1 Temperature And Super Conductivity. 1.1 Defining Temperature

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1 1 Tempeaue And Supe Conduiviy 1.1 Defining Tempeaue In ode o fully undesand his wok on empeaue and he elaed effes i helps o have ead he Quanum Theoy and he Advaned Quanum Theoy piees of he Pi-Spae Theoy as i builds upon hese seions. I will assume he eade is familia wih he Wave Wihin Wave (WaWiW) design paen. Also I will assume he eade is familia wih he Pi-Spae Hamilonian noaion used o develop he Navie-Sokes fomulas. Tempeaue and Pessue ae elaed. Using he WaWiW noaion hese ae seen as vibaions on he Nx(0) laye aleing (loweing/aising) he wave ampliude. Reall ha hese Nx(0) onsideed o be he loal waves. Tha is o say hese ae he waves whih ae easie o measue in ou ealiy (e.g. empeaue and pessue fo example). Fo example we have wo waves ou of phase onibuing o Eleiiy and he Gaviy field on he Nx(0) laye (see he Advaned Quanum Theoy). The espeive ohogonal waves ae he Magnei waves and he Tubulene waves. Fo he puposes of his seion on empeaue we ll fous on hee disin Nx(0) wave disubanes. Ng(0) = Gaviy Mass wave whih aies empeaue Ne(0) = Elei Chage wave whih aies empeaue Addiionally empeaue is aied hough he ehe by means of he Sandad Model Bozons whih wee evese engineeed ino he WaWiW noaion. This empeaue vibaion is also a he Nb(0) laye whee b is any abiay Bozon. Tempeaue is seen as a vibaion whih is a sign of empeaue. How an we model his Pi-Spae empeaue vibaion using he WaWiW noaion? Le s ake any abiay Nx(0) wave onaining one o moe hild-waves. Fo simpliiy he diagams I ll show jus assume one paen and one hild wave bu in ealiy he nesed waves go deepe. Howeve hese ae suffiien o show he piniple of empeaue in aion. Fo now we don ae if i s an elei wave a Bozon aying empeaue a Femion o whaeve. We jus wan o show Lowe Tempeaue and Highe Tempeaue and Zeo Tempeaue. Lae I ll dive ino he deail of he Classial Laws Bose-Einsein Condensae Supe Fluidiy and Supe Conduiviy e; based on hese simple Pi-Spae empeaue diagams. All of he uen behavio we an see and measue exend fom hese simple oneps wih some addiional deail. 1. Highe Tempeaue (Highe Ampliude Wave size.0 Nx(0) Ampliude)

2 . Lowe Tempeaue (Lowe Ampliude Wave size 0.5 Nx(0) Ampliude) 3. Absolue Zeo (Zeo Ampliude Wave size 0.0 Nx(0) Ampliude)

3 Noe how he las empeaue example looks quie diffeen. As a onsequene I ll show how exeme empeaue hange ales he vey behavio of subsanes in ou ealiy.` 1. Aveage Moleula Kinei Enegy In Pi-Spae Le s map he aveage moleula kinei enegy o Pi-Spae and solve a sample poblem. Le s do i lassially and using Pi-Spae Tansvese Kinei Enegy KE 3 kt 1 mv Solving fo veloiy v KE m Solving fo Veloiy in Pi-Spae we have he fom 3 kt v Cos ASin 1 m * Le s solve a poblem Find Tansvese KE and Aveage Veloiy T = 7 Degees Celsius = 300 Kelvin Mass Helium = 6.65*10^-7 Kg Solve fo Classi KE = (3/) ( ^ 3) (300)= Solving fo Veloiy Sq[(.0*(6.*10^-1))/(6.65*10^-7)] V = = 1.37*10^3 m/j

4 Solve fo Pi-Spae V = (Cos[ASin[ 1 - (((6.1*10^-1)/(6.65*10^-7))/( ^))]])*( ) Gives us V = = 1.37*10^3 m/j 1.3 Tempeaue and Pessue In Pi-Spae In Pi-Spae hea in a fluid is on he loal plane whee he pailes ae. In ode o map o Pi- Spae we need o deemine wha he effe of Tempeaue on he wave funion is. The ideal gas law shows us he elaionship. pv nrt Volume says he same so we see ha ineased pessue auses ineased empeaue. A lage pessue maps o a lage poenial whih in his ase means lage wave ampliude. Addiionally a lage Tempeaue also means lage wave ampliude. Noe: When hese lage ampliudes a on one o moe pailes hey make hem smalle and give hem ineased Kinei Enegy. I have aleady solved he veloiy fomulas fo hese ases. We onside he use of Bolzmann s Consan k pv nkt Le s onside he loal plane one moe. We add he pessue Poenial. Nex we add he empeaue. v Cos ASin p Loal

5 Loal T k p v ASin Cos The wave funion is expessed diely hee. In he wave-wihin-wave design paen we an daw his as follows 1. Highe Tempeaue/Pessue Poenial (Highe Ampliude Wave size.0 Nx(0) Ampliude) Tempeaue and Pessue ae a he Nx(0) laye. If we dop he Kinei Componen (we have aleady solved his fo he Aveage Tansvese Kinei Enegy and Pio Pessue) and fous on he poenial fo a losed sysem we ae lef wih he wo values equal o one anohe. The eason fo his is ha hey shae he same undelying wave funion wihin he losed sysem wih a fixed volume. T k p T k p This is he same as nkt pv

6 1.4 Chales Law Tempeaue And Volume In Pi-Spae Tempeaue and Volume ae elaed in Chales Law. The pessue says onsan. We end up wih he following expession. Also his an be expessed as V T V1 V T1 T So fa we have no expessed Volume in he Pi-Spae fomula. How does his law elae o Pi-Spae? v Cos ASin p T k We know pessue is onsan so we an dop he Pessue Loal v Cos ASin T k Loal Nex we need o show he elaionship beween empeaue and volume. In Pi-Spae empeaue ales he aea of a Pi-Shell/aom. Also in ealie seions I also showed ha a Gaviaional poenial o Pessue ales he aea of a Pi-Shell. In my Gaviy wok I showed ha Gauss Law fo Gaviy showed ha Gaviy ales he Volume of a Sphee. This is foundaional. D s D. da Theefoe any aea hange o an aom auses a Volume hange. Wha Gauss Divegene heoem saes is ha an aea hange o a Sphee also ales is volume. If you ae unsue of his please ead he Quanum Theoy do o Gauss Theoem. Theefoe empeaue hange ales he volume of a Pi-Shell o Mole of aoms.

7 volume T k v ASin Cos Loal This is how we desibe Chales Law in Pi-Spae. 1.5 Boyles Law Pessue And Volume In Pi-Spae Pessue and Volume ae elaed in Boyles Law. The wo values ae invesely popoional o one anohe when empeaue is onsan in a losed sysem. 1 1 v p v p Fom his we an see ha pessue and volume ae invesely popoional o one anohe. How an we epesen his in Pi-Spae? This is easonably easy. Le s ake a look a he Pi- Spae equaion. Loal T k p v ASin Cos We know empeaue is onsan so we an dop he empeaue. Loal p v ASin Cos We know densiy is mass divided by volume. Loal v m p v ASin Cos Whih give us

8 v Cos ASin pv m Loal Theefoe we an see pv is a onsan. This is how we see Boyle s Law in Pi-Spae. Please noe ha he expession on he lef is he Kinei Componen elaed o any paiula pessue. I ve aleady solved his in he Quanum Theoy do elaing o Pio. 1.6 Fis Law of Themodynamis In Pi-Spae This is a onsevaion of enegy fo Themodynami sysems. I is usually fomulaed by saing ha he hange in he inenal enegy of a losed sysem is equal o he amoun of hea supplied o he sysem minus he wok done by he sysem on is suoundings. The law of onsevaion of enegy an be saed: The enegy of an isolaed sysem is onsan. In Chemisy we have du dq dw du dq dw Le s map his o he Pi-Spae fomula. U is he oal enegy of he sysem. Q is he hea of he sysem and W is he wok done. In Pi-Spae we use a Hamilonian o expess he oal enegy of a sysem whih is zeo v Cos ASin p T Loal 0 k The expession on he lef is he Kinei Componen. The expessions on he igh ae he Themodynami omponens. These ae empeaue and PV (pessue volume). So we an lose he pas we don need and show ha hey epesen a onsan enegy value U in a losed sysem. p T k U In he ase of a losed sysem whee hee is being wok done by i we an expess a gain in one pa o he hea as a loss on he ohe. Theefoe we an say

9 This is analogous o T k p U du dq dw Also if an ouside empeaue gain due o heaing is added hen he oal enegy of he oveall sysem ineases. This is analogous o T k p U du dq dw In ase one is wondeing whee Pessue and Volume is I aleady showed This gives us pv m p This mahes T k pv m U du dq pdv Noe: The wave funion an be dopped if fo example he obje is symmei. Please ake a look a how he shape of he onaine affes he wave funion fo Venui and so on in he Quanum Theoy do. 1.7 Seond Law of Themodynamis in Pi-Spae Hea moves fom ho o old. This indiaes he pefeed veo omponen of he waves whih ay he hea. Theefoe lage ampliude waves move o smalle ampliude waves. 1. Highe Tempeaue (Highe Ampliude Wave size.0 Nx(0) Ampliude)

10 . Lowe Tempeaue (Lowe Ampliude Wave size 0.5 Nx(0) Ampliude)

11 In paie his is simila o he Piniple of Leas Aion exep we ae dealing wih wave lengh as opposed o wave ampliude. Waves move owads a plae whee he smalles waves ae. When an obje falls unde Gaviy i moves owads he ene of Gaviy. This is whee he waves ae he shoes. High pessue moves o low pessue. This is whee he aoms have he leas vibaions. Eihe he shoe wave lengh o he shoe wave ampliude is he pefeed veo omponen. Theefoe we an say ha minimizing Nx(0) is he pefeed veo omponen. 1.8 Dawing of Ho/Cold Paile in Pi-Spae Hee is a ough hand dawn piue of a ho and old paile.

12 One an see fom he piue ha ineased empeaue means lage wave ampliudes on he paile in quesion. This in un ineases he vibaional naue of he paile. I also anslaes ino an aea hange on he paile whih an be mapped o an aea/volume hange. Remembe in Pi-Spae ha aea hange anslaes o Classial Enegy so we an hen use his in he Tanslaional Kinei Enegy fomula I deived ealie.

13 1.9 Radiaion and Lases in Pi-Spae Now ha we have esablished he idea of empeaue in Pi-Spae being ampliude elaed we an apply his piniple o Eleomagnei Radiaion. We fous on he Elei wave whih is Ne(0). In he Advaned Quanum Seion we defined he Elei pa of he wave as follows. The Phoon is a gauge Bozon. This is pey saighfowad o model. We define jus he Ne(x) Elei wave fo now. The Nm(x) Magnei wave is he same jus ohogonal. I s Ne(0). In his ase he ampliude is 1. This is analogous o a empeaue. The lage he ampliude he lage he empeaue i aies. Phoons an be plaed on op of one anohe so ha hei ampliude ineases and heefoe hei empeaue ineases. This is how lases wok. Phoons ae plaed on op of one anohe and he ampliude keeps geing lage; fo example if you plaed wo one op of one anohe you would have ampliude. Phoons ae heefoe seen as adiaion. Visually hey fom a beam of ligh. The wavelengh deemined he olo. The ampliude deemines he bighness and sengh of he beam. Now in ode o undesand he meaning of he Waage of a lase hee is how i woks in Pi- Spae. Le s imagine one has a lase whih has powe of 1Was wha does his mean? Tanslaed ino a Pi-Spae undesanding his means ha he ampliude of he adiaion beam while ineaing wih a mass of one kilogam ove one seond auses he mass o lose aea of 1/^. This is analogous o one Newon pe seond. The aleed ampliude hange auses an aea hange and he obje appeas o bun/mel if suffiienly song.

14 1.10 Bose Einsein Condensae Bose and Einsein woked on a saisial model fo low empeaues and ealized ha Bozons ondensae a ula low empeaues. In his piee I will no evese enginee he mah bu I will explain puely in Pi-Spae a a onepual level wha is happening. Cold aom physis is pobably he mos ineesing aea of he Pi-Spae Theoy in ems of wha one an do wih mae in his sae. Reall ha empeaue equaes o an ampliude a he Nx(0) laye. This is he Loal wave. By his I mean hese ae he waves ha ae mos eadily measuable in ou ealiy. Fo example we have he Ne(0) wave whih is he elei wave and i an ay hea. Le s daw his one moe fo an aom wih he Ng(x) waves. Reall ha Ng(0) is he gaviy wave whih auses he aom o have diamee d. Inside ha is he Ng(1) whih is he Mass wave. Theefoe Mass geneaes a Gaviy field. I ve explained his in he Advaned Quanum Theoy do. Le s lowe he empeaue of he aom. 1. Lowe Tempeaue (Lowe Ampliude Wave size 0.5 Ng(0) Ampliude Mass Ng(1) is 10 ampliude 1)

15 Wha happens if we lowe he empeaue of he Ng(0) o absolue zeo?. Absolue Zeo (Zeo Ampliude Wave size 0.0 Ng(0) Ampliude jus mass 10 Ng(1)) We now only have he Ng(1) mass waves whih is wha emains in he ondensae. We no longe have a paile bu lumps of Non-Loal Ng(1) waves. The aom sill exiss and has mass and a diamee d. I sill exiss in ou ealiy bu has eneed a new sae. The aoms lump ogehe and fom a ondensae. In Pi-Spae he way we desibe his is ha we have eaed a Non-Loal Wave Pool. Namely we ae dealing wih mae whih has no Ng(0) waves. When we look a i i should appea dak as Ne(0) aies phoons. Also beause i has no Ng(0) waves is has only he popeies of Ng(1) whih is he mass waves and deepe. I will explain he moe ineesing popeies lae like why ligh sops in Bose Einsein ondensae and so on lae. Impoanly fom a Quanum Mehanial view poin no infomaion is los inside a ondensae. I is meely soed in he lowe Ng(1) and deepe waves o simply held in plae unil anspo is available (e.g. eheaing). The ondensaes ae he way o undesand wha happens in a Blak Hole fo example. The diffeene of ouse is ha he wavelengh is no shoened in a ondensae.

16 All we ae essenially doing wih a Condensae is emoving o dampening down he wave ampliude and effeing wave loss a a paiula laye. This is all I wan o explain hee fo now. Moe mah will ome lae. All a Bose Einsein ondensae does is lowe he Nx(0) ampliudes. This may no seem like suh a big deal bu in Pi-Spae i opens up a whole new way o do hings Non-Loally whih may seem o defy ou inuiion of wha ou ealiy is. Howeve all one is eally doing is ansfeing eihe mae o infomaion on Non-Loal waves; namely Nx(1++) waves. Wha i looks like (Non-Loal wave pool) The Loal waves Ng(0) fom aound he ondensae and we ge a lage wave lengh aound he ondensae. I gows as he empeaue dops. To a eain exen i has dopped ou of ou ealiy bu in Pi-Spae he pefeed em is a Non-Loal wave pool. (diagams espefully boowed fom Keele do) Bose Einsein Mah Bose Einsein ondensae uses a saisial disibuion model. Hee I ll show how he uen mah ies in wih Pi-Spae. The saisial model uses he Exponen

17 f ( E) Ae 1 E kt 1 Also we know fom Eule ha e ix os x isin x Reall fom he advaned quanum seion ha Ne(x) and Ng(x) wee modeled as ou of phase wih one anohe. This is anohe wave of saying osine x and sine x. Theefoe he exponen in his fom epesens Ne(x) and Ng(x). Now I also saed ha empeaue is based on he wave ampliude whih maps o he value A in Bose Einsein disibuion. I an be aied by eihe he Elei Wave Ne(x) o Ng(x). Theefoe we an ewie he Bose Einsein disibuion so i is ompaible wih Pi-Spae f ( E) A os E kt 1 isin E kt 1 This fomula epesens boh he Ne(x) and Ng(x) whih ae ou of Phase wih one anohe. Also i only epesens he loal Ne(0) and Ng(0) waves. Ampliude A is 0 when absolue empeaue is 0. Reall fom he pevious diagams ha Ng(0) is modeled as a Sin wave heefoe Nx( 0) Ai sin E kt We an dop he imaginay axis i as his maps o he uni ile whih is he pah of he wave aound he paile. Nx( 0) Asin E kt The Pi-Spae inepeaion of his is ha Ampliude A size maps o he empeaue. A zeo empeaue maps o a zeo ampliude size. This fomula only oves he Nx(0) laye and Pi-Spae models he deepe inne waves. 1.1 Slowing Ligh In a Condensae In Pi-Spae

18 I will disuss he high level heoeial way ha ligh an be fozen in a Condensae in Pi- Spae. Please ead he Advaned Quanum do if you ae unsue of Ne(x) and Ng(x). Ligh is a gauge bozon and so exiss puely on Ne(0) I is a massless wave and a so-alled Loal wave on Laye 0 In ode fo ligh o move hough a medium hee needs o be anspo waves on he espeive Ne(0) laye in he medium he waves ae moving hough. This an be povided fo example by an aom a oom empeaue whih onains Ne(0) waves. I an heefoe absob and hen e-emi Ne(0) waves. I an also pass hough a anspaen maeial whih onains Ne(0) waves. Le s desibe he well known EIT expeimen (Eleomagneially indued anspaeny) whee ligh is fozen. The maeial (ould be sodium aoms o silion) is ooled o nea absolue zeo Kelvin. Theefoe he aoms povide no anspo suppo fo ligh Ne(0). A oupling beam is shone on he old aoms whih allows hem o vibae whih maps o a paiula pobabiliy ampliude A. In Pi-Spae his maps o Ne(0) waves on he old aoms o have a speifi ampliude. In Quanum Mehanis his is he n > noaion bu is essenially a pobabiliy ampliude. Nex he pobe pulse is sen hough he old aoms. This is a guage bozon wih a paiula ampliude on he Ne(0) laye. The ampliudes of he oupling beam and pobe ae oheen o hey pass hough he Ne(0) oupling waves on he old aoms and he aoms beome anspaen.

19 Howeve nex he oupling beam is uned off. The old aom no longe has suffiien ampliude suppo fo ligh. Theefoe he pobe pules Ne(0) ligh waves have no anspo waves on he Ne(0) layes of he aom so he pobe ligh wais unil Ne(0) anspo is available in he medium. Theefoe we have fozen ligh. Also he less Ne(0) waves hee ae on he old aoms he slowe he Ne(0) ligh waves move. Imagine a high speed highway whee ula old holes in he oad oasionally open up and he as (ligh waves) have o wai unil hey fill bak in. Ne(0) ligh waves need o move hough Ne(0) anspo waves povided by he medium. I hee ae no anspo waves you have augh hem. Typially we do his by feezing aoms. Noe: If one undesand his piniple hee is no eason why one anno go a level deepe and ale/dampen he ampliude of waves whih ay mass waves. I will desibe moe on he heoeial impliaions of his lae and heoeial ool suff hee. Clealy in his ase ligh does no ay mass so i is a moo poin fo now a leas.

Lecture 9. Transport Properties in Mesoscopic Systems. Over the last 1-2 decades, various techniques have been developed to synthesize

Lecture 9. Transport Properties in Mesoscopic Systems. Over the last 1-2 decades, various techniques have been developed to synthesize eue 9. anspo Popeies in Mesosopi Sysems Ove he las - deades, vaious ehniques have been developed o synhesize nanosuued maeials and o fabiae nanosale devies ha exhibi popeies midway beween he puely quanum