. = . . . . . 1 ( . : . : . ... . . . .( ) =2 . . . . 0 . 97 % . ( ) . : . . . . =T=constant3 . J/s . . : 1 . P ) ( 4 . . ) Stefans law :( P = Power radiatedin W (J/s) = Stefan's Constant 5.67 x 10-8 W m-2 K-4 = A = Surface area of body (m) T = Temperature of body (K) 2 . ) maximum ( . max max . ) Wien( max :max.T = constantmax = Peak Wavelength (m) T = SurfaceTemperature(K) Constant = 2.898 x 10-3 mKThis rearranges to max = 2.898 x 10-3 / T T max . ) ( max. . max . max . 5 max: Some Blackbody Temperatures Region Wavelength(centimeters)Energy(eV)Blackbody Temperature(K)Radio > 10 < 10-5< 0.03Microwave 10 - 0.01 10-5 - 0.01 0.03 - 30Infrared 0.01 - 7 x 10-50.01 - 2 30 - 4100Visible 7 x 10-5 - 4 x 10-52 - 3 4100 - 7300Ultraviolet 4 x 10-5 - 10-73 - 1037300 - 3 x 106X-Rays 10-7 - 10-9103 - 1053 x 106 - 3 x 108Gamma Rays < 10-9> 105> 3 x 108 . . Rayleigh Jeans . :6 .( ) ( ) . ( ) . . . N ) 13 ( . Rayleigh Jeans 328cN ).................1 ( ) principle of equipartition( ) kT kT .( N kTcE328. Rayleigh-Jeans .7 Rayleigh-Jeans ( ) . ) UV-Catastrophe ( 0 . . 1900 ) Max Planck ( . )) 1.(( ( ) ( ) h .8 Ntotal : + + + + 03 2 1 0...ii totalN N N N N N Ni i. ) Boltzmann :( kT EikT EkTkT EkT EkT Eiiii ie N NeeeeeNN/0// 0///00 : 0/00/00 ikT h iikT Eii totale N e N N Ni h i Ei h . Etotal : ( ) ( ) ( ) ( )[ ] + + + + + + + + + + + + 0/00 0/0 3 2 1 03 2 1 003 3 2 2 1 1 0 0... 3 2 1 0... 3 2 1 0...ikT h itotali ikT Ei totaltotaliii totale i h N Ee N i h N i h N N N N h Eh N h N h N h N EE N E N E N E N E N Ei ( ) : E0=0hE1=1hE2=2hE3=3hE4=4hEPlanckdiscreteClassical Physicscontinuum90/0/0/00/0ikT h iikT h iikT h iikT h iee i he Ne i h NE : ( )( )1111111 1/2kT h x xxxxxeheheeheeeh E : kThx. :18/ 32 kT hehcE N E . 1 !: :. ( )43 34 5 43 34 40 033 34 4 33 340/ 32158158181818Tc hkc hT kdxe xc hT kdxe xh ckTEdxhkTdxhkTkThx substitutedehcEx x totalkT h total . 2 : Rayleigh-Jeans :: = c/ kT h . kT h x . x x ex 1 .kTckThhc ehcEkT h 3232/ 328 818 3: !102 ( ) . photoelectrons ) ( photoelectric effect .(: . ) Collector, C( A V . : vacuum++--e-e-e-e-AVAdjustable power supplyC11 . . . ) stopping potential .(: 1 . ) monochromatic( ) ( . 2 . ) threshold frequency ( . . 12 " " . :- . . 1905 h" " . " " " " " " h" " . frequencyKinetic energyThreshold frequency thMetal a Metal b-ea-eb-ec13Metal c E=h . 2. : . . ) work function.( E=e . : e hth : h e EE e hkinetickinetic+ + ) 13( h . h h h h h h h h h hvacuume-e-e-e-Eemetal14 . . . 4: 100. .. : Ee-=eV e 1.6 10- 19As V : E=1.610-17 J =100 eV. Ekinetic=mv2Ekinetic=1.610-17 J !. 5 :: 1 ( .2 ( 3 (( ) 200 nm .e-+-100 V15 6 ) : 100 W( ) monochromatic ( 560 nm . 100 W=100 J/s .: J Jnmms Js c hh E1998 34 1 8 3410 7 . 310 56010 0 . 3 10 6 . 656010 0 . 3 10 6 . 6 = s photonphoton Js J/ 10 7 . 2/ 10 7 . 310020119 163 ( ) Rutherford ( 1911 ) . Thomson ( : -particles ) .( . . ( ) . ( ) .( ) ) planetary model ( ) ( . 17 ) accelerated motion( ) Hertz.( - . . ) Bohr( . 4 ( ) Line Spectrum ( . 18 ( ) .( ) " " .( ) ) 3 - 10 kV( ) cathode ) ( anode( ) excited ( ) 1s.( " " . 19 . .

( ) . ( ) ! ) .( ) Lyman ) ( Balmer ) ( Paschen .( ) Rydberg( empirical equation ) :( 20

,_

22211 1 1n nR R 1.09678 107m-1 n1 n2 . . 5 ( ) Bohrs model ( ) :(" " 1 ( ) Coulombicforces .(2 ( . 3 ( h . h E EhE i f Ei Ef . 4 (( ) h/2 .,... 3 , 2 , 12 nhn L m r v :21 L ( ) p r v m r L . . -- : = : 22022 104 141rZer q qFrel Ze .= e= 1.6 10- 19As .Z .( ) = r .= r = 1 . : rvm a m F2 v= m= :22024 1rZervm ).............................. 2 ( :2204 1v mZer )...... 3 ( r mZev2024 1 )......... 4 () 2 :( ) ( rmvL22( )2 2 204 2202020202020224224 4444h ne Zvh nZeh nZeLZevZev LZev r v mZer v m ) 3 :(radius Bohre mhaZnae Z mh ne Zh nmZev mZere220020 22 204 22 2 20022024414 )..... 5 ( ) K ) ( V:( ) .( 2 204222 2 204 22 20202 2020202028) 6 ...(18 8 8 14 18 14 12121he mAnZAn hm e Zh nZe m ZerZeV K ErZeVrZer mZem v m Ketotal + ) 7 :( A ) J ( eV ) . 5 ) ( a0 ( .) 6 :( 1 . n = ) 6 .( 2 . ) Z( . ) n=1 ( 23-) . A ( -) 4A . (3 . n ) 5( ) 6 - .( 2 - 4 n) 6.( ) .(4 . n . ( ) . ) n2=nhigh) ( n1=nlow .( ( ) : e-allowedallowedNot allowed24( )

,_

,_

,_

,_

2 22 222 2222221 1 11 :1 1 111 1 1 1high lowhigh lowhigh low low highlow highn n h cAcZ atom Hydrogen Forn n h cZ Accc Fromn n hZ AnZ AnZ AhE Eh hE : c hAR ) R 21 .() Lyman, Balmer andPaschen series.( ) n=1( . ) n=2.( ) n=3.(Lyman seriesBalmer SeriesPaschen SeriesEnds at n=1 Ends at n=2Ends at n=3 UV visible Infra-red) 8 ( !: nlow=2 . ) Z=1 .( n = n=2.( ) 25 n=3 n=2 .( ) nm mm RRRnm mm RRRn nRhigh low5 . 656 10 565 . 610 09678 . 1 5365363653121 17 . 364 10 647 . 310 09678 . 14 44121 11 1 171 7 max2 2max71 7 min2 2min2 2

,_

,_

,_

) 9 (: !) 10 ( ) : Ionization energy( . n=1. n = .. 8 . 908 . 90 10 08 . 910 18 . 210 0 . 3 10 6 . 6. . 10 18 . 211110 18 . 210 18 . 21 1 1 18181 8 34182 218182 2 2 22atom H the ionize will nm Anynm mJs m JsEc h c hh EE I J EJ An nAn nZ A Ehigh low high low

,_

,_

,_

) 11 (: 60 nm .) 10( . . - = ) K=h I.E (. 26 . ) Sommerfeld ) ( Heisenberg( ) elliptical( n . He+, Li2+, Be3+, B4 +.27 1 ( ) ( . ) 22( . " " . 1923 ) Max Born.( ) Quantum mechanics .( ! . - ) Schrdinger( ) Matrix mechanics ( ) De Broglie( ) Wave mechanics .(2( ) ( ) Young( ) constructive & destructive interference( . 28 ) .( ) Compton ( 1922 ) momentum ( " " ) elastic collision .( . !: - . ) p=mv :(29( )v mhphc hc pc hh Ec p c c m c m E 2 ( ) !!!!!! ) 1 :( 60 10 / mm kg s Jsv mhs msmh km v353410 96 . 3778 . 2 6010 6 . 6/ 778 . 2360010000/ 10 . ." " ) 2 :( 450 nm .) 3 :( 100V .:

Angstrom mm kgs s Jv mhs mkgVAsmV evv m V eE Eeekinetic potential22 . 1 10 22 . 198 . 5926738 ) 10 11 . 9 (10 6 . 6/ 98 . 592673810 11 . 9100 ) 10 6 . 1 ( 2 2 2110313431192 ) X-ray ( 1927 ) Davisson ) ( Germer ( ) Diffraction( ." " . 30 . . h/2 . : 1 . ) b.( . 2 . : 222hn L r v mv mhn rv mh n r h/2 .3 . . 3( : 31222 221tyc xy x-axis. c . y . t x " " . time t 1t2t3 t4xY ) x( y ( ) . : x ( ) t. ) t1 ( y x x ) t1 t4(1. : 222 221t c x : ( )( ) t Q x k ie C t x , amplitude C Q k 22 : hEQhEhpkph22 11 . 32 : ( ) ( ) ( )( ) ( ) ) 2 . 2 ...(,) 1 . 2 .....( ) , (/ 2 / 2/ 2 / 2h t iE h p x ih t iE h p x ie t e C xt x t xe e C t xxx p x(px( x . ) x ) ( t .(: ( )( ) [ ]( ) t x Ett xiht xhE itt xehiEe Ctt xh iEt h ixpx,,2,2 ) , (2 ) , (/ 2 / 2

,_

H: ( ) ( ) t x E t x Ht ihH, ,) 3 . 2 .( ..........2 " " H 1 = 1 1 = " " 1 . ) " " O) ( mathematicaloperator.( H H. 4( . . p p . p p p )......2.4 ( p x,t) x :33) 5 . 2 .........(22) , ( 2 2) , (/ 2 / 2x ihp px iht xhpi ehpi e Cxt xx xxh ixpx h iEtx

,_

. : z ihpy ihpz y 22:

,_

+++ + + + z y x ihpp p p pp p p pz y xz y x 2 . :. Operator Laplacianz y xmhz y x mhmpKmpv m K Ekinetic222222222222222222 2228 8 22 21 ++

,_

++ : ( ) x x V V :. ) 6 . 2 ......(8 222VmhV K HV K E E Epotential kinetic total+ + + + ) 2.6 )( 2.3 :( ) 7 . 2 .....() , , , (2) , , , ( ) , , , ( ) , , , (82) , , , (8222222tt z y xiht z y x E t z y x t z y x Vmht iht z y x VmhH 1]1

+ + 34 ) 2.4 :.( 1 . ) 2.4 ( F F F F. F F . . 2 . ) 2.4.( F F ) expectation value( F : + + > < dd FF F** )........ 2.8 ( ) conjugate function ( ) complexfunctions( i. .] a+bi a b i 1 . i ) a+bi)(a+bi)=a2+2abi-b2 ) a+bi ( ) a-bi ( ) a2-b2( i. ) a-bi ( ) a+bi (. ) e-iy( eiy( i [. . F F O O : ( ) ( ) F O O F " " O " " F) 3 5 = 5 3.( F . . 5( ) 2.7 ) ( time-dependent .() stationary( (t ( ) time-independent)(2.9 :(35) 9 . 2 .......( ) , , ( ) , , ( ) , , (8) ( ) , , ( ) ( ) , , ( ) , , , (8) , , , ( ) , , , ( ) , , , (8222222222z y x E z y x z y x Vmht z y x E t z y x t z y x Vmht z y x E t z y x t z y x Vmh 1]1

+ 1]1

+ 1]1

+ - . . ) exact( . : ). single-valued ( x . ( ) ( ) x x=6 .-10 -8 -6 -4 -2 0 2 4 6 8 10x . ) ( ) .( . : dz dy dx dd z y x P d z y x P * ) , , ( ) , , (2 P ) d ) ( x,y,z .( 36 (( (probability density( :dz y x Pdz y x P ) , , (*) , , (2 ) ( . 2 . ) continuous( . . YX3 . ) quadratically integrable.( . : 100% 1. xyz(x, y, z)dxdydz37 d ) x,y,z( d z y x P2) , , ( d. 1 *2 + + d d) Normalization condition ( . 6 ) ( Orthogonality condition () O) ( f( f ) ( ) eigenvalue equation.( f ) ) ( eigenfunction( O ) eigenvalue .(f f O . : 1 . . 2 . ) degeneracy .( . .. n n nE HE H n ) n .( : xyz38n 11sE1s22sE2s32pxE2p42pyE2p52pzE2p63sE3s p. px py pz) degenerate .() non-degenerate :( i j i . 0 0* d di j i j 1* 2 + + d di i i) 4 :( dxd 22x dd.. . eikx. cos(kx ( . k. kx . e-ax2!: ( )ik eigenvalue yesf fdxde ik edxdikx ikx,. ) 5 :( eikx! ) 6 :( A . . =Ae-kx (0 x > ( . Asin(ax/L)(0 x L(39( ) ( )[ ] [ ]k AkAkAekAdx e Adx Ae Aekx kxkx kx2121 02 212 202202 20t ) 7 :( sin(ax/L ( ! ) 8 :( 2e-2x 4e-8x) 0 x > !( 7 ) ( Heisenberg Uncertainty Principle ( - ( = ) ) 37.( :( )( ) ( )( )( ) ( ) x xx xxxp x x px ihxihxxihxxihx p x px ihxx ihx p xx xx ihp 2 2 2 2 2 2 2 +

,_

+

,_

( ) . : 4hp q q p . ) 9 :( 1 2 m/s. . ( )( ) ( )ms m kgs Jqv mhphqv m v m p261 6 33410 6 . 210 2 10 1 14 . 3 410 6 . 64 4 40) microscopic :( ) 10 :( ) interval( 50 pm .( ) ( )s km s mm kgs Jvq mhvm pm q/ 1155 / 115489610 50 10 1 . 9 14 . 3 410 6 . 6410 50 5012 313412 : ( )( ) 422212ht EvEt v p qvEp p v Edp v v m d v dv v m dE v m Et v q t v q

,_

: . ( ) E. t . . . . : 2 2F F F 2F ( ) 2F ) 2.8 .(8( 41 i F F j F ) linear combination,ai+bj.( :( ) ( ) ( )( ) ( ). ion eigenfunct an is b ab a F F b F a b a F thenF F and F F butF b F a b F a F b a Fj ij i j i j ij j i ij i j i j i + + + + + + +42 43 1 ( ) free particle ( + V . . : ) 1 . 3 ( ) () (8) , , ( ) , , ( ) , , (8) , , ( ) , , ( ) , , (8222222222222222x Exxmhz y x E z y x z y x Vz y x mhz y x E z y x z y x Vmh E H 1]1

+

,_

++1]1

+ x k i x k ie B e A + x k ie A x k ie B ) 3.1 .(: A=0 B=0 .( ) 22 2 2 :) 2 . 3 (8) (8 8) (8:022 222 222222222 2 222k hppk he A k iihx ihpp p Momentummk hEx Emk he A kmhxxmhEnergye A k e A k ixe A k ixe ABxxx k ixx xx k ix k i x k i x k i x k i A=0 mk hE22 28 B=0 px 2k hpx . x-axis+-44) 3.2 ) ( not quantized ( k:. ) dx( 2) )( ( ) ( * ) ( **A e A e A e A e Adx Px k i x k i x k i x k i +) x ( :. . : A=B .( )( )( ) ( )( ) ( ) ( ) ( ) [ ]( )( )mk hEmk hkx Amk hkx A kmhx mhHkx A kxkx A kxkx Ax x and x x withkx i kx kx i kx Ax a i x a e Formula s Eulere e A e A e A e B e Ax a ix k i x k i x k i x k i x k i x k i22 222 222 2222222222288cos 28) cos( 28 8) cos( 2 ) sin( 2 cos 2) sin( ) sin( ) cos( ) cos(sin cos sin cossin cos : ' + + + + + + + ) 3.2 :.( [ ] [ ] ) ( cos 4 ) cos( 2 * ) cos( 2 **2 2kx A kx A kx Adx P ( ) ( ) A=B=1/2. B ( ) . ) nodes .(450x2nodes . ) 2Acos(kx (( ) Aeikx+Ae-ikx:(( ) ( ) ( )( ).) (.2 2.) ( ) sin(2) sin( 2 ) cos( 22eq eigenvalue notconst e e Ak hx ihe e A k i e A k i e A k ixe e Aeq eigenvalue not kx Aihx ihkx Axkx Apx ihp px k i x k ix k i x k i x k i x k i x k i x k ix x x + ) 2.8 ( 1 ) :(( ) ( ) + + ++ dx e e Ax ihe e A pdx p px k i x k i x k i x k ixx x 2**( ) ( ) dx e e Ak he e A px k i x k i x k i x k ix+ + 2*( ) ( ) dx e e Ak he e A px k i x k i x k i x k ix+ + 2( ) ( )( ) 0222 2 22 + + + + dx e ek hA pdx e e e ek hA px k i x k ixx k i x k i x k i x k ix46.( ) . ) 2k hpx ( ) 2k hpx ) " " .( superposition( 50% 50% .2 ( ) particle-in-a box ( " " :" " x=0 x=L ) V=0.( x=0 x=L ." " V = x L x 0 . . : x k i x k ie B e A + ) boundary conditions( : x-axis+-+ +V VV=0x=0 x=L47 . x=0 ) 20( ) 0 .(( )[ ][ ]) sin( 2) sin( ) cos( ) sin( ) cos() sin( ) cos( ) sin( ) cos(000 0kx A ikx i kx kx i kx Akx i kx kx i kx Ae e AB A B Ae B e Ax k i x k ik i k i+ + + + + . x=L ) 20( ) 0 .(( )) sin( 2... 3 , 2 , 1 0 sin0 ) sin( 2xLnA iLnk n L kn nL k A i t i : ) sin( 2 xLnA A 100 :%( )LALALxLnnLx dx xLndx xLndx xLnA dx xLnALL LL L2112422sin2 21 2cos 121sin2 cos 121sin1 sin 4 sin 420 0 02202 202 2t 1]1

,_

1]1

,_

,_

,_

,_

. : ) 3 . 3 (8 8822 22 22 2 222 2L mn hL mn hELnkmk hE 48) 3.3 ( ) quantized .( 50% 50 :%) 4 . 3 ......( ..........2Ln hpxt ) x,y ) ( x,y,z :(( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )2 2 222, ,2 222,222222 2, , 2222 2,222222222222228 88 8sin 2 sin 2 sin 2 sin 2 sin 2. , , ,88z y x n n n y x n nzzyyxxn n nyyxxn nzzzyyyxxxyyyxxxn n nL mhE n nL mhELnLnLnmhELnLnmhEzLnA yLnA xLnA yLnA xLnAz y x z y x y x y xz y x mhHy x mhHDimensions Three Dimensions Twoz y x y xz y x y x+ + +

,_

+ +

,_

+

,_

,_

,_

,_

,_

,_

++

,_

+ 49 . . ) degeneracy( . 2208 L mhE . : ( )2 2 20 , , z y x n n nn n n E Ez y x+ + . nx,ny,nz: E nznynx3 E01 1 16 E02 1 16 E01 2 16 E01 1 29 E02 1 29 E01 2 29 E02 2 15011 E03 1 111 E01 3 111 E01 1 312 E02 2 214 E03 2 114 E02 3 114 E03 1 214 E01 3 214 E02 1 314 E01 2 3( ) ( ) . ! ) 1 :( ) 3.3 (! ( ) ( )( ) ( )220120202020 1122020 22 281 2 1 21 2 18 8L mhn E n En E n n E n E n E E E EL mhE n EL mn hEnnn nnn + + + + + +++) 2 :( 500 nm . !E51( )( )( )( )( ) ( )( )nm mJ kgs JLE mhE mhLL mhEJJ EEE E E E E EJms m s Jnms m s J c henergy photon E E E2 10 210 47 . 1 10 11 . 9 810 6 . 68 8 810 47 . 12710 96 . 32727 3 610 96 . 310 500/ 10 0 . 3 10 6 . 6500/ 10 0 . 3 10 6 . 6920 31340022202019 63002020 3 6631998 348 343 663 ) 3 :( 1.0 10- 6g 10- 1cm/s 1 cm ) . n!( ! ( )( )( )( )( )( ) ( ) J J E n EJJnJm kgs JL mhEEEn n E EJ s m kg v m K Ennballballball35 55 19011955165522 92342200201623 9 210 3 . 3 10 44 . 5 10 06 . 6 1 210 03 . 310 44 . 510 510 44 . 510 10 0 . 1 810 6 . 6810 5 / 10 10 0 . 12121 + + ) . correspondence principle( ) n (. ) 4 :( x=0.49L x=0.51 L (( .. : 212 12xxx xdx P .52( )( )( ) % 404 . 049 . 0 51 . 0413729 . 1413729 . 1413729 . 151 . 0 sin2251 . 0 sin22413729 . 149 . 0 sin2249 . 0 sin22sin22sin221251 . 049 . 0251 . 049 . 0221 ,_

,_

,_

,_

,_

L LL LLdx dx PLL LLL LL LLL LxL LxLnLn Forxx .( ) 5 :( . . .( ) . ( )( ) ( )L L L xL nxnxLxLL L L xL nx n xLxLxLxLLxLA xLAx ddLxLAx ddxLAx ddx dd65,63,6161 221 2 303cos33,32,31, 03303sin03cos3sin03 3cos 23sin 4 23 3cos 23sin 20 22

,_

,_

,_

,_

,_

,_

,_

,_

,_

,_

: 53( )

,_

erf dy e dy e P Pdy e Pynyx totalnyx erf ) error function ( 2ye) normal distributionfunction ) ( Gauss distribution function ( ) standard deviation ) ( variance ) ( ..( 66 16% . n n ) n .( 4 3 2 1 0 n0.0785 0.0855 0.0951 0.1116 0.1573P) Zero-point energy ( . ! ) entropy( . ) 4.4( n 0 021 E . .( ) : - : 2 22121v m x k EE E Ekin pot+ + - : ( ) ( )2 22121v m x k E + - : ( ) x mvx xhp 2 2 4 ( )( )22218 21x mx k E+ - x x : 67( )( )( )( ) m m k mxm kmkk mxx mx k2 4 440142 22 222 22432 - x : ( )( ) 214141 28 2 21 18 2122222 + + + mm mmx mx k E . ) 7 :( 1H-35Cl ) 3 .(( ) J s Js h E20 1 13 340 010 85 . 2 10 65 . 8 10 6 . 62121 ) 8 :( D-35Cl 1H-35Cl .68 ( ) .: ( ....) : ) , , , , , ( ) , , , , , (82 2 2 1 1 1 2 2 2 1 1 1222z y x z y x E z y x z y x Vmh

,_

+ ) x1,y1,z1)( x2,y2,z2 ( . : ) , , ( ) , , ( ) , , , , , (2 2 2 1 1 1z y x Z Y X z y x z y x ) 5.1 () (X,Y,Z ) center ofmass" "( . ) translationalmotion . ( ) (x,y,z . ) 5.1 :( ) 2 . 5 ( .......... ) , , ( ) , , (8) , , ( ) , , ( ) , , ( ) , , (8222222z y x E z y x Vhz y x Z Y X E z y x Z Y X Vh

,_

+

,_

+ . 5.1- m (reduced mass( m1m2.(x,y,z)(X,Y,Z)center of mass692 12 12 13 2 11 1 1:...1 1 1 1m mm mm msystem particle two Form m m+ + + + + : enn en en emmm mm mm m+ ) 5.2 ) ( V( : rZeV024 : ) , , ( ) , , (4 802222z y x E z y xrZe h

,_

x y z r x y z r :. .................cos sinsin sincos++ zyx x r xrxr xr yr z ) x,y,z( ) r ( z x . x y z ) 22x 22y 22z ( r : xyzr(x, y, z)70222 2 222 2222222sin1sinsin1 1 +

,_

+

,_

,_

++ r r rrr r z y x: ) , , ( ) , , (4 sin1sinsin1 1802222 2 222 22 r E rrZer r rrr rh

,_

1]1

+

,_

+

,_

r . :0 ) , , (48) , , (sin1sinsin1 10222222 2 222

,_

+ +1]1

+

,_

+

,_

rrZeEhrr r rrr r) variable seperation( r : ( ) ( ) ( ) ( ) r R r , ,: ( ) ( ) ( )( ) ( ) ( ) 048sin1sinsin1 10222222 2 222

,_

+ + 1]1

+

,_

+

,_

r RrZeEhr Rr r rrr r: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) 048sinsinsin0222222 2 222

,_

+ + +

,_

+

,_

r RrZeEhrr Rrr Rrr Rrr r ( ) ( ) ( ) r R r2 sin2 :( )( )( )( )( )( )04sin 8 1sinsin sin0222 2 22222

,_

+ + +

,_

+

,_

rZeEhrrr Rrr r R : ( )( )( )( )( )( )220222 2 2221sinsin4sin 8 sin

,_

+

,_

+ +

,_

rZeEhrrr Rrr r R)5.3 (: r . r r r. . m2 m . ) -equation (71: ( )( ) ( )( ) 2222221m m)...5.4 ( ( )m ie A ) .( A : ( ) ( )( ) 211 2 11 1202 22020* A A d Ad Ae Ae dim im 0 2 ) . 76( . z )360=2.( . :. ( ) ( )( )( )...... 3 , 2 , 1 , 01 2 cos1 ) 2 sin( ) 2 cos(1222 2t t t + + +mmm i mee e A e A e Am im i im im im m m ) quantum number .() -equation () 5.4 ) ( 5.3 ( sin2 : ( )( )( )( ) 220222 22sinsinsin148 1 mrZeEhrrr Rrr r R+

,_

+

,_

+ +

,_

: ( )( )( )( )

,_

+

,_

+ +

,_

sinsin1sin 48 1220222 22mrZeEhrrr Rrr r R r l(l+1 ( :72( )( )( )( )) 1 ( sinsin1sin 48 1220222 22+

,_

+

,_

+ +

,_

l lmrZeEhrrr Rrr r R )5.5 (( )( )) 1 ( sinsin1sin22+

,_

+l lm ).... 5.6 () -equation ( ) Legendre( ) Legendre Polynomials .( l l m l .l mlt t t t ... 3 , 2 , 1 , 0..... 3 , 2 , 1 , 0 m l. m l m m m l .l ml,m( ( m( (0 0 12121 032 cos( ) 121t 132sin( ) 12ei t2 0( )1583 12cos 122t 1 154sin cos 12ei t2t 2 15162sin 122ei t) R-equation () 5.5 :(( )( )) 1 (48 10222 22+

,_

+ +

,_

l lrZeEhrrr Rrr r R : 732 2 204 28 n h e ZEn )....5.7 () 5.5 ) ( Laguerre ) ( LaguerrePolynomials ) ( Laguerre.( n l ) n-1 .(( ) 1 ... , 3 , 2 , 1 , 0... , 3 , 2 , 1 n ln Rn l n l. n l .n l Rn,l(r) 1 02 /2 / 302

,_

eaZ2 012 220322Zae

_,

//( ) 2 112 60322Zae

_,

//3 0( )19 36 60322 2Zae

_,

+//3 119 640322Zae

_,

//( ) 3 249 300322 2Zae

_,

// : 02ar Z a0 2200e mhae. R. ) d.( R R2 . 1s r ) exponentially .(74) spherical shell ( r. ( ) dr :on Distributi obability Radial R rdrr Pdr R r d R r Pdr r dPr 4) (4 ) (42 22 2 22 1s .75 1s, 2s,3s 2s, 3s . ) 1 :( 1s . ( )( )( )0202 2022222322302 22302220330322 /2 / 3022 20 220 220016 16 4414 42max 040 0 00 000a r r rae r ear e rdrde rdrdae ra drddrR r deaR ZeaZeaZeaZRimumdrR r dararararararar Z + +

,_

,_

,_

,_

,_

,_

a0 !! 76) orbital ( " " !! !! " " 90 % 90.% 10 !% ) 2 :( 1s . 1s 02 / 3011a r ZseaZ

,_

) 81 .(( )( )0202302203020210212 / 30166 . 2 : ,90 . 0490 . 0 4190 . 0 4 90 . 01 1000a r r for solve Integratedr e raPdr r eaPdr r P d Peaarrarrrsrsars 77 R n l m : m m l l n m l nR . ) 3 :( : n=3,l=2, m=1 . i a ria r a Zre eareearear ZaZeaZRR21cos sin21530 91621cos sin21530 916 230 9430 9400 03 /2 / 7021 2 31 1 23 /2 / 7023 /202 / 306 / 22 / 302 31 1 2 2 3 1 2 3

,_

,_

,_

i . )5.4( m2 m . i i:1 121211 12 21 1 m me em mi i i i ) 45 .(( ) ( )( ) [ ]( ) sin sin21cos) sin( cos( sin cos212 1211 11 1 + + + + + ii ie eyi ix i i x y i. x 1 ( ) coscos sin21530 91603 /2 / 7021 1 2 3 +a rear).....5.8 ( m=1 ) 1 -+ 1( i i . ) 76( 78 coscos sin zx) 5.8( x z. 3dxz . y 1 3dyz.( ) ) 4 :( 2px 2py 2pz. n 2px 3px 1s 2s n R . . 1s n=1 l=0 m=0. : 2 /2 / 300 0 11

,_

eaZ s ) fully symmetric.( pz z ) 4 ( . - 81- . Ylm Y ) angularfunction ) ( spherical harmonics( : m l l n m l nm m l l n m l nY RR n n ) 82 .()=: n-l-1 .(79 3p80) angular momentum ( ) p r L .( x ) xL( y ) yL ( z ) zL .( . x-y z. ) ( ) : .(x z y zp y L p x L z zL . - x-y z . z x xy yxpyypx81 x-y z ) A z ( z. z zL. z zL: x y zp y p x L

,_

xyyxihp y p x Lx y z 2

,_

+

,_

+

,_

++

,_

+

,_

+

,_

+

,_

+

,_

+

,_

,_

,_

+ + + + r r z z r zrzr r r y y r yryr r r x x r xrxxyr zz y xzr yz y x r r xx y x y x yz x z x z xz y z y z ysincossincos sin cossin sinsinsin cos coscos sintan coscos sin sincos sin, , ,, , ,, , ,12 2 212 2 2: ihLz2: xyzpypxLzz82

,_

,_

,_

,_

sin cot cos2cos cot sin222 ihLihLyzzyihLzxxzihLp z p y L p x p z Lp z p y L p x p z Lyxx yy z x z x yy z x z x y: 1]1

+

,_

+ + + + 222 2222 2 2 22 2 2 2sin1sinsin14 hLL L L LL L L Lz y xz y x ) zLyLxL ( ) L2 .( z ) zL ( :). 5.9 ( z zz z z zLihL RihRR L RihLihL L) (2) (22 2 ime21 2 212) (2mheiimhihLimz(5.10) 2hm Ll z) 5.10( m z ) zL) ( zL( L. l m m . L : 83( ) ( ) ( ) ( ) ( ) 1]1

+

,_

1]1

+

,_

1]1

+

,_

R L Lr R hLRhLhLL L2 2222 222222 222222 2222 2sinsinsin 4sin1sinsin14sin1sinsin14 )...5.11 ( R :( )( )( )( )( )( )( )( )22222222222222 22sinsin1sin 4sinsinsin141sin1sinsin14Lm hLm hLh1]1

,_

1]1

+

,_

1]1

+

,_

) 5.6.( ( ) [ ]2) 1 (14222hl l LL l lh + +)..... 5.12 ( l . : l 0 L n=1 . L h/2 2a a a a +2 . ( ) . z . zL)5.9( 2L) 5.11 ( . zL2L) L2 ( z (Lz ( ) . Lx ( .84 const Lx? xL ( ) Lx Lx Lz Lx L2 . Ly. L2 Lz Lx Ly) indeterminate ( L. s l=0 ml=0 L2=0 Lz=0. p l=1 p ( ) 2 / 2 2 / 1 1 1 h h +. ml p (ml=-1, 0,1( z ) 5.10 :.( x-y . Ly Lx z Lz.x-y planezml=1ml=-1ml=0LLLLzLz85) 5 :( d ).() 6 :( n l ml . n ) principal quantum number ( l ) angularmoment quantum number ( ) magnetic moment ( ! : A I . 86e- v x=v.t. 2r v t/2r v/2r ) . - e( ev/2r. ) I=Q/t ( I r ev 2 / . 2 22r v errv eA I l ) orbital motion( . )- magnetogyric ratio( : eelmer v mr v eL 2 2 )5.13 ( . )- 5.10( z ml:B l z LeBel lez z LmAmmh emh emhmmeL

,_

,2 24,10 274 . 944 2 2 ml B) Bohrsmagneton.( . rml 1 0-1external magnetic field87) 5.13( x y z ) 2,,,M M M Mz y x( 2,M Mz 2l z z l , x y )y xM M,..( p : p. B ) z( ) Emagnetic) ( B(: B Emag B m B B B B EB l z mag + cos cosx-y planeml=1LLzzx-y planeml=-1ml=1ml=0BBml 1 0-1external magnetic field88 p ) ( ) degeneracy ( p ) E0( :B m E EE E EB lmagnetic + + 00 ) nuclear magneticresonance, NMR ) ( electron spin resonance,ESR )( magnetic resonance imaging,MRI .( 1)spin motion ( 1921 ) Stern ) ( Gerlach ( . 1925 ) Goudsmit ) ( Uhlenbeck( ) spin angular moment ( ) finestructure . .( 1 1 .energyml 1 0-1No external magnetic fieldEnergyE0+BBE0E0-BB external magnetic field89 1929 ) Pauli ) ( Dirac( . ( ) . !" " : ( )21212 21t + + ss zm shm Shs s S S s Sz z ms . spin ms.90 ) spin magnetic moment( s: Be ez ssez sz e z see ssmh emh ehmmeS gfactor g electronic gS g electrons forS t

,_

t

,_

,_

4 2 212 22) ( 0023193314 . 2,,, ) 98 ( . 5s ( ) s ) l ( . ) 7 :( 1s . ) 75 84( 91) x,y,z r,, ( n,l,ml :. ( ) down spinup spin orz y xspinm l n spacespin space totall::, , space) space( spin) spin.( ) " " up ) " " ( down( spin .92 ) exact analytical solution( . . : 12 022 0222221 0221224 4 8 4 8rerZe hrZe hH +

,_

+

,_

- . . : ( )

,_

+

,_

2 0222 221 0221 224 8 4 8rZe hrZe hH : ( )2 2 2 2 1 1 1 1, , ) , , ( z y x z y x )6.1 (: e-1r1e-2r2r1293( ) ( )( ) ( )2 12 1 2 2 1 1 1 22 1 22 0222 221 11 0221 2222 1 2 12 0222 221 0221 222 1 2 14 8 4 84 8 4 8E E EE E EErZe hrZe hErZe hrZe hE HE H+ +

,_

+

,_

1]1

,_

+

,_

.( ) ) 5.7 :(eV En nn n hen he Zn h e ZE E E8 . 10811 18 28 82 122212 204 2222 204 2212 204 22 1

,_

+ + + ) - 79 eV( .- . : ( )( )2 2 2 1 1 1 1 11 1 1 1 1, , , , ,, ,z y x z y xz y x ) 6.1 ) ( approximation( ) perturbation theory .() shielding ) ( penetration () n( ) l( 94) 1s2 2s1( 2s1 . 1s + 3. 1s 2s . 2s + 1 .( - ) 2s 1s 2s ) .( 2p2s1s4r2R2r 2s 1s. 2s 2s + 1 +) 1.28 .( 2p 1s 2s 2s 2s + 1.02 . 2s 2p 2p. 3d 3p ) n=1, n=2 .(E EH-like atomMulti-electron atom3d3d3p3p2p2p1s 1s2s 2s3s 3s95x53d3p4r2R2r) Indistinguishibility principle ( . ) 1s2 .( n l ml ) Paulis exclusion principle -( " " " ." " " . " " " " " " ." " " " 50" " % 50 % ." "" " ) . 1s2 2s1( E1 1s 2s E2 - . 2s E2 1s E1 . 33% 2s 67 % 1s . ) 1s1 2s1( i :( ) ( )( ) ( ) 2 12 12 1 s s totaltotal 1s 2s 96 2s 1s. ( ) ( ) 1 22 1 s s total . : [ ] ) 1 ( ) 2 ( ) 2 ( ) 1 (212 1 2 1 s s s s space t t)6.2 ( 21 ) ( ) determinant :(( ) ( )( ) ( ) 2 12 12 21 1s ss s n ) Slater :() ( ) 2 ( ) 1 () ( ) 2 ( ) 1 () ( ) 2 ( ) 1 (2 2 21 1 1nnnn n n n .) Pauli Principle () 1s1 2s1 ( ) spin)spin,1 ... spin,4 spin,3 spin,4 :( ) [ ] ) 2 ( ) 1 ( ) 2 ( ) 1 (21 t tspin) space ) ( spin( ) exchange of electrons :.( spin space totaltotal totalP 121s2sEspin,1=(1)(2) spin,2=(1)(2) spin,3=(1)(2) spin,4=(1)(2) .97 12P ) exchange operator( . : [ ][ ][ ] [ ][ ] [ ] + + + + spin spinspin spinspin spinspin spinP PP PP PP P ) 1 ( ) 2 ( ) 1 ( ) 2 (21) 2 ( ) 1 ( ) 2 ( ) 1 (21) 1 ( ) 2 ( ) 1 ( ) 2 (21) 2 ( ) 1 ( ) 2 ( ) 1 (21) 1 ( ) 2 ( ) 2 ( ) 1 ( ) 1 ( ) 2 ( ) 2 ( ) 1 ( 12 1212 122 , 12 2 , 121 , 12 1 , 12) 6.2 ( : [ ][ ] + + + + space s s s s s s s s spacespace s s s s s s s s spaceP PP P ) 2 ( ) 1 ( ) 1 ( ) 2 ( ) 1 ( ) 2 ( ) 2 ( ) 1 (21) 2 ( ) 1 ( ) 1 ( ) 2 ( ) 1 ( ) 2 ( ) 2 ( ) 1 (212 1 2 1 2 1 2 1 12 122 1 2 1 2 1 2 1 12 12 : ++ spin space totalspin space totalspin space totalspin space total 2 ,1 ,) triplet state( ) S=s1+s2=1/2+1/2=1 ( ) singlet state( ) S=s1-s2=1/2 - 1/2=0.( ) S=1( ) S=0 .( .98) S( :( ) ( ) ( )( )( )... ,23, 1 ,21, 0.... ) 3 . 6 .....(212121... .....2 12 1 2 1t t + + + + + + + Ss s ShS S Sshs s sm m m M s s s Si i is sii s sii ) S ( ) is ( ) 6.3 ( S . ) triplet state( ) singlet state ( ) P ) ( r1=r2 ( . SSSs1s1s1s2s2s2z z zMs=ms1+ms2=1 Ms=0Ms=-1 .zSs1Spin=0Ms=0s2990r1-r20singlettripletp(r1,r2)) 1 :( 100 )Rotational motion () particle- on-a ring ( ) point particle( m x-y . 221 I Erot 1 I ) moment of inertia( :. ii ir m I2 mi ri . I=mr2.) 22 :( I L : ILErot22 ) 7.1 ( : x-y z . 1 1 ) 221v m Ekin ( .xmry101) , ( ) , (8) , , ( ) , , ( ) , , (8222222222y x E y xy x mhz y x E z y x z y x Vmh 1]1

,_

+1]1

+ x y r 222 2222221 1 +++r r r r y x r r : ) () (8) () (8) , ( ) , (1 18) , (82222222 22222 2222222222 EIhEr mhr E rr r r r mhy xy x mh

,_

++

,_

+ ( ) m ie A A m:. ( ) ( )) 2 . 7 (822 22 2 222Ih mEm e A m i m i e A m im i m i + ) 7.1 :( 2 82222 2h mIh m IE I Lrot ) 7.3 () 7.3 ) ( 5.10( x-y z ) 22 88 ( L ) 7.3( Lz m m=0, 1, 2, 3 ( ) ( ) 2 + ) 78 .() particle-on- a sphere ( m r. 102 ) V.( R ) 76 :(( ) ( ) ( )( ) ( ) ( )

,_

+

,_

+

,_

1]1

,_

++1]1

+ r ERr Rr r rrr r mhz y x E z y xz y x mhz y x E z y x z y x Vmh222 2 222 2222222222222sin1sinsin1 18) , , ( ) , , (8) , , ( ) , , ( ) , , (8) r ( r : ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )

,_

+

,_

,_

+

,_

r R Er mr R hr R E r Rr mh222 2 22222 2 22sin1sinsin18sin1sinsin18( )( ) ( ) ( ) ( ) ( )( )( )( )( )( )( )( )( )EhIEIhr R Er mr R h22222222 22222 2 228 1sin1sin1sin11sin1sin1sin18sin1sinsin18

,_

,_

+

,_

,_

+

,_

) 5.4 78( rxyz(x,y,z)103( )( )EhI m22228sin1sin1sin

,_

) 5.6 79 ( ( )( )) 1 (88) 1 ( sin1sin1sin222222+ +

,_

l lIhEEhIl lm m m l m lY m l . J l IhBJ J h B J JIhErot2228) 1 ( ) 1 (8+ + B ) rotational constant.( ) rotational quantum number( J=0, 1, 2 , . 2) 1 ( 2hJ J E I L + ) Rigid Rotor ( ) . dimensions( . ) 75 - 76 ( xyzr0.m1m2r1r2104. : r : ( )( )( )( )2222 2 21sinsin sin 8

,_

+ Ehr ) 112( ) 1 ( ) 1 (822+ + J J h B J JIhErot 2) 1 ( 2hJ J E I L + : 22 221 12r m r m r m Iiii+ r1 r2 m1 m2 . m1 m2 ) ( ) first momenta( 2 2 1 1r m r m 0 2 1r r r + ( )( )02 11202 12102 1210 2 2 1 11 0 2 1 1rm m mrrm m mrrm m mrr m m m rr r m r m+++ + ( )( )20202 12 11 220 22 12 12022 1122022 121r rm mm mm m rm mm mIrm m mm rm m mm I + ++

,_

++

,_

+105 : ) 1 :( ) O2 ( 120.8 pm ) J=1.( ( ) ( )Jsh h hJ J Lm kg m kg r Ikg gmolmol gNM mmmm mm mNMm m mavOavOO342 43212 26 2026 231 23111212 12 12 110 486 . 12221 1 ( 12) 1 (10 939 . 1 10 8 . 120 10 3285 . 110 3285 . 1 10 3285 . 110 02 . 6 299491 . 152 2 2 + + + 106( )Jm kgJsIhIhJ JIhErot262 43 223422222210 696 . 510 939 . 1 14 . 3 410 6 . 6428) 1 (8 + : 2 12 1 2 2 1 1r rm m r m r m . ) 2 :( 1H-35Cl 1.283 . J=2 . 22212IEI E107:1. Physical Chemistry, K.J. Laidler and J.H. Meiser, Houghton Mifflin Company, 3rd edition, 1999.2. Physical Chemistry, G.W. Castellan, Addison Wesley Publishing Company, 3rd edition, 1983.3. Atkins Physical Chemistry, P. Atkins and J. de Paula, Oxford University Press, 7th edition, 2002.4. Atomic Spectra, T.P. Softly, Oxford Science Publications, 5. Principles of Quantum Mechanics as applied to Chemistry and Chemical Physics, D.D. Fitts, Cambridge University Press, 2002.6. Quantum Chemistry, I.N. Levine, Prentice Hall, 5th edition, 1999.108 436/ / 1 . He+. 2 .. 10 V . ... 3 .. 5.7 V . . .4 . ) . Px.( . .. . A .5. !436/ / 1 . 0.1 nm .1 . . 2 . . 3 . ) 0 > x L ( . 2 . ) planar( 2.5 . : 1 . HOMO LUMO .2 . HOMO .3 . LUMO LUMO .( ) 3 . ) px.( 436/ / 1095 .. 10 V . . . 1nm !( ) 2 . 40( ) 1 nm. ( ) : . HOMO LUMO . . ) ( . HOMO LUMO . . HOMO () 3 . : . 2s . . 2s . . 2s .4. 2px) i .(5 . . d ).( . d ( ) 6 . . n l ml ms . 2p 2s 2p 2s . . . . 110