formulasearchengine - User contributions [en]

Apollo 9

2014-02-23T14:52:50Z

79.180.10.9: /* Mission highlights */ Happy birthday was removed- not relevant to story

Four or five years ago, a reader of some of my columns bought the domain name jamesaltucher.com and gave it to me as a birthday gift. It was a total surprise to me. I didn't even know the reader. I hope one day we meet. Two years ago a friend of mine, Tim Sykes, insisted I had to have a blog. He set it up for me. He even wrote the "About Me". I didn't want a blog. I had nothing to say. But about 6 or 7 months ago I decided I wanted to take this blog seriously. I kept putting off changing the "About Me" which was no longer really about me and maybe never was. A few weeks ago I did a chapter in one of the books in Seth Godin's "The Domino Project". The book is out and called "No Idling". Mohit Pewar organized it (here's Mohit's blog) and sent me a bunch of questions recently. It's intended to be an interview on his blog but I hope Mohit forgives me because I want to use it as my new "About Me" also. 1. You are a trader, investor, writer, and entrepreneur? Which of these roles you enjoy the most and why? When I first moved to New York City in 1994 I wanted to be everything to everyone. I had spent the six years prior to that writing a bunch of unpublished novels and unpublished short stories. I must've sent out 100s of stories to literary journals. I got form rejections from every publisher, journal, and agent I sent my novels and stories to. Now, in 1994, everything was possible. The money was in NYC. Media was here. I lived in my 10�10 room and pulled suits out of a garbage bag every morning but it didn't matter...the internet was revving up and I knew how to build a website. One of the few in the city. My sister warned me though: nobody here is your friend. Everybody wants something 
And I wanted something. I wanted the fleeting feelings of success, for the first time ever, in order to feel better about myself. I wanted a girl next to me. I wanted to build and sell companies and finally prove to everyone I was the smartest. I wanted to do a TV show. I wanted to write books 
But everything involved having a master. Clients. Employers. Investors. Publishers. The market (the deadliest master of all). Employees. I was a slave to everyone for so many years. And the more shackles I had on, the lonelier I got 
Much of the time, even when I had those moments of success, I didn't know how to turn it into a better life. I felt ugly and then later, I felt stupid when I would let the success dribble away down the sink 
I love writing because every now and then that ugliness turns into honesty. When I write, I'm only a slave to myself. When I do all of those other things you ask about, I'm a slave to everyone else 
Some links 
33 Unusual Tips to Being a Better Write 
"The Tooth 
(one of my favorite posts on my blog 
2. What inspires you to get up and start working/writing every day 
The other day I had breakfast with a fascinating guy who had just sold a piece of his fund of funds. He told me what "fracking" was and how the US was going to be a major oil player again. We spoke for two hours about a wide range of topics, including what happens when we can finally implant a google chip in our brains 
After that I had to go onto NPR because I firmly believe that in one important respect we are degenerating as a country - we are graduating a generation of indentured servants who will spend 50 years or more paying down their student debt rather than starting companies and curing cancer. So maybe I made a difference 
Then I had lunch with a guy I hadn't seen in ten years. In those ten years he had gone to jail and now I was finally taking the time to forgive him for something he never did to me. I felt bad I hadn't helped him when he was at his low point. Then I came home and watched my kid play clarinet at her school. Then I read until I fell asleep. Today I did nothing but write. Both days inspired me 
It also inspires me that I'm being asked these questions. Whenever anyone asks me to do anything I'm infinitely grateful. Why me? I feel lucky. I like it when someone cares what I think. I'll write and do things as long as anyone cares. I honestly probably wouldn't write if nobody cared. I don't have enough humility for that, I'm ashamed to admit 
3. Your new book "How to be the luckiest person alive" has just come out. What is it about 
When I was a kid I thought I needed certain things: a college education from a great school, a great home, a lot of money, someone who would love me with ease. I wanted people to think I was smart. I wanted people to think I was even special. And as I grew older more and more goals got added to the list: a high chess rating, a published book, perfect weather, good friends, respect in various fields, etc. I lied to myself that I needed these things to be happy. The world was going to work hard to give me these things, I thought. But it turned out the world owed me no favors 
And gradually, over time, I lost everything I had ever gained. Several times. I've paced at night so many times wondering what the hell was I going to do next or trying not to care. The book is about regaining your sanity, regaining your happiness, finding luck in all the little pockets of life that people forget about. It's about turning away from the religion you've been hypnotized into believing into the religion you can find inside yourself every moment of the day 
[Note: in a few days I'm going to do a post on self-publishing and also how to get the ebook for free. The link above is to the paperback. Kindle should be ready soon also. 
Related link: Why I Write Books Even Though I've Lost Money On Every Book I've Ever Writte 
4. Is it possible to accelerate success? If yes, how 
Yes, and it's the only way I know actually to achieve success. It's by following the Daily Practice I outline in this post: 
It's the only way I know to [http://photobucket.com/images/exercise exercise] every muscle from the inside of you to the outside of you. I firmly believe that happiness starts with that practice 
5. You say that discipline, persistence and psychology are important if one has to achieve success. How can one work on improving "psychology" part 
Success doesn't really mean anything. People want to be happy in a harsh and unforgiving world. It's very difficult. We're so lucky most of us live in countries without major wars. Our kids aren't getting killed by random gunfire. We all have cell phones. We all can communicate with each other on the Internet. We have Google to catalog every piece of information in history! We are so amazingly lucky already 
How can it be I was so lucky to be born into such a body? In New York City of all places? Just by being born in such a way on this planet was an amazing success 
So what else is there? The fact is that most of us, including me, have a hard time being happy with such ready-made success. We quickly adapt and want so much more out of life. It's not wars or disease that kill us. It's the minor inconveniences that add up in life. It's the times we feel slighted or betrayed. Or even slightly betrayed. Or overcharged. Or we miss a train. Or it's raining today. Or the dishwasher doesn't work. Or the supermarket doesn't have the food we like. We forget how good the snow tasted when we were kids. Now we want gourmet food at every meal 
Taking a step back, doing the Daily Practice I outline in the question above. For me, the results of that bring me happiness. That's success. Today. And hopefully tomorrow 
6. You advocate not sending kids to college. What if kids grow up and then blame their parents about not letting them get a college education 
I went to one of my kid's music recitals yesterday. She was happy to see me. I hugged her afterwards. She played "the star wars theme" on the clarinet. I wish I could've played that for my parents. My other daughter has a dance recital in a few weeks. I tried to give her tips but she laughed at me. I was quite the breakdancer in my youth. The nerdiest breakdancer on the planet. I want to be present for them. To love them. To let them always know that in their own dark moments, they know I will listen to them. I love them. Even when they cry and don't always agree with me. Even when they laugh at me because sometimes I act like a clown 
Later, if they want to blame me for anything at all then I will still love them. That's my "what if" 
Two posts 
I want my daughters to be lesbian 
Advice I want to give my daughter 

7. Four of your favorite posts from The Altucher Confidential 
As soon as I publish a post I get scared to death. Is it good? Will people re-tweet? Will one part of the audience of this blog like it at the expense of another part of the audience. Will I get Facebook Likes? I have to stop clinging to these things but you also need to respect the audience. I don't know. It's a little bit confusing to me. I don't have the confidence of a real writer yet 
Here are four of my favorites 
How I screwed Yasser Arafat out of $2mm (and lost another $100mm in the process 
It's Your Fault 
I'm Guilty of Torturing Wome 
The Girl Whose Name Was a Curs 
Although these three are favorites I really don't post anything unless it's my favorite of that moment 
8. 3 must-read books for aspiring entrepreneurs 
The key in an entrepreneur book: you want to learn business. You want to learn how to honestly communicate with your customers. You want to stand out 
The Essays of Warren Buffett by Lawrence Cunningha 
"The Thank you Economy" by Gary Vaynerchu 
"Purple cow" by Seth Godi 
9. I love your writing, so do so many others out there. Who are your favorite writers 
"Jesus's Son" by Denis Johnson is the best collection of short stories ever written. I'm afraid I really don't like his novels though 
"Tangents" by M. Prado. A beautiful series of graphic stories about relationships 
Other writers: Miranda July, Ariel Leve, Mary Gaitskill, Charles Bukowski, [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bags Outlet], Sam Lipsyte, William Vollmann, Raymond Carver. Arthur Nersesian. Stephen Dubner 
Many writers are only really good storytellers. Most writers come out of a cardboard factory MFA system and lack a real voice. A real voice is where every word exposes ten levels of hypocrisy in the world and brings us all the way back to see reality. The writers above have their own voices, their own pains, and their unique ways of expressing those pains. Some of them are funny. Some a little more dark. I wish I could write 1/10 as good as any of them 
10. You are a prolific writer. Do you have any hacks that help you write a lot in little time 
Coffee, plus everything else coffee does for you first thing in the morning 
Only write about things you either love or hate. But if you hate something, try to find a tiny gem buried in the bag of dirt so you can reach in when nobody is looking and put that gem in your pocket. Stealing a diamond in all the shit around us and then giving it away for free via writing is a nice little hack, Being fearless precisely when you are most scared is the best hack 
11. I totally get and love your idea about bleeding as a writer, appreciate if you share more with the readers of this blog 
Most people worry about what other people think of them. Most people worry about their health. Most people are at a crossroads and don't know how to take the next step and which road to take it on. Everyone is in a perpetual state of 'where do I put my foot next'. Nobody, including me, can avoid that 
You and I both need to wash our faces in the morning, brush our teeth, shower, shit, eat, fight the weather, fight the colds that want to attack us if we're not ready. Fight loneliness or learn how to love and appreciate the people who want to love you back. And learn how to forgive and love the people who are even more stupid and cruel than we are. We're afraid to tell each other these things because they are all both disgusting and true 
You and I both have the same color blood. If I cut my wrist open you can see the color of my blood. You look at it and see that it's the same color as yours. We have something in common. It doesn't have to be shameful. It's just red. Now we're friends. No matter whom you are or where you are from. I didn't have to lie to you to get you to be my friend 
Related Links 
How to be a Psychic in Ten Easy Lesson 
My New Year's Resolution in 199 
12. What is your advice for young entrepreneurs 
Only build something you really want to use yourself. There's got to be one thing you are completely desperate for and no matter where you look you can't find it. Nobody has invented it yet. So there you go - you invent it. If there's other people like you, you have a business. Else. You fail. Then do it again. Until it works. One day it will 
Follow these 100 Rules 
The 100 Rules for Being a Good Entrepreneur 
And, in particular this 
The Easiest Way to Succeed as an Entrepreneu 
In my just released book I have more chapters on my experiences as an entrepreneur 
13. I advocate the concept of working at a job while building your business. You have of course lived it. Now as you look back, what is your take on this? Is it possible to make it work while sailing on two boats 
Your boss wants everything out of you. He wants you to work 80 hours a week. He wants to look good taking credit for your work. He wants your infinite loyalty. So you need something back 
Exploit your employer. It's the best way to get good experience, clients, contacts. It's a legal way to steal. It's a fast way to be an entrepreneur because you see what large companies with infinite money are willing to pay for. If you can provide that, you make millions. It's how many great businesses have started and will always start. It's how every exit I've had started 
14. Who is a "person with true moral fiber"? In current times are there any role models who are people with true moral fiber 
I don't really know the answer. I think I know a few people like that. I hope I'm someone like that. And I pray to god the people I'm invested in are like that and my family is like that 
I find most people to be largely mean and stupid, a vile combination. It's not that I'm pessimistic or cynical. I'm very much an optimist. It's just reality. Open the newspaper or turn on the TV and watch these people 
Moral fiber atrophies more quickly than any muscle on the body. An exercise I do every morning is to promise myself that "I'm going to save a life today" and then leave it in the hands of the [http://www.adobe.com/cfusion/search/index.cfm?term=&Universe&loc=en_us&siteSection=home Universe] to direct me how I can best do that. Through that little exercise plus the Daily Practice described above I hope to keep regenerating that fiber 
15. Your message to the readers of this blog 
Skip dinner. But follow me on Twitter. 
Read more posts on The Altucher Confidential �
More from The Altucher Confidentia 
Life is Like a Game. Here�s How You Master ANY Gam 
Step By Step Guide to Make $10 Million And Then Totally Blow 
Can You Do One Page a Day?

Diophantus

2014-02-17T13:48:23Z

79.180.76.66: /* Margin writing by Fermat and Chortasmenos */

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Cobweb plot

2014-01-31T20:19:32Z

79.180.49.165: /* External links */

{{technical|date=November 2013}}
{{quantum mechanics}}

In [[quantum physics]], the '''spin–orbit interaction''' (also called '''spin–orbit effect''' or '''spin–orbit coupling''') is an interaction of a particle's [[spin (physics)|spin]] with its motion. The first and best known example of this is that spin–orbit interaction causes shifts in an [[electron]]'s [[energy level|atomic energy levels]] due to electromagnetic interaction between the electron's spin and the magnetic field generated by the electron's orbit around the nucleus. This is detectable as a splitting of [[spectral line]]s. A similar effect, due to the relationship between [[angular momentum]] and the [[strong nuclear force]], occurs for [[proton]]s and [[neutron]]s moving inside the [[Atomic nucleus|nucleus]], leading to a shift in their energy levels in the nucleus [[Nuclear shell model|shell model]]. In the field of [[spintronics]], spin–orbit effects for electrons in [[semiconductor]]s and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy.

==Spin–orbit interaction in atomic energy levels==
This section presents a relatively simple and quantitative description of the spin–orbit interaction for an electron bound to an atom, up to first order in [[perturbation theory (quantum mechanics)|perturbation theory]], using some semiclassical [[electrodynamics]] and non-relativistic quantum mechanics. This gives results that agree reasonably well with observations. A more rigorous derivation of the same result would start with the [[Dirac equation]], and achieving a more precise result would involve calculating small corrections from [[quantum electrodynamics]].

===Energy of a magnetic moment===
The energy of a magnetic moment in a magnetic field is given by:

:<math>\Delta H=-\boldsymbol{\mu}\cdot\boldsymbol{B},</math>

where '''''μ''''' is the [[magnetic moment]] of the particle and '''''B''''' is the [[magnetic field]] it experiences.

===Magnetic field===

We shall deal with the [[magnetic field]] first. Although in the rest frame of the nucleus, there is no magnetic field[needs citation], there ''is'' one in the rest frame of the electron. Ignoring for now that this frame is not [[inertial]], in [[SI]] units we end up with the equation

:<math>\boldsymbol{B} = -{ \boldsymbol{v} \times \boldsymbol{E} \over c^2},</math>

where '''''v''''' is the velocity of the electron and '''''E''''' the electric field it travels through. Now we know that '''''E''''' is radial so we can rewrite <math>\boldsymbol{E} = \left| E / r \right| \boldsymbol{r} </math>.
Also we know that the momentum of the electron <math>\boldsymbol{p} =m_\text{e} \boldsymbol{v} </math>. Substituting this in and changing the order of the cross product gives:

:<math>\boldsymbol{B} = {\boldsymbol{r}\times\boldsymbol{p}\over m_\text{e} c^2} \left | {E\over r}\right|. </math>

Next, we express the electric field as the gradient of the [[electric potential]] <math>\boldsymbol{E} = -\boldsymbol{\nabla}V</math>. Here we make the [[central field approximation]], that is, that the electrostatic potential is spherically symmetric, so is only a function of radius. This approximation is exact for hydrogen, and indeed hydrogen-like systems. Now we can say

:<math>\left | E\right| = {\partial V \over \partial r}={1\over e}{\partial U(r) \over \partial r},</math>

where <math>U=eV</math> is the [[potential energy]] of the electron in the central field, and ''e'' is the [[elementary charge]]. Now we remember from classical mechanics that the [[angular momentum]] of a particle <math>\boldsymbol{L} = \boldsymbol{r}\times\boldsymbol{p}</math>. Putting it all together we get

:<math>\boldsymbol{B} = {1\over m_\text{e}ec^2}{1\over r}{\partial U(r) \over \partial r} \boldsymbol{L}. </math>

It is important to note at this point that ''B'' is a positive number multiplied by ''L'', meaning that the [[magnetic field]] is parallel to the [[orbit]]al [[angular momentum]] of the particle, which is itself perpendicular to the particle's velocity. (Recall that according to most conventions, the vector cross product yields a vector which is (often symbolically) oriented out of the plane of its two constituent vectors).

===Magnetic moment of the electron===

The [[magnetic moment]] [[electron magnetic dipole moment|of the electron]] is

:<math> \boldsymbol{\mu}_S=- g_S \mu_B \frac{\mathbf{S}}{\hbar}.</math>

where <math>\boldsymbol{S}</math> is the spin angular momentum vector, <math>\mu_\text{B}</math> is the [[Bohr magneton]] and <math>g_\text{s}\approx 2</math> is the electron spin [[g-factor (physics)|g-factor]]. Here, <math>\boldsymbol{\mu}</math> is a negative constant multiplied by the [[spin (physics)|spin]], so the [[magnetic moment]] is antiparallel to the spin angular momentum.

The spin–orbit potential consists of two parts. The Larmor part is connected to the interaction of
the magnetic moment of the electron with the magnetic field of the nucleus in the co-moving frame of the electron. The second contribution is related to [[Thomas precession]].

===Larmor interaction energy===
The Larmor interaction energy is

:<math>\Delta H_\text{L} =-\boldsymbol{\mu}\cdot\boldsymbol{B}.</math>

Substituting in this equation expressions for the magnetic moment and the magnetic field, one gets

:<math>\Delta H_\text{L} = {2\mu_\text{B}\over \hbar m_\text{e} e c^2}{1\over r}{\partial U(r) \over \partial r} \boldsymbol{L}\cdot\boldsymbol{S}. </math>

Now, we have to take into account [[Thomas precession]] correction for the electron's curved trajectory.

===Thomas interaction energy===

In 1926 [[Llewellyn Thomas]] relativistically recomputed the doublet separation in the [[fine structure]] of the atom.<ref name=Thomas>L. H. Thomas, ''The motion of the spinning electron'', Nature (London), 117, 514 (1926).</ref> Thomas precession rate, <math>\boldsymbol{\Omega}_\text{T}</math>, is related to the angular frequency of the orbital motion, <math>\boldsymbol{\omega}</math>, of a spinning particle as follows <ref>L. Föppl and P. J. Daniell, ''Zur Kinematik des Born'schen starren Körpers'', Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 519 (1913).</ref><ref>[[Christian Møller|C. Møller]], ''The Theory of Relativity'', (Oxford at the Claredon Press, London, 1952).</ref>

:<math> \boldsymbol{\Omega}_\text{T} = \boldsymbol{\omega} (\gamma-1),</math>

where <math>\gamma</math> is the Lorentz factor of the moving particle. The Hamiltonian producing the spin
precession <math>\boldsymbol{\Omega}_\text{T}</math> is given by

:<math> \Delta H_\text{T} = \boldsymbol{\Omega}_\text{T} \cdot \boldsymbol{S}.</math>

To the first order in <math>(v/c)^2</math>, we obtain

:<math>\Delta H_\text{T} = - {\mu_\text{B}\over \hbar m_\text{e} e c^2}{1\over r}{\partial U(r) \over \partial r} \boldsymbol{L}\cdot\boldsymbol{S}. </math>

===Total interaction energy===
The total spin–orbit potential in an external electrostatic potential takes the form
:<math>\Delta H \equiv \Delta H_\text{L} + \Delta H_\text{T} = {\mu_\text{B}\over \hbar m_\text{e} e c^2}{1\over r}{\partial U(r) \over \partial r} \boldsymbol{L}\cdot\boldsymbol{S}. </math>
The net effect of Thomas precession is the reduction of the Larmor interaction energy by factor 1/2 which came to be known as the ''Thomas half''.

===Evaluating the energy shift===
Thanks to all the above approximations, we can now evaluate the detailed energy shift in this model. In particular, we wish to find a basis that diagonalizes both ''H0'' (the non-perturbed Hamiltonian) and ''ΔH''. To find out what basis this is, we first define the [[Azimuthal quantum number#Total angular momentum of an electron in the atom|total angular momentum]] [[Operator (physics)|operator]]

:<math>\boldsymbol{J}=\boldsymbol{L}+\boldsymbol{S}.</math>

Taking the dot product of this with itself, we get

:<math>\boldsymbol{J}^2=\boldsymbol{L}^2+\boldsymbol{S}^2+2\boldsymbol{L}\cdot \boldsymbol{S}</math>

(since '''''L''''' and '''''S''''' commute), and therefore

:<math>\boldsymbol{L}\cdot\boldsymbol{S}= {1\over 2}(\boldsymbol{J}^2 - \boldsymbol{L}^2 - \boldsymbol{S}^2)</math>

It can be shown that the five operators ''H''0, '''''J'''''2, '''''L'''''2, '''''S'''''2, and ''J''z all commute with each other and with Δ''H''. Therefore, the basis we were looking for is the simultaneous [[eigenbasis]] of these five operators (i.e., the basis where all five are diagonal). Elements of this basis have the five [[quantum number]]s: ''n'' (the "principal quantum number") ''j'' (the "total angular momentum quantum number"), ''l'' (the "orbital angular momentum quantum number"), ''s'' (the "spin quantum number"), and ''j''z (the "''z''-component of total angular momentum").

To evaluate the energies, we note that

:<math>\left \langle {1\over r^3} \right \rangle = \frac{2}{a^3 n^3 l(l+1)(2l+1)}</math>

for hydrogenic wavefunctions (here <math>a = \hbar / Z \alpha m_\text{e} c </math> is the [[Bohr radius]] divided by the nuclear charge ''Z''); and

:<math>\left \langle \boldsymbol{L}\cdot\boldsymbol{S} \right \rangle={1\over 2}(\langle\boldsymbol{J}^2\rangle - \langle\boldsymbol{L}^2\rangle - \langle\boldsymbol{S}^2\rangle)</math>
:<math>={\hbar^2\over 2}(j(j+1) - l(l+1) -s(s+1))</math>

===Final energy shift===

We can now say

:<math>\Delta E = {\beta\over 2}(j(j+1) - l(l+1) -s(s+1))</math>

where

:<math>\beta = \beta (n,l) = Z^4{\mu_0\over 4{\pi}}g_\text{s}\mu_\text{B}^2{1\over n^3a_0^3l(l+1/2)(l+1)}</math>

==Spin–orbit interaction in solids==

Crystalline solid (semiconductor, metal etc.) is characterized by its band structure. While on the overall scale (including the core levels) the spin–orbit interaction is still a small perturbation, it may play relatively more important role if we zoom in to bands close to the Fermi level (<math>E_\text{F}</math>). The atomic <math>\boldsymbol{L} \cdot \boldsymbol{S}</math> interaction for example splits bands which would be otherwise degenerate and the particular form of this spin–orbit splitting (typically of the order of few to few hundred millielectronvolts) depends on the particular system. The bands of interest can be then described by various effective models, usually based on some perturbative approach. An example of how the atomic spin–orbit interaction influences the band structure of a crystal is explained in the article about [[Rashba effect|Rashba interaction]].

===Examples of effective Hamiltonians===

Hole bands of a bulk (3D) zinc-blende semiconductor will be split by <math>\Delta_0</math> into heavy and light holes (which form a <math>\Gamma_8</math> quadruplet in the <math>\Gamma</math>-point of the Brillouin zone) and a split-off band (<math>\Gamma_7</math> doublet). Including two conduction bands (<math>\Gamma_6</math> doublet in the <math>\Gamma</math>-point), the system is described by the effective eight-band [[Luttinger–Kohn model|model of Kohn and Luttinger]]. If only top of the valence band is of interest (for example when <math>E_\text{F}\ll \Delta_0</math>, Fermi level measured from the top of the valence band), the proper four-band effective model is

:<math>H_\text{KL}(k_\text{x},k_\text{y},k_\text{z})=-\frac{\hbar^2}{2m}\left[(\gamma_1+{\textstyle\frac52 \gamma_2}) k^2 -
2\gamma_2(J_\text{x}^2k_\text{x}^2+J_\text{y}^2k_\text{y}^2
+J_\text{z}^2k_\text{z}^2) -2\gamma_3 \sum_{m \ne n}J_mJ_nk_mk_n\right]</math>

where <math>\gamma_{1,2,3}</math> are the Luttinger parameters (analogous to the single effective mass of a one-band model of electrons) and <math>J_{\text{x},\text{y},\text{z}}</math> are angular momentum 3/2 matrices (<math>m</math> is the free electron mass). In combination with magnetization, this type of spin–orbit interaction will distort the electronic bands depending on the magnetization direction, thereby causing [[Magnetocrystalline anisotropy]] (a special type of [[Magnetic anisotropy]]).
If the semiconductor moreover lacks the inversion symmetry, the hole bands will exhibit cubic Dresselhaus splitting. Within the four bands (light and heavy holes), the dominant term is

:<math>H_{\text{D}3}=b_{41}^{8\text{v}8\text{v}}[(k_\text{x}k_\text{y}^2-k_\text{x}k_\text{z}^2)J_\text{x}+(k_\text{y}k_\text{z}^2-k_\text{y}k_\text{x}^2)J_\text{y}+(k_\text{z}k_\text{x}^2-k_\text{z}k_\text{y}^2)J_\text{z}]
</math>

where the material parameter <math>b_{41}^{8\text{v}8\text{v}} = -81.93 \,\text{meV} \cdot \text{nm}^3</math> for GaAs (see pp. 72 in Winkler's book, according to more recent data the Dresselhaus constant in GaAs is 9 eVÅ3;<ref>J. J. Krich and B. I. Halperin, Phys. Rev. Lett. '''98''', 226802 (2007)</ref> the total Hamiltonian will be <math>H_\text{KL}+H_{\text{D}3}</math>). [[2DEG|Two-dimensional electron gas]] in an asymmetric quantum well (or heterostructure) will feel the [[Rashba effect|Rashba interaction]]. The appropriate two-band effective Hamiltonian is

:<math>
H_0+H_\text{R} = \frac{\hbar^2 k^2}{2m^*} \sigma_0 + \alpha (k_\text{y} \sigma_\text{x} - k_\text{x}\sigma_\text{y})
</math>

where <math>\sigma_0</math> is the 2 × 2 identity matrix, <math>\sigma_{\text{x},\text{y}}</math> the Pauli matrices and <math>m^*</math> the electron effective mass. The spin–orbit part of the Hamiltonian, <math>H_\text{R}</math> is parametrized by <math>\alpha</math>, sometimes called the Rashba parameter (its definition somewhat varies), which is related to the structure asymmetry.

Above expressions for spin–orbit interaction couple spin matrices <math>\boldsymbol{J}</math> and <math>\boldsymbol{\sigma}</math> to the quasi-momentum <math>\boldsymbol{k}</math>, and to the vector potential <math>\boldsymbol{A}</math> of an ''AC'' electric field through the Peierls substitution <math>{\boldsymbol{k}}=-i\nabla-(\frac{e}{\hbar c}){\boldsymbol{A}}</math>. They are lower order terms of the Luttinger–Kohn <math>\boldsymbol{k}\cdot{\boldsymbol{p}}</math> expansion in powers of <math>k</math>. Next terms of this expansion also produce terms that couple spin operators of the electron coordinate <math>\boldsymbol{r}</math>. Indeed, a cross product <math>(\boldsymbol{\sigma}\times{\boldsymbol{k}})</math> is [[invariant]] with respect to time inversion. In cubit crystals, it has a symmetry of a vector and acquires a meaning of a spin–orbit contribution <math>{\boldsymbol{r}}_{\text{SO}}</math> to the operator of coordinate. For electrons in semiconductors with a narrow gap <math>E_G</math> between the conduction and heavy hole bands, Yafet derived the equation
:<math> {\boldsymbol{r}}_{\text{SO}}=\frac{\hbar^2g}{4m_0} \left(\frac{1}{E_G}+\frac{1}{E_{G}+\Delta_0}\right)(\boldsymbol{\sigma}\times{\boldsymbol{k}})</math>
where <math>m_0</math> is a free electron mass, and <math>g</math> is a <math>g</math>-factor properly renormalized for spin–orbit interaction. This operator couples electron spin <math>{\boldsymbol{S}}=\frac{1}{2}{\boldsymbol{\sigma}}</math> directly to the electric field <math>\boldsymbol{E}</math> through the interaction energy <math>-e({\boldsymbol{r}}_{\text{SO}}\cdot{\boldsymbol{E}})</math>.

==See also==

* [[Angular momentum coupling]]
* [[Stark effect]]
* [[Zeeman effect]]
* [[Kugel–Khomskii coupling]]
* [[Angular momentum diagrams (quantum mechanics)]]
* [[Spherical basis]]
* [[Rashba effect ]]
* [[relativistic angular momentum]]

==References==
<references/>
Y. Yafet, in: {\it Solid State Physics}, ed. by F. Seitz and D. Turnbull (Academic, NY), v. {\bf 14}, p.1.

E. I. Rashba and V. I. Sheka, in: {\it Landau Level Spectroscopy} (North Holland, Amsterdam) 1991, p.131.

==Textbooks==
* {{cite book | author=E. U. Condon and G. H. Shortley|title=The Theory of Atomic Spectra | publisher=Cambridge University Press |year=1935 |isbn=0-521-09209-4 }}

* {{cite book | author=D. J. Griffiths|title=Introduction to Quantum Mechanics (2nd edition) | publisher=Prentice Hall |year=2004}}

* {{cite book |title=Quantum Mechanics: Non-Relativistic Theory, Volume 3 |last=Landau |first=Lev |authorlink=Lev Landau |coauthors= L. M. Lifshitz |chapter=<math>\S</math>72. Fine structure of atomic levels}}

* {{cite book |title=Fundamentals of Semiconductors|last=Yu|first=Peter Y.|coauthors=Manuel Cardona}}

* {{cite book |title=Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems|last=Winkler|first=Roland}}

{{DEFAULTSORT:Spin-orbit interaction}}
[[Category:Atomic physics]]
[[Category:Magnetism]]
[[Category:Spintronics]]

Ehresmann connection

2014-01-22T10:17:53Z

79.180.126.193:

In [[geometry]], '''Stewart's theorem''' yields a relation between a lengths of the sides of the [[triangle]] and the length of a [[cevian]] of the triangle. Its name is in honor of the Scottish mathematician [[Matthew Stewart (mathematician)|Matthew Stewart]] who published the theorem in 1746.<ref>M. Stewart ''Some General Theorems of Considerable Use in the Higher Parts of Mathematics'' (1746) "Proposition II"</ref>

== Theorem ==
Let <math>a</math>, <math>b</math>, and <math>c</math> be the lengths of the sides of a triangle. Let <math>d</math> be the length of a [[cevian]] to the side of length <math>a</math>. If the cevian divides <math>a</math> into two segments of length <math>m</math> and <math>n</math>, with <math>m</math> adjacent to <math>c</math> and <math>n</math> adjacent to <math>b</math>, then Stewart's theorem states that
:<math>b^2m + c^2n = a(d^2 + mn).\,</math>
[[Apollonius' theorem]] is the special case where d is the length of the [[Median (geometry)|Median]].

The theorem may be written somewhat more symmetrically using signed lengths of segments, in other words the length ''AB'' is taken to be positive or negative according to whether ''A'' is to the left or right of ''B'' in some fixed orientation of the line. In this formulation, the theorem states that if ''A'', ''B'', and ''C'' are collinear points, and ''P'' is any point, then<ref>Russell</ref>
:<math>PA^2\cdot BC+PB^2\cdot CA-PC^2\cdot AB - BC\cdot CA\cdot AB =0.\,</math>

== Proof ==
[[File:Stewarts theorem.svg|right|300px|Diagram of Stewart's theorem]]
The theorem can be proved as an application of the [[law of cosines]]:<ref>Follows Hutton & Gregory or, more closely, PlanetMath.</ref>

Let ''θ'' be the angle between ''m'' and ''d'' and ''θ′'' the angle between ''n'' and ''d''. Then ''θ′'' is the supplement of ''θ'' and cos ''θ′'' = −cos ''θ''. The law of cosines for ''θ'' and ''θ′'' states
:<math>
\begin{align}
c^2 &= m^2 + d^2 - 2dm\cos\theta \\
b^2 &= n^2 + d^2 - 2dn\cos\theta' \\
&= n^2 + d^2 + 2dn\cos\theta.\, \end{align}
</math>

Multiply the first equation by ''n'', the second equation by ''m'', and add to eliminate cos ''θ'', obtaining
:<math>
\begin{align}
&b^2m + c^2n \\
&= nm^2 + n^2m + (m+n)d^2 \\
&= (m+n)(mn + d^2) \\
&= a(mn + d^2), \\
\end{align}
</math>
which is the required equation.

Alternatively, the theorem can be proved by drawing a perpendicular from the vertex of the triangle to the base and using the [[Pythagorean theorem]] to write the distances ''b'', ''c'', and ''d'' in terms of the altitude. The left and right hand sides of the equation then reduce algebraically to the same expression.<ref>This is a overview of the proof in Russell.</ref>

== See also ==
* [[Mass point geometry]]

==References==
<references/>
5. I.S Amarasinghe, Solutions to the Problem '''43.3''': Stewart's Theorem(''A '''New Proof''' for the Stewart's Theorem '''using Ptolemy's Theorem'''''), ''Mathematical Spectrum'', Vol '''43(03)''', pp.138 - 139, 2011.

6. A. Ostermann, G. Wanner, Further Results in Euclidean Geometry: '''''Problem 14 of Exercises 4.11''''', '''Geometry by Its History'''(''Springer books''), '''pp. 112''', 2012.

==Further reading==
*{{cite book |title=A Course of Mathematics|first1=C.|last1=Hutton|first2=O.|last2=Gregory |publisher=Longman, Orme & co.|year=1843|page=219|volume=II|url=http://books.google.com/books?id=9-4GAAAAYAAJ&pg=PA219#v=onepage}}
* {{cite book |title=Pure Geometry
|first=John Wellesley|last=Russell|publisher=Clarendon Press|year=1905
|chapter=Chapter 1 §3: Stewart's Theorem
|url=http://books.google.com/books?id=r3ILAAAAYAAJ|oclc= 5259132}}

==External links==
* {{MathWorld|title=Stewart's Theorem|urlname=StewartsTheorem}}
* {{PlanetMath|title=Stewart's Theorem|urlname=StewartsTheorem}}
* {{PlanetMath|title=Proof of Stewart's Theorem|urlname=ProofOfStewartsTheorem}}

[[Category:Euclidean plane geometry]]
[[Category:Triangle geometry]]
[[Category:Articles containing proofs]]
[[Category:Theorems in plane geometry]]

Dinic's algorithm

2014-01-20T10:34:25Z

79.180.69.87: /* Special cases */

In [[mathematics]], a '''coreflexive relation''' is a [[binary relation]] that is a subset of the [[identity relation]].<ref>Fonseca de Oliveira, J. N., & Pereira Cunha Rodrigues, C. D. J. (2004). Transposing Relations: From Maybe Functions to Hash Tables. In Mathematics of Program Construction (p. 337).</ref> Thus if ''a'' is related to ''b'' (''aRb'') then ''a'' is equal to ''b'' (''a = b''), but if ''c'' is equal to ''d'' (''c = d'') it does not necessarily hold that ''c'' is related to ''d'' (''cRd'').

In [[mathematical notation]], this is:

:<math>\forall a, b \in X,\ a R b \Rightarrow \; a = b.</math>

The [[identity relation]] is coreflexive by definition. Any relation that is coreflexive is thus a subset of the identity relation.

For example, consider the relation ''R'' as "equal to and odd". Over the set of positive integers, the relationship ''R'' holds over the pairs {(1, 1), (3, 3), ...} but does not hold over {(2, 2), (4, 4), ...}.

==Notes==
<references />

[[Category:Mathematical relations]]

Schrödinger equation

2009-07-14T08:51:32Z

79.180.13.91: /* Single particle in three dimensions */

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