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# Physical Chemistry CHEM 3510

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This 19 page Class Notes was uploaded by Rebeka Zemlak MD on Wednesday October 21, 2015. The Class Notes belongs to CHEM 3510 at Tennessee Tech University taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/225695/chem-3510-tennessee-tech-university in Chemistry at Tennessee Tech University.

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Date Created: 10/21/15

CHEM 3510 Fall 2006 Unit IV Applications of Quantum Theory A The Particle in a Box 1 Introduction a The particleina box refers to the quantum mechanical treatment of b One tries to solve the Schrodinger equation to obtain the wavefunctions and the allowed energies for the particle c The timeindependent threedimensional Schrodinger equation iii Vxyz yxyz Eyxyz 2 6x2 6y2 622 2 Particle in a onedimensional box or onedimensional motion a Consider only a onedimensional motion it along x coordinate 3 drop y and z in the equation above b Consider that the particle experiences no Potential energy potential energy between the position 0 anda Vx0for0 SxSa c The Schrodinger equation for this problem 0 h2 d 211 3 E a a 2m dxz 1 w H w u d The mathematical solution of this equation is yx AcoskxBsinkx l 2 Where k 2mE 27239 thE 22 Ehk 872392m CHEM 3510 Fall 2006 e Apply the boundary conditions and the normalization condition to determine A and B qO 0 3 A 0 ya 0 3 Bsinka 0 3 ka 147239 where n 12 f The allowed energy levels are 2 2 h 14 3E 2 wherenl2 ma 100 Classically allowed 9 energies where n is called D The energy is 8 El The energy levels increase as h28m a2 D The energy separation between the allowed 3 energy levels increases D The energy levels and the energy separation between the energy levels increases as g Quantum numbers appear naturally when the boundary conditions are put in the Schrodinger equation like in the string problem and are not introduced ad hoc like in Plank model for blackbody radiation or Bohr model of hydrogen atom h Determining the wavefunctions E Find B by setting the condition that the function 1x is normalized I 1 xwxdx 1 0 a 3B2jsin2 l3Bz l3BZ 0 a 2 a o B is called CHEM 3510 Fall 2006 D The normalized eigenfunctions x are given by l 2 Lynx3 sin whereOSxSaandn12 a a 543 2 1 i Solving the Schrodinger equation for the particle in a box problem gives a set of allowed energies or eigenvalues and a set of wave functions or eigenfunctions j Representations including the energies and the wave functions a and the probability densities bc for the first few levels for the particle in a box N 4 waitquot 5 maul 91 n2 3 8mm 39 mar l 772 1 NIH fr 3m 39 n11 CHEM3510 k El 0 Fall 2006 I39 40 i rr T p 39 b d can b freely in a molecule also called Example Butadiene has an adsorption band at 451x10 cmquot As a simple approximation consider butadiene as being a onedimensional box oflength 578 A 578 pm and consider the four 7 electrons to occupy the levels calculated using the particle in a box model B o The electronic excitation is given by AE h 2 32 e 22 Smea o The calculated excitation energy i 3 i 454x104 cm 1 115 compared Very well With the experimental Value quot o This simple can be quite successful The probability of nding the particle between x1 andxz is given by 2 a I mom mix I1 El Ifx10andxza2then Prob 03x SaZ12 for alln El Forn 1 Prob 03x 3114 ltProb 114 Sx 3112 D As It increases for example It 20 these 2 probabilities become equal Generalizing the probability density becomes uniform as This is an illustration ofthe that says that quantum mechanics results and classical mechanics results tend to agree in the limit of large quantum numbers D The largequantumnumber limit is called CHEM 3510 Fall 2006 p The eigenfunctions are ill11xlnxdx 0 Where m i n 0 El Example look at ylx and 113x a a 0 J39yfxl3xclx EJ39sin sin3 xabc 0 0 a0 a a q Average values and variances for the position and the momentum of the particle a 2 a mzx a 7 Average value ofposition x JynOc x yn xdx chsin2 dx 0 a 0 a 2 a 2 2 2 2 a a 7 Average value ofpos1t10n square x gt 11 x x z1n xdx 0 3 271272392 2 2 2 a 2 2712 7 Variance in position ax x x 2 2n7239 3 2 2 12 7 Standard deviation in position o x 1x2 x2 2a 3quot 2 717239 27m a nmc mac 7 Average value ofmomentum p ih 2Jsin cos dx 0 a 0 a 2 2 2 2 2 2 2 2 2 h h 7Also p2 2mE 2mquot 2quot 2 a 2ma a a nznz 3 12 7 Finally axap 2 gt g Heisenberg Uncertainty Principle 3 Particle in a twodimensional box or twodimensional motion a The Schrodinger equation for this problem 2 2 2 3 h Mira W Ewxy 2 6x2 6y2 Potential a b Allowed energies or eigenvalues h2 n 7 quotxny g a2 b Z Particle con ned to surlace CHEM 3510 Fall 2006 b The wavefunctions or eigenfunctions n 11xy X xY y gummy 7W a a 4 Particle in a threedimensional box or threedimensional motion a The Schrodinger equation i 52W 52V 52v Eyxyz L l 2m 6x2 332 622 b Solutions 2 2 2 nx x ny y nzm x Z Slll 111 5111 annynzlt y a b c 2 2 2 2 E J x quoty z quotxnynz gm a2 5 2 02 c Average values of the position and the momentum of the particle 11 b c A a b c A A A A rJdedyJdzz xyzRzxyz31EJEk where RXiYJZk 0 0 0 ded id gtkx z13 x z0 Where 13 ihrii39ikil P y w y 11 y Kax Jay 62 211II1I212III2I 3 0 0 0 1Hnr1 Numeracy 19 35 1 313 l33 3 d The case of a cubic box a b c 13 H i lzijl394lllm 3 7 d332l139 3923 3 El Three sets of quantum numbers H 1311th gum 1 111 1 33912 Ir give same energy u an 1 6722 3 l1 qa1111l31ni3 3 E211 E121 E112 2 I 9 221u12u22y 3 8ma 2 1111 1 CHEM 3510 Fall 2006 6722 CI The energy level E is 8ma 3722 CI The energy level E111 2 is 8ma El Degeneracy is equal to 0 Once the symmetry is destroyed then degeneracy is lifted 5 Separation of variables a The Hamiltonian for the particle in a 3dimensional box 2 2 2 2 192 aaa Hx my 192 21 5x2 ay2 622 El It can be written as a sum of terms where 2 2 2 2 2 2 A h a A h a A h a H H x Z Qj y 2may23 2maZ2 o The operator is said to be b The eigenfunctions llnxnynz are written as a product of eigenfunctions of each operator H x H and H Z and the eigenvalues Enxnynz are ya written as a sum of the eigenvalues of each of the operator H x H y and H Z c This is a general property in quantum mechanics If the Hamiltonian or an operator in general can be written as a sum of terms involving different coordinates ie the Hamiltonian is separable then the eigenfunctions of H is a product of the eigenfunctions of each operator constituting the sum and the eigenvalues of H is a sum of eigenvalues of each operator constituting the sum d Example H H1s192w 2 wnmo w sww and Enm En Em where 1910mm Emma and 192wwmw Emumm CHEMKSIU FallZEIEI B The Harmonic Oscillator 1 Introduction 1 b The harmonic oscillator in classical mechanics ereree f meg k2 Huuke39stWwhErekxs the farcecanmm 2 r 7 e Huuke39s Law cumbmed wtth Newmh39s equauun m k2 n m 1 1 x eThe sehheh quhs equanun m e Acusm where h eJZ quot011770 k m WI A eThepetehhel energy quhe usmllatur Nae 8 1m ees2 ht V 2 WI emehhehemehgyefmeesnuetm 19quot Hquot eThetete1 energy quhe usmllatur E V K 7th The tete1 ehegyts cunserved tt is transferred between Kznd V C The harmonic motion in a diatomic molecule 7 cehsae the meyemeht uf atums ufmasses m1 and M dzx t m 2 em e x ezn Mt quot Jamt rt39m39d 152 k 7 72 1 1 m d 2 x n e 1 t 7 By summmg the twu equanun abuve t 2 VI X o where Mmtm2 and X M he M thEreXxs the cemerthe mass coart mte e Subheetmg equatmn z hymea by m hem equanun 1 dwxded m 2 u kx0 where Keefe e2 and he Wquot it quotI m2 Where h 15 the quotshared me quhe system and ms the relanve commute The movement ofa twobody system can be reducedto the movement of a onebody system with a mass equal to the of the twobody system a CHEM 3510 Fall 2006 2 The quantummechanical harmonic oscillator a The Schrodinger equation for quantummechanical harmonic oscillator 2 2 h d Z Vxwx Ellx 2 dx where Vx kx2 Potential V 2 3 d wx2 dxz h2 The eigenvalues are En h nlhmnl hvnl u 2 2 2 where n012 a E Vi E 2 1 The wave functions eigenfunctions are E ka Jyx 0 Dispiacement x k 2 zjnx Aana12xeiwc 2 where a 2 E 14 1 a D The normallzation constant 1s glven by An 2 7 The wave functions form D D The H quot011 2x are polynomial functions called Hermite polynomials where H quotg in a n3911 degree polynomial in 5 0 Here are the first few Hermite polynomials Ho51 H1E2 H2lt54272 H3E8 3712 H4E16 4748 212 H5E32 57160 3120 even polynomials x ix odd polynomials x i ix D Continuous odd functions properties f 0 0 and I f xdx 0 7A CHEM3510 P 0 2 Fall 2006 The normalized wave functions and the probability density a e 1 mm Numl 2539 l l i quot i wry VAL QD LM N v W H r x J 1 w ml A Mmf a ML 1 i I or gm w n 2 S u 39 The existence of D The minimum energy the groundstate energy is not zero even for n 0 D This energy is called ZPElhv 2 D It is a result in concordance with the Uncertainty Principle that says one cannot determine exactly both the position for example x L for p y The quantummechanical harmonic oscillator model accounts for the IR spectrum ofa diatomic molecule D It is a model for Vibrations in diatomic molecules D The transitions between Various levels in harmonic oscillator model follow An 1 D The quantummechanical harmonic oscillator predicm the existence of only one frequency in the spectrum of a diatomic the frequency called AE Vobs En1 TEH Fa112006 gtAEh i nllj nlj 1 i y 2 2 27 y l k N l k Vobs and Vobs 27 y 27w y 0 The quantity x from above is in this case the difference between the interatomic distance during the vibration and the equilibrium distance x l 10 0 If the fundamental vibrational frequency is known one can determine the force constant as k 27w 70bs 2 2 Vobs 2 Typical values of 17GbS are The probability density is different than 0 bigger than 0 even in regions Where E lt V El This is equivalent to a negative kinetic energy and is an example of a property of quantum mechanical particles that is noneXistent in classical mechanics CI The quantum mechanical particles have the property of nonzero probability in regions forbidden by classical mechanics The average value of position and momentum for the harmonic oscillator ltxgt Jwxxwltxdx 2 ltxgt 0 ltpgt I wxgt ihdiwltxgtdx 2 ltpgt o 00 x Note that vibrational quantum number n is usually labeled as v in other textbooks Fall 2006 CHEM 3510 C The Rigid Rotator 1 Introduction 3 The rigid rotator or rigid rotor refers to the quantum mechanical treatment of The quantum mechanical rigid rotator is a model for a rotating diatomic molecule c Treatment of rotation in diatomic molecules 7 For the center ofmass mm m2r2 l 7 Velocities v1 r1 272ert Fla v2 r2 272391r0t rzw Where a Z vrot is the I 39 m 7 w quot i I Mich angular speed l aiinaxs 1 2 1 2 L2 139 2 g 2 2 21 m1m2 K1net1c energy K Emvlz m m1 m2 7 Moment of inertia I mlrlz m2r22 ur2 where r r1 r2 and u 7 Similar to moment of inertia for a single rotating particle I mr2 CI The movement of a twobody system can be replaced by the movement of a onebody system Where the mass is replaced with the reduced mass 2 Quantummechanical rigidrotator model The Schrodinger equation 72 a VV0HK V2 2 2 2 2 V2a26262 6x 6 62 2 16 23 1 isingi 2 12 62 6 9 up r sin 9 6 mg r r2 6V 6 quot aquot r2 sint9 6 9 b The special case when r is constant ie the rotator is rigid ll93 EY9 Where Y 6 are the rigidrotator wave functions and are also known as CHEM 3510 Fall 2006 32 1 a a 1 62 E s1nt9j Sin2 66 7Y9 EY9 7 Further rearrange E 2sz 66 66 32 r12 7 The rigid rotator wave functions Y 6 will be given later 2 3 sin6isin6a Ya sin26Y 0 where 66 c By solving the equation above one can determine that 8 must satisfy 8 J J 1 1 The discrete set of allowed energy levels is given by 2 EJ 22 1JJ1 where J 012 6 Each level has a degeneracy given by f The for the rigidrotator model 0 Only transitions between adjacent states are allowed g The energy and frequency for transition between levels 2 AEZEJ1 EJ h 2Jlhl 47 I L 4721 h The frequency of these transitions is about 1010 1011 Hz which is in gtV J1 the microwave range of electromagnetic radiation gt i Usually write the frequency in terms of V ZBJ1 where B 2 L2 87 I El Typical values are CHEM 3510 Fall 2006 j In terms of wavenumbers V2 Jl Where E 87ch El Typical values are k From experimental 1N3 one can determine the moment of inertia I then the bond distance r in a diatomic molecule 1 M2 317359 J 5555 Rotational energy leveEs w ail 45 SLE 21339 Syech39um fl j 5 a 3 a 25quot m 639 1 Note that rotational quantum numberJ is sometimes ie in our textbook labeled as 1 CI The quantum number I is usually used for the orbital angular momentum of an electron orbiting around a nucleus and the quantum number J is usually used for rotating molecules 60 CHEM 3510 Fall 2006 D Tunneling 1 d Also true is that the transmission e Chemical reaction can occur Tunneling is the quantum mechanics property of the wavefunction to be nonzero in classically forbidden zones Examples a Penetration through an energy barrier of energy V Incident wave Transmitted wave e an d a e E D The wavefunction and its derivatives should be continuous WaveVu Ctmn x c The transmission probability T is higher as D the barrier is narrower D the incident energy is closer to the height D the mass of the particle is smaller Heavy 5 a a on Wavemncuan Transmission prubahxmy T N Transmission pmbabihky T V 0 u 02 04 05 03 10 x lncident energy Ev probability T lt 1 even whenE gt V exclusively through tunneling even if involves heavier atom ie carbon tunneling riaa39mim c39o39miiii CHEM 3510 Fall 2006 E Appendix 1 Cartesian and spherical coordinates a The position of a point can be specified by using Cartesian coordinates xyz or by using spherical coordinates r 6 x b Relations between Cartesian and spherical coordinates 2 2 2 1 xrs1n6cos r x y Z yrsin6sin and cos z Zrcos6 Vx 3 2 tan yZ c Other relations dV dxdydz r2 sin 6drd6d oo oo oo oo7r27r j j j dxdydzz j j j r2sin 6drd6d oo oo 00 0 00 oo oo oo oo 7 27 j j j Fxyzdxdydz jrzdrjsmede jd Fr6 OO 00 oo 0 0 0 2 2 2 5x ay 52 1 a 2 a 1 a a 1 52 2 r 2 SM 2 r 6r 6r 62 r 5111659 56 r r s1n 6 5 V6 62 CHEM 3510 Fall 2006 2 Comparison between linear and circular motion gt Linear Motion Angular Motion Speed Angular speed v V ms 0 radS Mass Moment of inertia m I mr2 Linear momentum pmv Angular momentum LIcomvr Kinetic energy 2 K zlmvz 2p 2 2m Rotational kinetic energy 1 2 L2 K 101 2 21 Momentum as a vector pmv Angular momentum as a vector L 10 Lrxp Components Lx ypz 2p y Ly pr xpz L2 xpy ypx CHEM 3510 Fall 2006 F Unit Review 1 Important Terminology particleinabox quantum number normalization constant probability density freeelectron model Correspondence Principle classical limit separable Hamiltonian harmonic oscillator force constant reduced mass Hermite polynomials zeropoint energy 64 CHEM 3510 Fall 2006 selection rule IR spectrum Vibration fundamental Vibrational frequency tunneling rigid rotator moment of inertia spherical harmonic functions degeneracy microwave spectroscopy rotational constant spherical coordinates angular momentum 65

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