vessel into which we funnel our ignorance in using the approximation that the molecule is behaving in a hydrogenlike manner, and it is surprisingly useful in predicting experiments. Currently my research involves studying electronic Rydberg states of molecular nitrogen, N_{2}, and looking at heavy Rydberg states of the hydrogen molecule, H_{2} to gain a better understanding of the physics of certain states that have been experimentally observed in both systems.
where ħ is the reduced Planck constant, μ is the reduced mass of the electronnucleus system, Z is the number of positive charges in the nucleus that the electron is orbiting, e is the charge of a proton, τ is the circle constant, ε_{0} is the vacuum permittivity, and ψ is the wavefunction. Solving this equation (which is nontrivial and is usually done after a semester of Advanced Quantum Mechanics) yields a surprisingly simple formula for the energies of the atom,
,
where h is Planck's constant, c is the speed of light, m_{e} is the rest mass of the electron, and n is any integer larger than or equal to 1. The constant R_{∞} is known as the Rydberg constant, named after Swedish physicist Johannes Rydberg, the scientist who discovered a formula to predict the specific colors of light hydrogen (or any hydrogenlike atom) would absorb or emit. Indeed, the formula I gave, E_{n}/hc, is equivalent to the inverse wavelength, or spatial frequency, of light that it takes for the atom in its n^{th} energy state to free the electron of its atomic bond. Indeed, this was a puzzle in the early 20^{th} century. Why was it that hydrogen (and other atoms) only absorbed and emitted specific colors of light? White light, as Isaac Newton showed, is comprised of all visible colors of light, and when you split up that light using a prism or similar device, you get a continuous rainbow. This was not the case for light emitted or absorbed by atoms.
The equation above was first derived by Niels Bohr, who approached solving this problem not from using the Schrödinger equation, but from looking at the electron's angular momentum. If electrons could be considered wavelike, as quantum mechanics treats them, then he figured that the orbits of the electron must be such that an integer number of electron wavelengths fit along the orbit.
Left: Allowed orbit. Right: Disallowed orbit. Image: Wikimedia commons
This condition requires that
The wavelength of the electron is inversely related to its momentum, p = mv, via Planck's constant, λ = h/p. The other relation we need is from the physics of circular motion, which says that the centripetal force on an object moving in a circular path of radius r is mv^{2}/r. Equating this to the Coulomb force holding the proton and electron together, we get
Plugging this into the quantization condition, along with some algebra, yields the energy equation.
What's incredible is that hydrogen's energy spectrum has a closedform solution, since most problems in physics can't be solved to produce such solutions, and while this equation only works exactly for oneelectron atoms, it can be modified to work for socalled Rydberg atoms and molecules, where a single electron is highly excited (large n) and orbits a positive core, which need not be a nucleus, but a nonpointlike structure. In my lab, we consider two types of Rydberg molecules.
The example on the left is an electronic Rydberg molecule, while the one on the right is called an ionpair Rydberg state, where a negative ion acts as a "heavy electron" coorbiting a positive ion. To model the energies of these kinds of states, we use a modified energy equation.
where I.P. represents the ionization energy of the electron, and the new quantity δ is known as the quantum defect. It's a number that, for electronic Rydberg states, has a magnitude that's usually less than 1, while for ionpair states can be quite large (around –60 or so in some cases); it in some sense contains information of how the core ion, e.g. H_{2}^{+}, is oriented, how the electron is spread over space, how its polarized, and so on. It's a vessel into which we funnel our ignorance in using the approximation that the molecule is behaving in a hydrogenlike manner, and it is surprisingly useful in predicting experiments. Currently my research involves studying electronic Rydberg states of molecular nitrogen, N_{2}, and looking at heavy Rydberg states of the hydrogen molecule, H_{2} to gain a better understanding of the physics of certain states that have been experimentally observed in both systems.
So, I decided to take the Pottermore Sorting quiz with all questions available on some third party quiz site, because while I love the Thunderbird, the house's traits do not really fit me (I can be adventurous, but let's face it: I'm a schooler). So, I took the full Pottermore quiz, and my results were somewhat predictable.
Indeed, Horned Serpent is my Ilvermorny house by a decent margin, with my second place house being Pukwudgie, then Wampus, and Thunderbird being my least compatible house. (lolololololol)
My Hogwarts house, however, is not Ravenclaw, despite my scholarly ways. I'm a Gryffindor. Ravenclaw was indeed my next most compatible, followed by Hufflepuff and finally, by a large margin, Slytherin.
These results make much more sense to me. Remember, kids, larger sample sizes are better.
Indeed, Horned Serpent is my Ilvermorny house by a decent margin, with my second place house being Pukwudgie, then Wampus, and Thunderbird being my least compatible house. (lolololololol)
My Hogwarts house, however, is not Ravenclaw, despite my scholarly ways. I'm a Gryffindor. Ravenclaw was indeed my next most compatible, followed by Hufflepuff and finally, by a large margin, Slytherin.
These results make much more sense to me. Remember, kids, larger sample sizes are better.
I had a whirlwind of a time these last few weeks. I'll try to break it down simply.
Week of May 23: Attended the DAMOP conference in Providence, RI to present my research in poster form. It was all right, but probably my least favorite DAMOP thus far. (Last year it was in Columbus, OH, which meant I got to see friends during the week, and the year before it was in Madison, WI, which was an absolute joy because Madison is a rarity in that it's a city I actually kinda like.) My roommate and I left the conference on Thursday to attend a wedding, which happened Thursday evening. The ceremony was one of my favorites I've attended thus far.
The following day I left on a plane to go to Columbus to attend the bachelor party, rehearsal, and subsequent wedding of a friend I've known and kept in touch with since 6th grade. It was wonderful.
Week of May 30: Went back to my childhood home and had a Memorial Day dinner with family and friends, played the role of babysitter with Tekulo (we earned major brownie points, both with the kids and their mothers). The next day I chilled at home and spent time with my family, with the evening punctuated by an awesome bonfire that was probably against local fire codes. Saw my next door neighbor unexpectedly and got a chance to catch up. Wednesday featured more catching up with some college friends who live about 40 minutes away from my parents' place. We played some Harry Potter Trivial Pursuit and watched YouTube videos.
Thursday was the beginning of a roadtrip from Ohio to Wisconsin, with the ultimate goal to meet up with KK and hang out for a few days. Among the people we saw along the way were two of my fellow grad students who were working in Chicago, my roommate's (Friend: Toa of Friendship) cousin, KK, and friends of Friend: Toa of Friendship's girlfriend (did you keep track of that? ). It was awesome, but rather short, and we'll definitely have to do something again.
I got back to my apartment Wednesday night and am now enjoying time off relaxing before I go back to the lab on Monday to start up summer research mode! Huzzah!
Week of May 23: Attended the DAMOP conference in Providence, RI to present my research in poster form. It was all right, but probably my least favorite DAMOP thus far. (Last year it was in Columbus, OH, which meant I got to see friends during the week, and the year before it was in Madison, WI, which was an absolute joy because Madison is a rarity in that it's a city I actually kinda like.) My roommate and I left the conference on Thursday to attend a wedding, which happened Thursday evening. The ceremony was one of my favorites I've attended thus far.
The following day I left on a plane to go to Columbus to attend the bachelor party, rehearsal, and subsequent wedding of a friend I've known and kept in touch with since 6th grade. It was wonderful.
Week of May 30: Went back to my childhood home and had a Memorial Day dinner with family and friends, played the role of babysitter with Tekulo (we earned major brownie points, both with the kids and their mothers). The next day I chilled at home and spent time with my family, with the evening punctuated by an awesome bonfire that was probably against local fire codes. Saw my next door neighbor unexpectedly and got a chance to catch up. Wednesday featured more catching up with some college friends who live about 40 minutes away from my parents' place. We played some Harry Potter Trivial Pursuit and watched YouTube videos.
Thursday was the beginning of a roadtrip from Ohio to Wisconsin, with the ultimate goal to meet up with KK and hang out for a few days. Among the people we saw along the way were two of my fellow grad students who were working in Chicago, my roommate's (Friend: Toa of Friendship) cousin, KK, and friends of Friend: Toa of Friendship's girlfriend (did you keep track of that? ). It was awesome, but rather short, and we'll definitely have to do something again.
I got back to my apartment Wednesday night and am now enjoying time off relaxing before I go back to the lab on Monday to start up summer research mode! Huzzah!
So much more enjoyable. I'm on Chapter 10 of Conquest (decided to switch it up), and already I feel much more positive. The characters have fun quirks and interactions, the plot is interesting, and I'm invested in finding out what happens next.
Glad that this version is my physical copy.
Glad that this version is my physical copy.
Thoughts on Fates: Birthright
Posted by
Akano
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Video Games
Mar 18 2016
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427 views
Fire Emblem, Fates, Birthright and 2 more...
I've been playing Fire Emblem Fates for the last couple weeks on Hard, Classic mode. I'm currently on Chapter 15: Wolfskin of Birthright (got the Conquest cartridge, bought the other two as DLC). My thoughts so far:
 Story: Eh. It should be good; it has all the prerequisites to be good. You make a choice between two warring families (which puts tension on everything and everyone), the history between Nohr and Hoshido seems complex and longstanding, the circumstances of your upbringing are frustrating/unfair, but all in all the story just seems to happen. It has so many things ready to set itself in motion, but instead is very stilted and lacks flow.
 Characters: Hit and miss for me. Some characters are phenomenal (like Kaden, he's fantastic. Also, Jakob is so wonderfully British. I mean, Nohrian.), while others are just sorta meh. Also, Azura's A support with the female Avatar is the least satisfying support ever. They could have made it a lot more interesting/developing. This and the previous category have made me think to myself, "Man, I just want to play Awakening/Blazing Sword," which doesn't reflect well on this game.
 Classes: Interesting/why? There are some new Hoshido classes that I think are cool (Kinshi knight!) and others that I prefer the classic classes to (Diviner << Mage; I really don't like Orochi and her horoscope nonsense. Mages at least learn their magic from study, not dubious astrology pseudoscience...). I think I'll enjoy playing Conquest a bit more for this reason, among others.
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About Me
Akano
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Name: Akano
Real Name: Forever Shrouded in Mystery
Age: 29
Gender: Male
Likes: Science, Math, LEGO, Bionicle, Comics, Yellow, Voice Acting, Pixel Art, Video Games
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning
Twitter: @akanotoe
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