( n x ( ) Last Time Exam 3 results. Question. 3-D particle in box: summary. Modified Bohr model. 3-D Hydrogen atom. r n. = n 2 a o
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1 Last Time Exam 3 esults Quantum tunneling 3-dimensional wave functions Deceasing paticle size Quantum dots paticle in box) This week s honos lectue: Pof. ad histian, Positon Emission Tomogaphy Tue. Dec Physics 08, Lectue 6 1 Exam aveage = 76 Aveage is at / bounday ouse evaluations: Rzchowski: Thu, Dec. 6 Montauli: Tues, Dec. 11 ount Tue. Dec Physics 08, Lectue F Phy08 Exam 3 Aveage D A Range A 3-D paticle in box: summay Thee quantum numbes n x,n y,n z ) label each state n x,y,z =1,, 3 integes stating at 1) Each state has diffeent motion in x, y, z Quantum numbes detemine p x = h h = n x " nx L Momentum in each diection: e.g. Enegy: E = p x m + p y m + p z m = E o ome quantum states have same enegy n x + n y + n z ) How many 3-D paticle in box spatial quantum states have enegy E=18E o?. 3 D. 5 E. 6 E = E o n x + n y + n z ) n x,n y,n z ) = 4,1,1 ), 1,4,1 ), 1,1,4) Tue. Dec Physics 08, Lectue 6 3 Tue. Dec Physics 08, Lectue D Hydogen atom Modified oh model oh model: Resticted to cicula obits Found 1 quantum numbe n Enegy E n = " 13.6 ev, obit adius n Fom 3-D paticle in box, expect that H atom should have moe quantum numbes n = n a o Expect diffeent types of motion w/ same enegy Diffeent obit shapes A mall angula momentum These obits have same enegy, but diffeent angula momenta: L = " p ) Rank the angula momenta fom lagest to smallest: a) A,, b),, A c),, A d), A, e), A, ig angula momentum Tue. Dec Physics 08, Lectue 6 5 Tue. Dec Physics 08, Lectue 6 6 1
2 Angula momentum is quantized obital quantum numbe ) Angula momentum quantized L = h l l +1, is the obital quantum numbe Fo a paticula n, has values 0, 1,, n-1 =0, most elliptical =n-1, most cicula Fo hydogen atom, all have same enegy Tue. Dec Physics 08, Lectue 6 7 Obital mag. moment Obital chage motion poduces magnetic dipole µ Popotional to angula momentum µ = µ L /h ) µ = µ l l +1 ) µ " eh m = 0.97 #103 A m ) Obital magnetic dipole µ uent electon Each obit n,l has ame enegy: E n = "13.6/n ev Diffeent obit shape angula momentum): L = h l l +1) Diffeent magnetic moment: µ = µ L /h ) Tue. Dec Physics 08, Lectue 6 8 Obital mag. quantum numbe m Diections of obital ba magnet quantized. Obital magnetic quantum numbe m anges fom - to in intege steps 1) diffeent values Detemines z-component of L: L z = m l h This is also angle of L Fo example: 1 gives 3 states: Fo a quantum state with, how many diffeent oientations of the obital magnetic dipole moment ae thee?. 3 D. 4 E. 5 Tue. Dec Physics 08, Lectue 6 9 Tue. Dec Physics 08, Lectue 6 10 ummay of quantum numbes Fo hydogen atom: n : descibes enegy of obit descibes the magnitude of obital angula momentum m descibes the angle of the obital angula momentum Hydogen wavefunctions Radial pobability Angula not shown Fo given n, pobability peaks at ~ same place Idea of atomic shell otation: s: 0 p: 1 d: f: 3 g: 4 Tue. Dec Physics 08, Lectue 6 11 Tue. Dec Physics 08, Lectue 6 1
3 Full hydogen wave functions: uface of constant pobability n=: next highest enegy 1s-state n =1,, m l = 0 pheically symmetic. Pobability deceases exponentially with adius. hown hee is a suface of constant pobability s-state p-state p-state n =,, m l = 0 n =,, m l = 0 n =,, m l = ±1 ame enegy, but diffeent pobabilities Tue. Dec Physics 08, Lectue 6 13 Tue. Dec Physics 08, Lectue 6 14 n=3: two s-states, six p-states and ten d-states 3s-state n = 3,, m l = 0 3p-state 3p-state n = 3,, m l = 0 n = 3,, m l = ±1 n = 3, l =, m l = 0 n = 3, l =, m = ±1 l n = 3, l =, m = ± l Tue. Dec Physics 08, Lectue 6 15 Tue. Dec Physics 08, Lectue 6 16 Electon spin ew electon popety: Electon acts like a ba magnet with and pole. Include spin Quantum state specified by fou quantum numbes: n, l, m l, m s ) Thee spatial quantum numbes 3-dimensional) Magnetic moment fixed One spin quantum numbe but possible oientations of magnet: up and down Descibed by pin up spin quantum numbe m s m s = +1/ pin down m s = "1/ z-component of spin angula momentum z = m s h Tue. Dec Physics 08, Lectue 6 17 Tue. Dec Physics 08, Lectue
4 Quantum umbe How many diffeent quantum states exist with n=?. 4 D. 8 = 0 : s m : m s = 1/, -1/ = 1 : p 6 m l = +1: m s = 1/, -1/ m : m s = 1/, -1/ m l = -1: m s = 1/, -1/ How many diffeent quantum states ae in a 5g n=5, =4) sub-shell of an atom? A D. 16 E. 14 =4, so +1)=18. In detail, m l = -4, -3, -, -1, 0, 1,, 3, 4 and m s =+1/ o -1/ fo each. 18 available quantum states fo electons Thee ae a total of 8 states with n= Tue. Dec Physics 08, Lectue 6 19 Tue. Dec Physics 08, Lectue 6 0 Putting electons on atom Electons obey Pauli exclusion pinciple Only one electon pe quantum state n,, m, m s ) unoccupied occupied Hydogen: 1 electon n=1 states one quantum state occupied n =1,,m l = +1/) Atoms with moe than one electon Electons inteact with nucleus like hydogen) Also with othe electons auses enegy to depend on Helium: electons two quantum states occupied n =1,,m l = +1/) n =1,,m l = "1/) n=1 states Tue. Dec Physics 08, Lectue 6 1 Tue. Dec Physics 08, Lectue 6 Othe elements: Li has 3 electons " n = m m s = + 1 # & m m s = " 1 " n = m m s = + 1 # & m m s = " 1 " n = m m s = + 1 # & m m s = " 1 m l = "1 m s = + 1 m l = "1 m s = " 1 n=, 8 total, 1 occupied Electon onfiguations Atom onfiguation H 1s 1 He 1s 1s shell filled Li 1s s 1 n=1 shell filled - noble gas) e 1s s s shell filled n=1 states, total, occupied one spin up, one spin down " n =1 # n =1 & m m # m s = +1/& m s = "1/ Tue. Dec Physics 08, Lectue 6 3 1s s p 1 etc e 1s s p 6 p shell filled n= shell filled - noble gas) Tue. Dec Physics 08, Lectue 6 4 4
5 The peiodic table Atoms in same column have simila chemical popeties. H 1s 1 Li s 1 a 3s 1 K 4s 1 Quantum mechanical explanation: simila oute electon configuations. e s Mg 3s a 4s c 3d 1 Y 3d 8 moe tansition metals p 1 Al 3p 1 Ga 4p 1 p i 3p Ge 4p p 3 P 3p 3 As 4p 3 O p 4 3p 4 e 4p 4 F p 5 l 3p 5 4p 5 He 1s e p 6 A 3p 6 K 4p 6 Excited states of odium a level stuctue 11 electons e coe = 1s s p 6 closed shell) 1 electon outside closed shell a = [e]3s 1 Outside 11 th ) electon easily excited to othe states. Tue. Dec Physics 08, Lectue 6 5 Tue. Dec Physics 08, Lectue 6 6 Emitting and absobing light Zeo enegy Optical spectum Photon emitted hf=e -E 1 n=4 n=3 n= n=1 E 3 = " ev E = " 13.6 ev E 1 = " ev n=4 n=3 n= Photon absobed hf=e -E 1 n=1 E 3 = " ev E = " 13.6 ev E 1 = " ev Optical spectum of sodium Tansitions fom high to low enegy states Relatively simple 1 electon outside closed shell 589 nm, 3p -> 3s a Photon is emitted when electon dops fom one quantum state to anothe Absobing a photon of coect enegy makes electon jump to highe quantum state. Tue. Dec Physics 08, Lectue 6 7 Tue. Dec Physics 08, Lectue 6 8 5
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