DEGENERACY AND ALL THAT

Transcription

1 DEGENERACY AND ALL THAT Te Nature of Termodyamics, Statistical Mecaics ad Classical Mecaics Termodyamics Te study of te equilibrium bulk properties of matter witi te cotext of four laws or facts of experiece tat relate measurable properties, like temperature, pressure, ad volume etc. It is importat to uderstad tat from te viewpoit of termodyamics te microscopic ature of matter is irrelevat, tat is, termodyamics would apply equally well if matter formed a cotiuum. I additio, termodyamics is a measuremet or laboratory based sciece ad is ot a brac of metapysics. Statistical Mecaics Statistical Mecaics is a statistical approac to solvig te classical body problem i order to study te same bulk properties of matter as termodyamics but doig so at te microscopic level. I tis way Statistical Mecaics allows a uderstadig of te equilibrium properties of matter at a molecular level. Statistical Mecaics makes eavy use of Termodyamics, Classical Mecaics ad Quatum Mecaics for its developmet ad ece te perquisites for tis course. Classical Mecaics Te Classical Mecaical approac to studyig te body problem ivolves solvig six simultaeous differetial equatios for eac particle i te system. Tis assumes oe kows te iitial positio q(t) ad mometum p(t) of eac particle at time t o. Sice te bulk properties of te system of iterest are temselves fuctios of te q ad p, i.e., G=G[p(t),q(t)] we ca te do a time average of te form 1 t0 G Gobs G[ p( t), q( t)] dt t0 were is log eoug to esure G is idepedet of, i.e., fluctuatios are egligible. Util te early 1950 s, suc computatios were largely beyod reac but sice te wit te advet of computers, te subect of molecular simulatio or molecular dyamics as progressed to te poit were te classical mecaical computatios of te bulk equilibrium properties of all states of matter are ofte cosidered more accurate ta te actual experimetal measuremets of te same properties. Quatum Mecaics: Te body problem i quatum mecaics is similar to te problem i classical mecaics altoug te formalism is quite differet. Here oe attempts to solve te body Hamiltoia, H E s s s s for a system s, of particles wic, provided te particles are distiguisable ad o-iteractig, we ca write, s i1 i1 ( qt, ) Hs Hi( p, q, t) i Page 1

2 E s Ei i1 Te more geeral situatio is were te system cosists of idistiguisable ad o-iteractig particles, i wic case te system wave fuctio is ot a simple product over te sigle particle wave fuctios as i te case of distiguisable particles, but is istead give by te determiat:! (1) ( ) 1 1 (1) ( ) were te symbol o te determiat deotes te distictio betwee Fermio ad Boso systems. Tis distictio arises aturally because it seems all kow particles are oe or te oter. Tus: - Bosos (potos, mesos ad atoms wit a eve umber of fermios) sym - Fermios (electros, protos, eutros ad atoms wit a odd umber of fermios) atisym For fermios te occupatio umber of ay quatum state is 0 or 1 (Pauli Exclusio Priciple) wile bosos ave o suc restrictio. Icidetally, if a collectio of fermios are sufficietly far apart so tere is o wave fuctio overlap, te tese particles ca be treated as distiguisable ad te Pauli priciple does ot old. Fially, quatum mecaical averages for system properties is give by te usual expressio Quatum Mecaical Degeeracy G G G d * ˆ obs s ss Degeeracy plays a fudametal role i te developmet of Statistical Mecaics ad so we will remid of you of wat you probably already kow sice te prerequisite for te course icludes a course i Quatum Mecaics. Cosider a sigle free particle costraied to move i tree dimesios i a cubic box of legt L o a side. Te potetial i te box is zero everywere ad te potetial outside te box is ifiite (wat is te purpose of tis costrait?). Te wave equatio for tis so called particle i a tree dimesioal box problem is give by: [ V ( r )] E m For problems i wic te potetial eergy ca be separated i terms of Cartesia coordiates we ca write V( x, y, z) V ( x ) V x V x Page

3 ad E Ex Ey Ez Te, te assumptio of a solutio of te form: ( x, y, z) X ( x) Y( y) Z( z) leads to te separatio of te tree dimesioal secod order partial differetial equatio ito tree idetical ordiary secod order differetial equatios, oe for eac coordiate, x, y, ad z, ad of te form: d X m E x V x X x [ ( )] ( ) 0 dx Solutio leads to tree idetical expressios for te eergy of te form: E x x x were x is a o-zero iteger ml x wit te wave fuctio ( x) / L si x x L Here x is te particle quatum umber ad L x is te x dimesio of te box. Te tree dimesioal solutio follows immediately sice te eergies add ad te wave fuctios multiply givig E ad were agai te s are o-zero itegers. ( ) x y z Te correspodig wave fuctio is te product give by, x x y y z z ( x, y, z) ( x) ( y) ( z) 8 / V si si si Lx Ly L z Tis brigs us te poit of tis discussio, amely te cocept of degeeracy. Defiitio of Degeeracy: If a subset of wave fuctios, 1,, k, give, we substituted back ito Ĥ E te same x value for E, we say E is k-+1 degeerate. Uderstadig te cocept of degeeracy is critical to uderstadig statistical mecaics. Page 3

4 Example: For te groud state of our particle i a 3 dimesioal box we ave x = y = z = 1 ad is clearly 6 o-degeerate. However, if x = ad y = z =1 we ave E correspodig to te followig tree wave fuctios: x y z 1 8 / V si si si Lx Ly L z x y z 8 / V si si si Lx Ly L z x y z 3 8 / V si si si Lx Ly L z (,1,1) (1,,1) (1,1,) 6 All tree give te same value of E o substitutio back ito te wave equatio. We say te tat te eergy level 6 / is 3 fold degeerate. You migt wis to sow tat te eergy level i wic E=14 is 6 fold degeerate. States vs Levels 6 I te previous case of E we saw tat te eergy level 6 ad a degeeracy of 3 correspodig to tree differet wave fuctios. I oter words tere exists a oe to oe correspodece betwee a eergy state ad its wave fuctio, i.e, E = 6 for 1,,3 Te importat poit ere is tat i tis course we will typically refer to a systems eergy state, ot its eergy level. I oter words, if we say tat a particular eergy level is 10 fold degeerate we mea tat tere are 10 wave fuctios or alteratively 10 eergy states, E 1, E, E 10 correspodig to tat particular eergy level. Tere will also be times we we ave to use levels as i te evaluatio of te electroic partitio fuctio, but i tose cases we will usually iclude te degeeracy, g i i our formulism. Desity of States for Large for te Particle i a Box All material particles, atoms, electros etc. are eiter Fermios or Bosos ad strictly speakig, te form of statistics we use to describe tese systems sould reflect tat fact. However, provided tat te average quatum state available to a particle is uoccupied, tat is, c 1 were c is te average occupatio umber of particle state, te we ca alteratively use te Maxwell Boltzma equatio tat allows us to factor te system partitio fuctio ito te product of sigle particle partitio fuctios for relatively easy computatio. Te validity of tis approximatio usually depeds o te fact tat traslatioal motio aloe esures tat tis coditio olds. Furtermore, sice te Page 4

5 model for traslatioal motio is te tree dimesioal particle i a box it beooves us to look at te questio of te desity of quatum states for a typical particle. If oe calculates te degeeracy of eac eergy level for a particle i a cubic box for, say a doze or so values of, tey will see tat te degeeracy does ot vary uiformly i ay predictable way wit icreasig E except to say tat wile te tred is erratic, it does icrease wit E ad becomes more uiform ad dese for large E. I particular, we see tat tree dimesioal eergy expressio is actually a equatio of a spere of radius R ad wose values of R( x, y, z ), we plotted, are cofied to 1/8 quadrat of te spere because all of te s must be positive ad o-zero. Wy? x y z R Note tat we ave writte E for cosistetly wit te text. For large values of te eergy te desity of poits essetially fills te volume of te quadrat. We ca ow treat R or as a cotiuous variable so we ca calculate te umber of lattice poits cosistet wit a eergy wic is essetially te volume of te eigt quadrat of te spere. We ca ow write for te umber of eergy states, ( ) / ( ) V * R ( R ) spere / ad calculate te umber of states i a ti sell of tickess. Tus, 4 3/ 1/ (, ) ( / ) ( ) Takig T=300K, m=10-5 kg, L=10 m ad ε=0.01ε we fid tat te degeeracy of a sigle particle (atom, molecule, etc.) movig i 3 dimesios at room temperature i a typical room 10 m o a side is approximately wic is a uge degeeracy. Tis meas tat te average eergy level correspodig to 3/kT as associated wit it rougly eergy states all wit equal probability from te poit of view of our particle. We ow look at ow te degeeracy cages we te system cotais N o-iteractig particles. Te system eergy te follows from te sigle particle expressio by obvious extesio, E s 8 N 3N ( ) x, y, z, 1 ml 1 Agai, by aalogy wit sigle particle case, we ave a equatio for a N dimesioal yperspere of radius R wose volume is give by V N N/ N R N 1 ad te correspodig degeeracy for te etire system of N particles takes te form, Page 5

6 3 N / 1 ml (3 N /1) E E E E (, ) ( N 1) (3 N / ) Here (N) is te usual gamma fuctio defied as wit te property tat 1 x x e dx 0 ( ) ( 1) ( )! However, as large as te sigle particle degeeracy was foud to be it is completely isigificat compared to a system of N particles i te same volume. Usig te same coditios as for te oe N dimesioal case we fid our N particle degeeracy to be o te order of O (10 ) or typically, 0 10 O (10 ). Now tere s a umber to be reckoed wit! Te value of tis discussio will become evidet we we get to capter 4 at wic time we will revisit tis issue. DW McClure, Emeritus Professor of Cemistry/Pysics Portlad State Uiversity 4/5/010 Page 6