Theoretical Chemistry I Quantum Mechanics
Theoretical Chemistry I Quantum MechanicsAxel Groß16 October 2008
PrefaceTheoretical Chemistry 1 Quantum MechanicsProf. Dr. Axel GroßPhone: 50–22819Room No.: O25/342Email: mContents1. Introduction – Wave Mechanics2. Fundamental Concepts of Quantum Mechanics3. Quantum Dynamics4. Angular Momentum5. Approximation Methods6. Symmetry in Quantum Mechanics7. Theory of chemical bonding8. Scattering Theory9. Relativistic Quantum MechanicsSuggested Reading: J.J. Sakurai, Modern Quantum Mechanics, Benjamin/Cummings 1985 G. Baym, Lectures on Quantum Mechanics, Benjamin/Cummings 1973 F. Schwabl, Quantum Mechanics, Springer 1990III
PrefaceCriteria for getting the Schein: not specified yetThese lecture notes are based on the class “Theoretical Chemistry I – Quantum Mechanics” in the winter semester 2007 at the University of Ulm(see http://www.uni-ulm.de/theochem/lehre/)I am very grateful to Maximilian Lein who provided a LATEX version of the originalnotes which have been the basis for this text; furthermore, he created many of thefigures. Without his efforts this version of the lecture notes would not have beenpossible.Ulm, October 2007IVAxel Groß
Contents1Introduction - Wave mechanics1.1 Postulates of Wave Mechanics . . . . . . . . . . . . . . . . . . . . .1.2 Simple problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.1 Piecewise constant potentials . . . . . . . . . . . . . . . . .1.2.2 A simple atom: square well potential . . . . . . . . . . . . .1.2.3 Transmission-Reflection Problems . . . . . . . . . . . . . . .1.2.4 A simple reaction: transmission through a potential barrier1.2.5 Appendix: Transmission through a potential barrier . . . . .1. 1. 2. 2. 3. 6. 9. 122 Fundamental Concepts of Quantum Mechanics2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2 Kets, Bras, and Operators . . . . . . . . . . . . . . . . . . . . . . . .2.2.1 Kets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.2 Bra space and inner product . . . . . . . . . . . . . . . . . . .2.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.1 Multiplication of operators . . . . . . . . . . . . . . . . . . . .2.3.2 Outer Product . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.3 Base Kets and Matrix Representations . . . . . . . . . . . . .2.3.4 Eigenkets as Base Kets . . . . . . . . . . . . . . . . . . . . . .2.3.5 Resolution of the Identity, Completeness Relation, or Closure2.4 Spin 1/2 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5 Measurements, Observables And The Uncertainty Relation . . . . . .2.5.1 Compatible Observables . . . . . . . . . . . . . . . . . . . . .2.5.2 Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . .2.5.3 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . .2.5.4 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . .2.6 Position, Momentum, and Translation . . . . . . . . . . . . . . . . . .2.6.1 Digression on the Dirac Delta function . . . . . . . . . . . . .2.6.2 Position and momentum eigenkets . . . . . . . . . . . . . . .2.6.3 Canonical Commutation Relations . . . . . . . . . . . . . . .2.7 Momentum-Space Wave Function . . . . . . . . . . . . . . . . . . . .2.7.1 Gaussian Wave Packets . . . . . . . . . . . . . . . . . . . . . .2.7.2 Generalization to three dimensions . . . . . . . . . . . . . . .2.7.3 Appendix: Position representation of the momentum 3353637373 Quantum Dynamics3.1 Time Evolution and the Schrödinger Equation .3.1.1 Time Evolution Operator . . . . . . . . .3.1.2 Derivation of the Schrödinger Equation3.1.3 Formal Solution for U (t, t0 ) . . . . . . .3.1.4 Schrödinger versus Heisenberg Picture .3.1.5 Base Kets and Transition Amplitudes . .39393940404447.V
Contents3.1.6 Summary . . . . . . . . . . . . . . .3.2 Schrödinger’s Wave Equation . . . . . . . .3.3 Harmonic Oscillator . . . . . . . . . . . . .3.3.1 Heisenberg Picture . . . . . . . . . .3.4 Harmonic Oscillator using Wave Mechanics3.4.1 Symmetry of the Wave Function . . .3.5 Ensembles and the density operator . . . . .456VI.48485155565758Angular Momentum4.1 Rotations and Angular Momentum . . . . . . . . . . . . . .4.2 Spin 12 Systems and Finite Rotations . . . . . . . . . . . . .4.3 Eigenvalues and Eigenstates of Angular Momentum . . . . .4.3.1 Matrix Elements of Angular Momentum Operators .4.3.2 Representations of the Rotation Operator . . . . . .4.4 Orbital Angular Momentum . . . . . . . . . . . . . . . . . .4.5 The Central Potential . . . . . . . . . . . . . . . . . . . . . .4.5.1 Schrödinger Equation for Central Potential Problems4.5.2 Examples for Spherically Symmetric Potentials . . .4.6 Addition of Angular Momentum . . . . . . . . . . . . . . . .4.6.1 Orbital Angular Momentum and Spin 21 . . . . . . .4.6.2 Two Spin 12 Particles . . . . . . . . . . . . . . . . . .4.6.3 General Case . . . . . . . . . . . . . . . . . . . . . .4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.7.1 Connection between L2 and the Laplacian . . . . . .4.7.2 Properties of the spherical harmonics . . . . . . . . .6363656870717273747577777879818182Approximation Methods5.1 Time-Independent Perturbation Theory:Non-Degenerate Case . . . . . . . . . . . . . . . .5.1.1 Harmonic Oscillator . . . . . . . . . . . .5.2 Degenerate Perturbation Theory . . . . . . . . . .5.2.1 Linear Stark Effect . . . . . . . . . . . . .5.2.2 Spin-Orbit Interaction and Fine Structure5.2.3 van-der-Waals Interaction . . . . . . . . .5.3 Variational Methods . . . . . . . . . . . . . . . .5.4 Time-Dependent Perturbation Theory . . . . . . .85.Symmetry in Quantum Mechanics6.1 Identical Particles . . . . . . . . . . . . . . . . . . . . . . .6.2 Two-Electron System . . . . . . . . . . . . . . . . . . . . .6.3 The Helium Atom . . . . . . . . . . . . . . . . . . . . . . .6.3.1 Ground State . . . . . . . . . . . . . . . . . . . . .6.3.2 Excited States . . . . . . . . . . . . . . . . . . . . .6.4 Symmetry of molecules: Group theory and representations6.4.1 Irreducible presentations . . . . . . . . . . . . . . .6.4.2 Group characters . . . . . . . . . . . . . . . . . . .6.4.3 An example: the N2 H2 molecule . . . . . . . . . .8589899092939496.103. 103. 106. 107. 107. 108. 109. 110. 111. 113
Contents7 Theory of chemical bonding7.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.2 Born–Oppenheimer Approximation . . . . . . . . . . . . . . .7.3 The H 2 molecule . . . . . . . . . . . . . . . . . . . . . . . . .7.4 The H2 molecule . . . . . . . . . . . . . . . . . . . . . . . . .7.4.1 Molecular orbitals . . . . . . . . . . . . . . . . . . . .7.4.2 Heitler-London method . . . . . . . . . . . . . . . . .7.4.3 Covalent-ionic resonance . . . . . . . . . . . . . . . .7.5 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . .7.6 Molecular orbitals and symmetry properties . . . . . . . . . .7.7 Energy levels of diatomic molecules: vibrations and rotations.115. 115. 116. 117. 121. 122. 123. 124. 125. 127. 1328 The8.18.28.38.4many-electron problem and quantum chemistryHartree Theory . . . . . . . . . . . . . . . .Hartree–Fock Theory . . . . . . . . . . . . .Post–Hartree–Fock Methods . . . . . . . . .Density Functional Theory . . . . . . . . . .9 Scattering Theory9.1 Wave Packets . . . .9.2 Cross Sections . . . .9.3 Partial Waves . . . .9.4 Born Approximation.137. 137. 141. 145. 149.155. 155. 157. 157. 158VII
ContentsVIII
1 Introduction - Wave mechanicsWe will start by recalling some fundamental concepts of quantum wave mechanicsbased on the correspondence principle.1.1 Postulates of Wave Mechanics1. The state of a system is described by its wave function Ψ(x, t) The probabilitydensity is defined as2ρ(x, t) Ψ(x, t) (1.1)2 Ψ(x, t) d3 x describes the probability to find the particle at time t in the volumeelement d3 x at x.2. Physical observables correspond to operators that act on the wave function. Forexample, the momentum p and the energy E are represented by the followingderivativesp ,iE i . t(1.2)(1.3)3. Starting from the Hamilton function H of classical mechanics, the time-dependentSchrödinger equation is given by p2 V (x) i Ψ(x, t) E 2m t 2 2 V (x) Ψ(x, t) ,2m(1.4)i.e.i Ψ(x, t) H Ψ(x, t) t(1.5) 2 2 V (x)2m(1.6)with the Hamiltonian H 4. Energy eigenstates are given by the time-independent Schrödinger equation(H E) Ψ(x, t) 0(1.7)1
1 Introduction - Wave mechanicsFigure 1.1: Illustration of a piecewise constant potential.1.2 Simple problemsAlthough the time-dependent and the time-independent Schrödinger equations lookrather simple, their solution is often not trivial. They correspond to second-orderpartial differential equations. Here we will consider some simple problems. For thesake of simplicity, we consider piecewise continuous potentials. Nevertheless, we willbe show the variety of different solutions that are possible in quantum mechanics, suchas bound states, scattering and tunneling. In spite of the fact that the chosen potentialsare rather simple, we will see that the solutions of corresponding time-independentSchrödinger equation can be quite involved. Later we will learn other techniques suchas algebraic formulations that make the solution much easier.1.2.1 Piecewise constant potentialsHere we will derive the general form of the solution of the time-independent Schrödingerequation for a piecewise constant potential, i.e., potentials that have steps but that areotherwise flat. Such a potential is illustrated in Fig. 1.1. For every flat region of thepotential labeled by the index i in Fig. 1.1, the time-independent Schrödinger equation(1.7) in one dimension can be rewritten as 2 d 2xi 1 x xi Vi Ψi (x) EΨi (x),2m dx2 d2 Ψi (x)2m(1.8) 2 E Vi Ψi (x)2dx We now assume that E Vi . Then we can define the wave vectorp2m(E Vi )k , (1.9)so that the Schrödinger equation simply becomesd2 Ψi (x) k 2 Ψ(x) .dx2(1.10)This is a well-known differential equation in physics and chemistry. Its solution justcorresponds to a plane waveΨi (x) e ikx ,2(1.11)
1.2 Simple problemswhere the sign presents a wave traveling to the right, i.e., in positive x-direction,and the sign a wave traveling to the left in negative x-direction.For E Vi , the wave vector (1.12) becomes purely imaginary, i.e. k iκ withp2m(Vi E). (1.12)d2 Ψ(x) κ2 Ψ(x) .dx2(1.13)κ The Schrödinger equation is given byNow the solutions correspond to exponentially rising and exponentially vanishingfunctionsΨ(x) e κx .(1.14)Finally we need to know how the solutions are connected at the steps. If the potential step is finite, then the second derivative Ψ00 makes a finite jump at xi , so that bothΨ and the first derivative Ψ0 are continuous at xi . In other words,Ψi (xi ) Ψi 1 (xi )Ψ0i (xi ) Ψ0i 1 (xi )(1.15)for Vi Vi 1 . For an infinite jump in the potential, i.e., for an infinitely highpotential wall, Ψ0 makes a finite jump, but Ψ is still continuous.1.2.2 A simple atom: square well potentialNow we consider a particle in one dimension that can move freely for x a, but isconfined by infinite potential walls. This means that the potential is given by(0 , x aV (x) .(1.16) , x aThis potential is usually refered to as the “particle in a box”. It can also be regardedas a very simple model for the electron of the hydrogen atom that is kept close to theproton. Since the potential is constant for x a, the solution just corresponds toa superposition of plane waves (1.11). For x a, the wave function has to vanishbecause of the infinite potential. At x a, the first derivative of the wave functionmakes a jump, but the wave function itself is continuous, i.e.,(c1 e iqx c2 e iqx , x aΨ (1.17)0, x aFor x a, the solutions correspond to a superposition of waves travelling to the rightand to the left. Such a superpsoition leads to standing waves which can be written assine and cosine functions, sin qx and cos qx, respectively. On the otherhand, the factthat the wave functions have to vanish for x a means that Ψ(x) has also to vanish3
1 Introduction - Wave mechanicsFigure 1.2: Square-well potentialat x a, or in other words, they sine and cosine functions have to have a node there.This leads to the following conditions for the allowed wave vectors q:nπ,2nπsin qa 0 qa ,2cos qa 0 qa n 2k 1(1.18)n 2k(1.19)Hence the allowed eigenfunctions are given by(cos nπn 2k 12a xΨn sin nπxn 2k2a(1.20)The corresponding eigenenergies areEn 2 nπ 2 2 qn2 2m2m 2a(1.21)The ground state function is given by cosine function that has no node between aand a. The further eigenstates alternate between sine and cosine function, and eachhigher eigenstate has one more node than the directly lower lying state. Note that theeigenenergies (1.21) grow with the quantum number n as En n2 . This is different from the case of the hydrogen atom where the eigenenergies are proportional to 1/n2 .Now we will make the system a little bit more realistic by considering walls of finiteheight. The square well potential is then given by(0 , x aV (x) V0 θ( x a) V0 0 real number .(1.22)V0 , x awhere θ(x) is the Heaviside step function which is defined as(0, x 0Θ(x) 1, x 0(1.23)The resulting potential is shown in Fig. 1.2. Bound states exist for 0 E V0 . Nowwe are confronted with a typical quantum phenomenon. Classically, particles can notpenetrate into potential regions that are higher than their energy. However, quantum4
1.2 Simple problemsmechanically, there is a non-zero probability to find a particle in potential regions thatare finite but larger than the energy of the particle. The particles can still move freelyfor x a, but now they penetrate into the potential walls. The time-independentSchrödinger equation becomes 2mE002Ψ q Ψq x a(1.24)p 2m(V0 E)Ψ00 κ2 Ψκ x a(1.25) For x a, the solutions are again oscillatory, i.e., the are plane waves that can berepresented by exponential functions with purely imaginary exponent or as a combination of sine and cosine functions. In contrast, for x a the basic solutions are alinear combination of exponentially rising and vanishing functions(c1 e iqx c2 e iqx , x aΨ (1.26)c e κx c e κx , x aThe solutions have to vanish for x , or in other wordsZ Ψ(x) 2 dx .(1.27) Therefore one can only have the exponentially decreasing part for x a and theexponentially increasing component for x a. One can only say that e κx is notnormalizable for x a and analogously e κx not for x a. Furthermore, since V (x)is an even potential, the solutions can be characterized according to their symmetry,i.e., the solutions are either even or odd as we will see in later chapters. This meansthat they can be represented by cosine or sine functions, respectively. If we have evensymmetry, the solution will be(A cos qx x aΨ(x) (1.28)e κ x x aFor odd symmetry, we get(B sin qxΨ(x) e κ x x a x a(1.29)It is interesting to note that even such a simple example as the square well potentialis not that easy to solve. We will illustrate this in the following. Assume first that Ψhas even symmetry. Continuity at x a requiresA cos qa e κa(1.30)Ψ0 has to be continuous, too. From that, we get(1.31) If we devide (1.31) by (1.30), we obtaintan aq κq(1.32)5
1 Introduction - Wave mechanicsNow we introduce the dimensionless parameter 2mV0λ a, (1.33)so that Eq. (1.32) becomestan aq κκa qqa pλ2 (qa)2 qa(1.34)This is a transcendental equation that cannot be solved analytically.Now assume odd symmetry. Bq cos qa B sin qa cot qa κ qpλ2 (qa)2qa(1.35)Again a transcendental equation that can only be solved graphically. For the graphicalsolution we first not that κ/q behaves likes 1/q for small q. Furthermore:apκ 0 for qa 2mV0 λq (1.36)In Fig. 1.3 the graphical solution is sketched. For three different values of V0 κ/qis plotted as a function of qa together with tan qa and cot qa. Every crossing pointof the curves corresponds to a solution. Since κ/q diverges for qa 0, there is atleast one crossing point with tan qa. The lowest energy state is always even. Whenλ increases by π/2, there is another crossing point and hence one additional state.Hence the total number of states is given by 2a 2mV02λNS (1.37)ππ with [α] nearest integer greater than α. Even and odd states alternate.It is always instructive to look a limiting cases whose solution is exactly known. Ifthe potential wall V0 grow to infinity, we recover the particle-in-a-box problem. Firstof all, we see from (1.37) that NS V0 , i.e., for inifinite potential walls we alsoget infinitely many states, as is also reflected in Eq. (1.21). Furthermore, when V0increases, the curves for λ in Fig. 1.3 become higher and more flat. This means thatthey will cross tan qa and cot qa at values closer and closer to qa nπ/2, which alsocorresponds to the case of the particle in a box.1.2.3 Transmission-Reflection ProblemsTransmission-reflection problems occur in chemistry when two particles meet eachother. Here we treat such a problem as an one-dimensional potential stepV (x) V0 θ(x)6(1.38)
1.2 Simple 22π5π/23π7π/2qaλ3λ2π/23π/2πFigure 1.3: Graphical solution for the even and odd solutions of the square well problemFor x 0, the potential is 0, for x 0, the potential is V0 (see Fig. 1.4). TheSchrödinger equation for x 0 and x 0 is given by2mEd2 Ψ 2 Ψ k 2 Ψ2dx d2 Ψ2m(E V0 ) Ψ k 02 Ψdx2 2x 0(1.39)x 0(1.40)Let E V0 . Suppose a particle is incident from the left.ΨI (x) e ikx re ikx ik0 xΨII (x) te(1.41)(1.42)r and t are the reflection and transmission amplitudes. We will now discuss the socalled probability flux which is given byj(x) d Ψ Ψmdx(1.43)7
1 Introduction - Wave mechanicsj inj transj reflV0xx 0Figure 1.4: Potential step and transmission coefficient T (E) for a potential step ofheight V0 0.5.For the particular problem of the potential step, we obtaind Ψ IΨImdx (e ikx r eikx )ik(eikx re ikx )m 2 ik 1 r re i2kx r ei2kx {z}mjI (x) purely imaginary k2(1 r ) jin jreflm00 jII (x) t e ik x (ik 0 )teik xm k 0 2 t jtrans m(1.44) (1.45)We will now define R and T , the reflection and transmission coefficient.jref2 r jink0 2jtrans t T jinkR (1.46)(1.47)Due to particle conservation we have jin jrefl jout , which leads to k k 0 22(1 r ) t mm R T 1jI jII(1.48)If the potential is Hermitian, then the number of particles is conserved. If it is nonHermitian, the potential must have an imaginary part not identically to zero. Imaginary potentials can describe the annihilation of particles.It is left as an exercise to calculate the transmission coefficient for the potential step.8
1.2 Simple problemsThe result is given bypT (E) (E V0 )E 2 ,pE (E V0 )4(1.49)which is also plotted in Fig. 1.4. There is another typical quantum phenomenon visiblein Fig. 1.4: For energies slightly la
2. Fundamental Concepts of Quantum Mechanics 3. Quantum Dynamics 4. Angular Momentum 5. Approximation Methods 6. Symmetry in Quantum Mechanics 7. Theory of chemical bonding 8. Scattering Theory 9. Relativistic Quantum Mechanics Suggested Reading: J.J. Sakurai, Modern Q
1. Introduction - Wave Mechanics 2. Fundamental Concepts of Quantum Mechanics 3. Quantum Dynamics 4. Angular Momentum 5. Approximation Methods 6. Symmetry in Quantum Mechanics 7. Theory of chemical bonding 8. Scattering Theory 9. Relativistic Quantum Mechanics Suggested Reading: J.J. Sakurai, Modern Quantum Mechanics, Benjamin/Cummings 1985
The exercises of „Quantum Mechanics, The Theoretical Minimum“ page 1 of 105 Preface This file contains the exercises of „Quantum Mechanics, The Theoretical Minimum“ and is specific in this respect. On the other hand, the topics generally deal with quantum mechanics and
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