授業の目的 【日本語】
Goals of the Course(JPN) | | |
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授業の目的 【英語】
Goals of the Course | | This tutorial course is a companion course to the Quantum Mechanics III lecture course. The two should be taken together.
The purpose of this course is to cultivate students’ calculation skills and deepen their understanding of quantum mechanics -- i.e. Nature.
This unit is the second half of a full-year course. Building on Quantum Mechanics II, students will learn quantum mechanics at an advanced undergraduate level. The course will build physical intuition of Nature on the quantum scale while improving students’ ability to express physical intuition in mathematical terms and to solve problems. Students will learn not to be overwhelmed by mathematical symbols, and to discern the simplicity of physical principles expressed by them.
Quantum Mechanics III (together with this tutorial course) is an essential step towards higher level study in Condensed Matter Physics, High Energy Physics, Materials Science and many areas of Applied Physics. If taken together with Quantum Mechanics II, Statistical Physics II and Statistical Physics III, students will be well prepared to level up to Condensed Matter Field Theory and Quantum Field Theory in Year 4 or graduate study. Chemistry students have the opportunity to understand foundations of chemistry at a fundamental level and would gain an advantage over their peers.
The main areas covered are:
1. The connection between symmetries and conservation laws
2. The Theory of Angular Momentum
3. Spin and Paramagnetic Resonance
4. The Variational Approach
5. Perturbation Theory: Time Independent Nondegenerate; Time Independent Degenerate; Time Dependent
6. Scattering Theory |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | |
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到達目標 【英語】
Objectives of the Course | | Students will be adequately prepared with regards to their knowledge of quantum mechanics to undertake further studies in Sc-lab, St-lab, E-lab, H-lab, QG Lab, R-lab, TB-lab and other, experimental labs in both the Department of Physics and Department of Applied Physics at Nagoya University. A knowledge of the principles is essential for students interested in experimental physics and theoretical physics. Students from other disciplines can also benefit from the deep treatment of quantum phenomena.
Topics include: Symmetry and conservation laws; theory of angular momentum (including addition of angular momentum); solution of rotationally invariant problems; the (spinless) hydrogen atom; Spherical tensor operators and selection rules: Wigner-Eckart Theorem; The Variational Method and WKB Approximation; Time-independent perturbation theory (non-degenerate and degenerate cases); Time-dependent perturbation theory (including scattering off a potential that is a perturbation of the vaccuum); introduction to scattering theory. |
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授業の内容や構成
Course Content / Plan | | The tutorial course contents is defined by the lecture course contents, which is outlined below. The precise order and content of the lectures might vary slightly.
Participants are expected to solve problems relating to the lecture course content, and present their solutions in the tutorial course (Physics Tutorial IVb). Students will also submit "tutorial assignments", which are also set by the Quantum Mechanics III lecturer and count towards the tutorial course grade.
The course covers Shankar Chapts 11-19 (some parts omitted); as well as parts of Cohen-Tannoudji et. al, Sakurai, Merzbacher and Gottfried. Some topics are covered in tutorials.
Lecture 1. The Copenhagen Interpretation and Galilean Invariance: Wigner's Theorem; linear and antilinear operators; the time-reversal operator and time-reversal symmetry. Translational invariance and its consequences. Active and passive views. Transformation of operators. Infinitesimal translations; Finite translations. Correspondence with translations in Euclidean space. Translational invariance defined. Consequence: a certain conservation law, and universality of experiments performed at different locations.
Lecture 2. Invariance and conservation laws cont’d. Time translation invariance; parity invariance; resultant conservation laws. Formal correspondence between generators of infinitesimal canonical transformations and generators of infinitesimal unitary transformations.
Lecture 3. Rotational invariance and its consequences. Rotations in Euclidean space do not commute: derivation of commutation relations between generators of infinitesimal rotations in Euclidean space. Consequence: commutation relations for operators defined on Hilbert space (and quantum mechanics). Conservation of angular momentum.
Lecture 4. Rotational invariance and angular momentum. Rotations in 2D: correspondence between those in Euclidean space and Hilbert space. Identifying the generator of infinitesimal rotations in Hilbert space. Active and passive views. Consistency checks: composition of translations and rotations in Hilbert space and Euclidean space. Lie algebra. The eigenvalue problem of L_z. Angular momentum in 3D and the eigenvalue problem of J^2 and J_z. Matrix representation: block diagonal forms and partitioning of Hilbert space.
Lecture 5. Rotational invariance and angular momentum cont’d. Finite rotation operators. Irreducible representations. Orbital angular momentum eigenfunctions in the coordinate basis. Solution of rotationally invariant problems. The free particle in spherical coordinates.
Lecture 6. Solution of rotationally invariant problems cont’d. Radial equation, reduced radial equation, boundary conditions. The (spinless) hydrogen atom in coordinate basis; quantization condition. Eigenfunctions. Also in momentum basis.
Lecture 7. The (spinless) hydrogen atom cont’d. Cause of “unexpected” degeneracy. Comparison with experiment; reasons for deviations. Fine structure corrections; hyperfine structure corrections. Spin. Mathematical representation of spin: spinors and their generalization. Response of vector field Ψ(r) under infinitesimal rotation produces two kinds of generators of infinitesimal rotations, corresponding to orbital and intrinsic angular momentum operators. Kinematics: properties of the Pauli spin matrices. Spin dynamics. Classical magnetic moment suggests form of spin magnetic moment operator. Derivation of Bohr magneton in the Coulomb gauge. Time evolution of spinors.
Lecture 8. Addition of angular momentum. Clebsch-Gordon coefficients. Addition of L and S.
Lecture 9. Spherical tensor operators and selection rules: Wigner-Eckart Theorem. Cartesian tensor operators and irreducible tensor operators. Explanation of “accidental” degeneracies.
Lecture 10. The Variational Method and WKB Approximation. Tunneling amplitudes; bound states.
Lecture 11. Time-independent Perturbation Theory (non-degenerate case). 1st order and 2nd order energy corrections; 1st order correction to wave function. Dipole selection rule. Example: Stark effect.
Lecture 12. Time-independent Perturbation Theory (degenerate case). Time-dependent Perturbation Theory. Transition rate. Sudden perturbation. Adiabatic perturbation. Auger (Radiationless) Transition in helium.
Lecture 13. Time-dependent Perturbation Theory cont’d. Periodic perturbation. Fermi’s Golden Rule. Gauge transformations, invariance of QM under gauge transformation. Photoelectric effect in the hydrogen ground state. Non-examinable: non-Abelian gauge field theories.
Lecture 14. Elementary introduction to scattering theory. Definition of scattering cross-sections. Calculation of the cross-section using probability currents. Expression for the cross-section in terms of scattering amplitude. Born series, Born approximation. Physical interpretation.
Lecture 15. Scattering theory cont’d. Scattering from a central potential using the method of partial waves. Higher orders in perturbation theory: Schroedinger picture, Interaction picture, Heisenberg picture. |
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履修条件
Course Prerequisites | | Calculus I; Calculus II; Linear Algebra I; Linear Algebra II; Mathematical Physics I; Mathematical Physics II; Quantum Mechanics II; or Consent of Lecturer.
Students must have passed Quantum Mechanics II to take Quantum Mechanics III. |
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関連する科目
Related Courses | | | It is strongly advised that students taking Quantum Mechanics III concurrently enroll in Physics Tutorial IVb. |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Tutorial preparation and tutorial performance count for 50% of the grade. Tutorial assignments count for the other 50%. |
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不可(F)と欠席(W)の基準
Criteria for "Fail (F)" & "Absent (W)" grades | | The "Absent (W)" grade is reserved for students who withdraw by May 31. After that day, a letter grade will be awarded based on marks earned from all assessment during the semester.
If Quantum Mechanics III is NOT A COMPULSORY SUBJECT and the student plans never to take Quantum Mechanics III in the future, then a late withdrawal request will be considered. |
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参考書
Reference Book | | 1. Sakurai, J. J., Napolitano, Jim J., Modern Quantum Mechanics (2nd Ed.), Addison-Wesley, 2010. (This book complements, and at times supersedes, the treatment in Shankar.)
2. Cohen-Tannoudji, C., Diu, B., Laloe, F., Quantum Mechanics, Wiley, 1991. Vol. 1 and Vol. 2.
3. Merzbacher, E., Quantum Mechanics, 3rd Ed., Wiley, 1998.
(Merzbacher was one of the best teachers of quantum mechanics.)
4. Landau L. & Lifshitz L., Quantum Mechanics: Non-Relativistic Theory, 3rd Ed., Butterworth-Heinemann (1981).
5. Messiah, A., Quantum Mechanics (2 Volumes), Dover, 2015. (Highly recommended, classic alternative reading. Cheap to buy.)
6. Gottfried, K. and Yan, T.-M., 2004, Quantum Mechanics: Fundamentals, Springer.
(Most of this book is too hard for undergraduates but several sections are at the right level and very clear. Consult this book as an authoritative reference.) |
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教科書・テキスト
Textbook | | | Shankar, R., 1994, Principles of Quantum Mechanics, 2nd ed., Kluwer Academic/Plenum. |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | This course is part of your training to be a professional researcher. You are expected to revise the lecture notes, read and work through the textbook, and solve assignment problems outside lecture hours. You cannot learn physics by only attending lectures. The exams will consist of questions covering both lecture notes and assignments.
Students must be willing to work hard if they want to achieve a good, internationally competitive level. |
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注意事項
Notice for Students | | Plagiarism (e.g. copying solutions that you have found on the Internet) is an act of academic dishonesty. Cheating in exams (e.g. having lecture notes, assignment solutions or online references open on your computer screen during an online exam) is a serious offence. Copying other people's solutions and claiming them as your own is also an act of academic dishonesty. Nagoya University has a strict policy towards academic dishonesty:
"Acts of academic dishonesty are prohibited during exams, for reports and assignments. If acts of academic dishonesty are discovered, you may be subject to discipline, which may affect your ability to graduate on time."
The punishment for serious breaches (such as cheating in an exam or repeated plagiarism despite a warning) is the loss of all grades from all subjects during the semester and cancellation of any scholarships received.
Even if your course is difficult and it is to be expected that you'll find it hard to finish assignments, it is far better that you submit an honest effort than take the dishonest path. Remember, to be on course for a "B" you only need to score over 70% in the assignments -- and if you paid attention in the tutorials, you would have seen nearly all the problems done for you (in my subjects, at least).
There's no excuse for cheating. |
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他学科聴講の可否
Propriety of Other department student's attendance | | | Students from any department are welcome as long as they have the necessary grounding in mathematics and physics. |
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他学科聴講の条件
Conditions for Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | |
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履修の際のアドバイス
Advice | | It is strongly advised that students Quantum Mechanics III concurrently enrol in Physics Tutorial IVb.
Students must be willing to work hard if they wish to achieve a good, internationally competitive level.
At the end of this course the lecturer will provide upon request a reading list for students wishing to proceed to the "next level up" in quantum field theory and theoretical condensed matter physics. Students should have a solid understanding of the material covered in both Quantum Mechanics and Statistical Physics courses. |
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授業開講形態等
Lecture format, etc. | | Face to face lectures and tutorials are compulsory (other than in exceptional circumstances; e.g. COVID infection). However, in order to record a video of the lecture -- including student interaction with each other and with the lecturer -- the lectures will simultaneously be carried out online using MS Teams. Students are therefore requested to bring their laptop or tablet to the lecture room. Make sure it has a microphone. Bring an electrical cord.
Live lectures via MS Teams (face-to-face and online). Before the start of semester students should ensure that they have correctly installed MS Teams using their THERS (国立大学法人東海国立大学機構 ) email account.
NUPACE students should contact Professor John Wojdylo before the start of semester for assistance with installing Teams correctly. |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | All lectures will be live face-to-face and online via MS Teams. Face-to-face attendance is compulsory (barring exceptional circumstances such as COVID infection).
A lecture video will be available immediately after each lecture to help with student revision.
The lecturer will be available at most times during the day to answer questions via Teams chat. |
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