Quantum Mechanics in Chemistry CHEM 3890 Honors Physical Chemistry I. Fall 2013 Bard 132 Monday, Wednesday, Friday 11:15am-12:05pm.

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Quantum Mechanics in Chemistry CHEM 3890 Honors Physical Chemistry I Quantum physics is mind-bending, but it is also one of the most successful scientific theories ever put forward.

1 Quantum Mechanics in Chemistry CHEM 3890 Honors Physical Chemistry I Quantum physics is mind-bending, but it is also one of the most successful scientific theories ever put forward. New Scientist Fall 013 Bard 13 Monday, Wednesday, Friday 11:15am-1:05pm Instructor: Office Hours: M. Luke McDermott Baker B38 Friday :30 pm and by request Teaching Assistants: TA#1 Office Hour Recitation Section TA# Office Hour Recitation Section Grading: Letter Grades Only, 4 credits I. Rationale: From the College: CHEM 3890 is an introduction to the quantum mechanics of atoms and molecules. The fundamental principles of quantum mechanics are introduced, and applications of the theory to atomic and molecular structure are covered in detail. CHEM 3900 is a continuation of CHEM 3890 and discusses the thermodynamic behavior of macroscopic systems in the context of quantum and statistical mechanics. After an introduction to the behavior of ensembles of quantum mechanical particles (statistical mechanics), kinetic theory and the laws of thermodynamics are covered in detail. Translation: CHEM 3890 provides the science student with a rigorous foundation for Physical Chemistry. More generally, CHEM 3890 allows students to understand why quantum mechanics matters, how the subject developed, and where they see it daily. Prerequisites: MATH 130 or MATH 310 or MATH 0; PHYS 08; CHEM 080 or CHEM 160 or permission of instructor. 1

2 II. Course Aims and Objectives: Aims By the end of this course, students will have: Presented why Quantum Mechanics was needed to supplement Classical Mechanics Described the development of Quantum mechanical theory through significant historical breakthroughs Solved Schrödinger s wave equation for constrained particles Explained the origin of general chemistry principles such as molecular orbitals Understood state-of-the-art Physical Chemistry research Specific Learning Objectives: Students will demonstrate a mastery of fundamental principles of Quantum Mechanics by learning vocabulary and mathematical methods, solving classic problems, diagramming historically significant experiments, and analyzing current scientific articles in Physical Chemistry. Specifically, students should be able to: Remember fundamental experiments and theories in quantum mechanics. Understand how quantum mechanics supplements classical mechanics. Apply Schrödinger s wave equation to classic problems: (1) the Particle in a Box; () The Harmonic Oscillator and the Rigid Rotor; (3) The Hydrogen Atom; and (4) Multielectron Atoms. Analyze complex problems related to the above classic problems and create paths to solving these complex problems. Evaluate modern research in quantum mechanics, especially molecular modeling. Create mathematical computer code to solve problems quantum mechanically. III. Format and Procedures: Class time is centered on finding answers. Students are expected to come to class ready to solve problems. To interact with the rest of the class, you will need a web-connected device (e.g. smartphone, tablet, laptop). If you do not have access to such a portable device, we can loan you one for each class. Using these devices to interact (see sidebar), class time will follow this cycle: Learning Catalytics Brain wave studies show that sitting passively in a lecture can induce as little brain activity as watching television. To counteract the passive nature of lectures, bring your web-connected device to class every day and log on to our LCatalytics.com class session. During class, we will have all kinds of interactive questions, where you will answer through the website. Then you will have a chance to work with your neighbors to convince each other who is right. Then put your new (or old) answer into the question again. With Learning Catalytics, we can give you instant feedback and find out where the class should go. Make a username now and log on with our course session ID: CHEM3890. Our session password is QMfall013.

1. I will explain a Concept. I will pose a Question on the Concept 3. You will each make an educated guess to Answer the Question on our Learning Catalytics website (learningcatalytics.com) 4.

I will review and summarize the Concept Other class activities, which you will produce in groups for class time (see Section VII.): Debates on quantum mechanics interpretations.

3 1. I will explain a Concept. I will pose a Question on the Concept 3. You will each make an educated guess to Answer the Question on our Learning Catalytics website (learningcatalytics.com) 4. You will attempt to convince others around you of the correct answer 5. You will adjust your educated guess to Answer the Question 6. I will go over the correct Answer 7. I will review and summarize the Concept Other class activities, which you will produce in groups for class time (see Section VII.): Debates on quantum mechanics interpretations. Perspectives on historical scientists. IV. Time Commitment: Per week: 3 hours: Class time is your opportunity to interact with other students and me. 1-3 hours: Recitation and TA Office Hours are your greatest resource for having questions answered. TAs will lead recitations so that you will do problems in groups during the session and the TA will work with you as your group encounters problems. 1-3 hours: Group homework sessions are the best way to complete problem sets. Teaching and learning from peers is proven to help students learn more effectively than interaction with professors. V. My Assumptions Quantum Mechanics is taught in many fields, but in this course, you will learn the chemist s perspective. At the end of this syllabus, you will find an outline of what part of quantum mechanics history we will cover with this course, as well as how quantum mechanics relates to the rest of chemistry. Also attached is a formula sheet that will let you see all of the equations that you will know how to use at the end of the course. VI. Required Texts - Introduction to Quantum Mechanics, nd edition, by D.J. Griffiths This book is short, readable, and full of helpful equations and problems (Solution manuals can be purchased). - Physical Chemistry: A Molecular Approach, by D.A. McQuarrie & J.D. Simon (University Science Books). will need this book for Chem 3900 as well. VII. Assignments Problem Sets A large portion of the learning for this class will be done in the context of problem sets. The reality is that quantum mechanics is best understood and used as a rigorous, mathematical tool. 3

4 In modern research, scientists use the same equations as you to solve their research problems. When you see how the equations we discuss in class accurately describe molecular-level problems, the power of quantum mechanics will be clear to you. The goal of these problem sets is to empower you and help you are now able to calculate how chemistry works at the most fundamental level. The Final Exam The final exam is your opportunity to test the variety of your abilities at the end of this class. When you walk out of the exam, my hope is that you will feel excited by how much you are capable of. Group Presentations and Journal Article Review These assignments bring you into the historical and modern community of scientists. When scientists talk about breakthroughs, they often discuss it in the context of who made the breakthrough. Or scientists talk about how their own work in the context of how it relates to other scientists work. Your presentations and reviews will show you the history and community of quantum mechanics. VIII. Grading Procedures: 1. 7 Problem Sets (40%) These problems can be time-consuming, but collaborative work and computer coding can lead to memorable learning experiences. Above all: work together. One Final Exam (30%) The concepts tested by problem set will be tested again during the final exam. The problems will not be in exactly the same format, as no computers will be Research shows that practice tests are one of the most effective learning strategies. available during the exam, so take the practice exams that will be available. 3. Group Presentation (0%) You will be assigned to one of 1 groups. 10 groups will be assigned an important scientist in quantum mechanics. They will make a 0-minute presentation including his most significant work and an assessment of class understanding. Points will be awarded for completeness, creativity, and class understanding. Two groups will be assigned to take part in a 45-minute debate on a significant perspective on quantum mechanics. Points will be awarded for clarity and completeness. Bonus points are available for exceptional presentations. 4. Journal Article Review (10%) You will summarize two scientific journal articles, one of which must be a review article. Points will be awarded for connections drawn to course material. IX. Academic Integrity Each student in this course is expected to abide by the Cornell University Code of Academic Integrity. Any breach of academic integrity will receive a zero. 4

5 For this course, collaboration is allowed for all assignments. However, each character of mathematical code, each word of explanation, and all submitted work must be written by your own hands. However, this permissible cooperation should never involve one student having possession of a copy of all or part of work done by someone else, in the form of an attachment, file, or hard copy. X. Accommodations for students with disabilities In compliance with the Cornell University policy and equal access laws, I am available to discuss appropriate academic accommodations that may be required for student with disabilities. Requests for academic accommodations are to be made during the first three weeks of the semester, except for unusual circumstances, so arrangements can be made. Students are encouraged to register with Student Disability Services to verify their eligibility for appropriate accommodations. 5

XI. Tentative Course Schedule Lecture 1 Historical Development of Quantum

Machinery of Quantum Mechanics (Mathematics) Operators, Eigenvalues,

Schrödinger Equation Solving the Schrödinger Equation Particle in a Box Free

Momentum and Magnetic Fields Barrier Problems The Harmonic Oscillator The

Variational Principle Perturbation Theory Lectures 6-30 Chemical Bonds

6 XI. Tentative Course Schedule Lecture 1 Historical Development of Quantum Mechanics Lectures -4 Experimental Evidence for Quantum Mechanics Lectures 5-7 Machinery of Quantum Mechanics (Mathematics) Operators, Eigenvalues, Eigenfunctions Journal Article Review Due Lectures 6-1 The Wavefunction and the Schrödinger Equation Solving the Schrödinger Equation Particle in a Box Free Particle Planck s explanation of blackbody radiation Lectures Angular Momentum and Magnetic Fields Barrier Problems The Harmonic Oscillator The Hydrogen Atom Lectures 19-1 Presentations Lectures -5 Approximation Methods Variational Principle Perturbation Theory Lectures 6-30 Chemical Bonds Spectroscopy Lectures Presentations and Debate Lectures Final Exam Review Journal Article Review Due Final Exam Quantum Tunneling 6

XII. Quantum Mechanics Map 1850-1900 Boltzmann, Maxwell, Hertz, Rydberg, Röntgen, Curie (1903 Nobel Prize), Rutherford Statistical Mechanics, Molecular Orbitals 1900-1910 Planck, Lewis, Einstein,

7 XII. Quantum Mechanics Map Boltzmann, Maxwell, Hertz, Rydberg, Röntgen, Curie (1903 Nobel Prize), Rutherford Statistical Mechanics, Molecular Orbitals Planck, Lewis, Einstein, Rutherford, Taylor Energy Quantization, Multiple Bonds (Octet Rule), Photoelectric Effect, Wave-Particle Duality Millikan, Bohr Oil Drop Experiment, Avogadro constant, quantum atomic levels, Stern, Gerlach, de Broglie, Bose, Pauli, Hund, Heisenberg, Fermi, Dirac, Schrödinger, Neumann, Born, Raman, Spin, De Broglie Wavelength, Bosons, Fermions, Bose-Einstein statistics, Pauli exclusion principle, Hund s Rule, Heisenberg Uncertainty Principle, Fermi-Dirac statistics, wave equations, quantum mechanics in terms of Hermitian operators on Hilbert spaces, Copenhagen interpretation (probability and wavefunction collapse) 1930-now Not covered in detail! 7

8 Quantum Mechanics Formulas Constants h h π Expectation Values and Operators ˆp = i h De Broglie Einstein Relations E = hω p = hk Dispersion Relations ω light (k) = ck ω electron (k) = hk M Heisenberg Uncertainty Principle p x x h/ E t h/ x = x x Schroedinger Equation h Ψ(r, t) m Ψ(r, t) + V (r, t)ψ(r, t) = i h t Time-Independant SWE Ψ(r, t) = ψ(r)ϕ(t) h m ψ(r) + V (r)ψ(r) = Eψ(r) ϕ(t) = e iet/ h Probability Current Density Free Electron Beam j = i h m (Ψ Ψ Ψ Ψ ) ψ(x) = A 1 e ikx + A e ikx ; k = R = A A A 1 A 1 m e (E V 0 ) h Ê = i h t Ĥ = h m + V (r) ρ(r, t) = Ψ = Ψ (r, t)ψ(r, t) f = Ψ (r, t) ˆf Ψ(r, t) dv = Ψ ˆf Ψ V Harmonic Oscillator V (x) = 1 mω x ψ 0 (x) = mω 1/4 e x /L ψ 0 (Q) = 1 1/4e Q π h π h L = mω E 0 = hω ( E n = n + 1 ) h Q x/l ) ( d dq + Q ψ = E ψ = E ĥψ 0 â + 1 ( d dq + Q â 1 ( d dq + Q H n (Q) = QH n 1 (Q) (n 1)H n (Q) ) ) ψ n (Q) = A n H n (Q)e Q/ A n = A n 1 n [ ] n 1 ψ n (Q) = Qψ n 1 (Q) ψ n (Q) n 1

9 Hydrogen Atom e V (r) = 4πɛ 0 r E n = R n Quasi Classical Approximation Wave Packets ψ = e i/ h R p(x) dx T QC = e / h R b a p(x) dx = e / h Ψ(x, t) = 1 + a(k) exp[i(kx ωt)] dk π Ψ(x, t) = 1 + A(k, t)e ikx dk π A(k, t) = 1 + Ψ(x, t)e ikx dx π Φ(p, t) = 1 h A(k, t); p = hk Expansion Principle and Hilbert Space Ψ = a n = n Ψ, a n n n=1 Ĥ n = E n n Stationary Perturbation Theory Degenerate Ĥ = Ĥ(0) + Ŵ Ĥ (0) n = E n (0) n N ψ a n n n=1 H 11 H 1... H 1N a 1 a 1 H 1 H... H N a = E a.. H N1 H N... H NN a N a N H mn = m Ĥ n H 11 E H 1... H 1N H 1 H E... H N = 0 H N1 H N... H NN E Nondegenerate ψ = u + m u ψ = u + φ φ u = 0, ψ u = 1 E n = E n + n Ŵ n E = E u + u Ŵ φ m Ŵ n m E E m n ψ n m E E u + E () E () = m u m Ŵ u E u E m Ψ Ψ = a n = 1 n=1 A = n a n A n Ψ = n n = n n a 1 a. ˆP nn = ˆ1

Quantum Mechanics CHEM/ENCH 313

Quantum Mechanics CHEM/ENCH 313 Instructor: Soran Jahangiri Instructor Contact Information Email: soran.jahangiri@chem.queensu.ca Office: Chernoff Hall, Room 313 Office Hours: Monday 2:30PM - 3:30PM, Tuesday