学部・大学院区分
Undergraduate / Graduate | | 理学部 | | 時間割コード
Registration Code | | 0680430 | | 科目区分
Course Category | | 専門科目
Specialized Courses | | 科目名 【日本語】
Course Title | | [G30]量子力学２ | | 科目名 【英語】
Course Title | | [G30]Quantum Mechanics II | | コースナンバリングコード
Course Numbering Code | | | | 担当教員 【日本語】
Instructor | | WOJDYLO John Andrew ○ | | 担当教員 【英語】
Instructor | | WOJDYLO John Andrew ○ | | 単位数
Credits | | 2 | | 開講期・開講時間帯
Term / Day / Period | | 秋 月曜日 ５時限
Fall Mon 5 | | 授業形態
Course style | | 講義
Lecture | | 学科・専攻
Department / Program | | | | 必修・選択
Compulsory / Selected | | | See the “Course List and Graduation Requirements for your program for your enrollment year. |
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授業の目的 【日本語】
Goals of the Course(JPN) | | | | 授業の目的 【英語】
Goals of the Course | | This 2nd course in quantum mechanics is the first half of a full-year Year 3 course. The main goal of this course is first to make von Neumann's Hilbert Space approach to quantum mechanics seem intuitively simple to the student; then to enhance their understanding of Nature by considering topics including:
• why the force felt by a quantum particle is different to the force felt by a classical particle in the same potential, and what the conditions of the quantum-classical crossover are;
• what the spread of possible paths taken by a quantum particle in the neighbourhood of a classical path is likely to be, and what the conditions of the quantum-classical crossover are;
• various manifestations of the classical limit being reached in the limit of high quantum number, even when the system is inherently quantum mechanical (such as the simple harmonic oscillator);
• interference effect between identical particles in the quantum mechanical limit and the contrast with distinguishable particles;
• correlation of measurements of entangled particles and why this implies non-locality (via the CHSH (Clauser-Horne-Shimony-Holt) Inequality);
• quantum teleportation and other "spooky" quantum effects that have been confirmed by experiment.
Through learning how to represent fascinating phenomena such as these mathematically, students learn to express with ease their new physical intuition using mathematical language: the mathematical language becomes second nature and students learn not to be overwhelmed by the mathematical symbols and to discern the simple physical principles expressed by them. By breaking down seemingly complex calculations in strange new territory into many simple steps -- by building the habit of working through each line in the textbook and testing and firming up their understanding by solving numerous problems -- students extend their problem-solving and theory-building abilities. This approach also instills critical thinking, as students make it a habit to verify statements for themselves and not just believe everything they are told.
This is a challenging course intended mainly for students who wish to pursue research in theoretical physics. Students need to be strong in both physics and mathematics and be prepared to work hard if they want to succeed in it.
At the end of Quantum Mechanics III next semester students will be adequately prepared with regards to their knowledge of quantum mechanics to undertake further studies in St-lab, Sp-lab, J-lab, R-lab, TB-lab, E-lab, H-lab, QG-lab and other, including experimental, labs in both the Department of Physics and Department of Applied Physics (e.g. Tanaka Lab), as well as chemistry and computational biology labs at Nagoya University. For chemistry majors, Quantum Mechanics II and III provide a powerful boost to their skills-set which can open many doors.
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NOTICE TO PROSPECTIVE NUPACE (exchange) STUDENTS
The G30 Physics Program has a small and highly selective student intake, in any year comprising the best few percent of applicants. We are not like regular physics programs in which students may select from a number of different streams (such as students mainly interested in physics research; students mainly interested in teaching physics; students mainly interested in engineering and not fundamental physics; etc.). We only have one stream, and due to the small class size and limited resources, we specifically focus on preparing students for research in theoretical physics, which is what most of our students need in Year 4 and beyond. The course level is comparable to that at some of the best universities.
We do not have time to support NUPACE students who are not strong in physics and mathematics: there is no time for remedial lecturing. Due to the difficulty of some of the topics covered this semester, and the mathematical dexterity required to master them, from our experience with exchange students it is unlikely that exchange students who are not currently achieving an A grade in both physics and mathematics will be able to keep up, they will struggle and likely lose interest resulting in a waste of their time and ours.
Therefore prospective NUPACE students should note that in principle, only applicants averaging 80% or above in their undergraduate physics and mathematics courses will be permitted to take this course. Students should have a desire to undertake research in theoretical physics. Students who are unsure about taking the course may email the lecturer. If necessary, in order to catch up to the required level, prospective NUPACE students will be asked to undertake a self-study course before coming to Nagoya University and during the semester, called "Guided Independent Study -- Further Studies in Quantum Mechanics", which would count as credit towards their NUPACE Exchange Completion Certificate. (Students need at least 15 points to receive this certificate.) Students would be assigned questions to solve from the textbook and would have to submit their solutions to the lecturer to show that they have done the necessary preparation.
DO NOT ARRIVE AT NAGOYA UNIVERSITY EXPECTING THE YEAR 3 G30 PHYSICS COURSES TO CONTRIBUTE TO YOUR DEGREE BECAUSE YOU MIGHT NOT BE PERMITTED TO ATTEND THE COURSE. YOU MUST OBTAIN PRIOR APPROVAL FROM THE COURSE LECTURER. (This holds particularly for QM2, QM3, SP2 and SP3.)
You must be willing to work hard -- this course in the G30 Physics Program is not primarily a cultural experience opportunity in Japan. It requires a substantial time commitment. |
| | 到達目標 【日本語】
Objectives of the Course(JPN)) | | | | 到達目標 【英語】
Objectives of the Course | | The first three lectures (based on Susskind1) are an overview of classical mechanics, focusing, via Poisson brackets and Lagrangian and Hamiltonian mechanics, on the connection between symmetry and conservation laws. Thus students are introduced to the viewpoint at the heart of present-day physics, one that is essential next semester in Quantum Mechanics III. An extra optional lecture (Goldstein) describes in more depth the structure of classical phase space and generators of infinitesimal canonical transformations. Lecture 4 (Susskind2) is an intuitive justification for the mathematical objects required to describe nature at the atomic scale. Next, in order to make inroads into Shankar, we choose to consider carefully the Mathematical Tools of QM (Cohen-Tannoudji, Chapter 2), such as Dirac notation and its usage, then the Postulates of Quantum Mechanics (Cohen-Tannoudji, Chapter 3), which is really just a more in-depth treatment of Lecture 4. After a lecture covering the rest of Shankar Chapter 1 – normal coordinates; Hermitian operators in infinite dimensions; the basics of Hilbert’s 1910 abstract formulation of PDE boundary value problems – we find ourselves already in Shankar Chapter 5, with the advantage, compared to standard treatments, of a strong grasp of the machinery of quantum mechanics. The rest of the course follows Shankar closely, apart from Lectures 14 and 15, which, based on Susskind2, expand on ideas encountered in Chapter 10 by considering qubits, tensor product spaces and operators, the density operator, the reduced density operator, pure states and mixed states, the meaning of measurement, and so on, in an exciting setting at the forefront of current research.
Overall, this semester students will gain a solid grounding in basic quantum mechanics. Problem-solving is an integral part of the course: students should attend fortnightly tutorials (Physics Tutorial IIIb) where they will discuss many of the assignment questions and receive hints for solutions. Weaker students are particularly encouraged to attend tutorials and submit assignments. It is recommended that students also enroll in Statistical Physics II concurrently, where they will complement their knowledge with theory of many-particle systems, both classical and quantum.
Lectures will be recorded and made available on MS Teams immediately after each lecture. |
| | 授業の内容や構成
Course Content / Plan | | Course contents:
The course consists of the equivalent of 15 lectures of examinable material, based on Shankar Chapters 1-10, which constitutes a standard one semester Year 3 topic coverage. In addition, a number of additional sessions will be offered to explain new concepts step by step or to explore quantum phenomena that are easily within reach of the core material -- somewhat like seminars with discussion encouraged. The intention is to help students grasp the abstract content presented in the textbooks, much of which strikingly contradicts classical intuition; and to see with the mind's eye the amazing quantum reality
Shankar Chapters 1-10; or Susskind1 and Susskind2. Some topics are more fully explored in tutorials.
Lecture 1. [1] Symmetries and Conservation Laws. What is a state in classical mechanics? How do states evolve? State space, phase space. Why do trajectories never intersect? Newtonian mechanics. Formulation in terms of energy. The Lagrangian. Principle of Least Action. Euler-Lagrange equations. Cyclic coordinates and conserved quantities. (Susskind1)
Lecture 2. [1] Symmetries and Conservation Laws cont’d. We seek a better way to characterize the connection between symmetries and conservation laws. Poisson brackets. Continuous symmetries. Generators of infinitesimal transformations. Angular momentum is the generator of infinitesimal rotations. Linear momentum is the generator of infinitesimal translations. The Hamiltonian is the generator of infinitesimal time translations. The PB of the Hamiltonian with the generator determines a conservation law if G generates a transformation that leaves the total energy invariant. (Susskind1)
Lecture 3. [0.75] Canonical Transformations: transformations of phase space coordinates (not necessarily infinitesimal) that leave "the physics" unchanged. They map trajectories (i.e. a solution of the equations of motion) into physically equivalent (e.g. rotated) trajectories. (Shankar, Goldstein) NONEXAMINABLE: passive and active transformations. (Shankar, Goldstein)
Optional Lecture 3B. A closer look at: canonical transformations; generators of infinitesimal canonical transformations; symmetry and conservation laws; classical Liouville's Theorem. Phase space is like a flowing incompressible fluid. The flow is a symmetry transformation generated by the Hamiltonian. (Goldstein Ch 8 and 9.)
Lecture 4. [1] Mathematical Tools of QM: A First Look. What kind of mathematics do we need to describe QM experiments? (Based on Susskind2.)
Optional Lecture 4B Mathematical Tools of QM. Introduction. Discrete basis, continuous basis. Orthonormality relations, closure relations. (Cohen-Tannoudji, Chapter 2)
Lecture 5 [1] Mathematical Tools of QM. Dirac notation: ket, bra. Dual space. Discrete basis, continuous basis. Orthonormality relations, closure relations. (Same as last lecture, but in Dirac notation.) (Cohen-Tannoudji, Chapter 2)
Lecture 6. [1] Mathematical Tools of QM. Change of basis using Dirac notation: discrete/continuous basis. Matrix elements of operators. Psi in r basis, p basis: change of basis here is a Fourier transform. Eigenvalue equations and observables. Degenerate, non-degenerate eigenvalues. Orthogonality of eigenspaces belonging to different eigenvalues. Hermitian operators have real eigenvalues. The concept of "observable": e.g., the projection operator. (Cohen-Tannoudji, Chapter 2)
Lecture 6B. [1] Mathematical Tools of QM. Simultaneous diagonalization of two Hermitian operators: non-degenerate case; degenerate case. Block diagonal matrix. Functions of operators: differentiation, integration. Two useful, easy theorems. (Cohen-Tannoudji, Chapter 2; Shankar)
Lecture 7. [0.5] Mathematical Introduction. Some operators in infinite dimensions: X and K operator matrix elements in X and K bases. Commutation operator [X,K]. Hermiticity in infinite dimensions: necessary and sufficient conditions. (Domain of unbounded operators.) NONEXAMINABLE: Meaning of diagonalization of Hermitian operators: normal modes/stationary states. Example: two masses on three springs in one dimension. Example: string clamped at both ends. (Shankar p. 46-54, 57-73.)
Lecture 7B [1] Postulates of Quantum Mechanics (in-depth reprisal of Lecture 4). Quantum state. Reduction (collapse) of the wave packet; role of the projection operator; probability of results of measurement. [Time evolution of a system. (Susskind2 4.12, 4.13)] Quantization rules. Compatible, incompatible observables and the commutator operator. Imprecise measurements. (Cohen-Tannoudji p.213-225; 231-236; 263-266)
Lecture 8. [1] Postulates (cont'd) and Simple Problems in One Dimension. Why is a quantum ensemble necessary? (Shankar p. 125-127) Expectation value and uncertainty (Shankar p. 127-129). Example 4.2.4: Gaussian wave fn. (Shankar p. 134-141) How to extract experimental information from a wave function: probability that a particle has position between x and x+dx; probability that a particle has momentum between p and p+dp; uncertainty in position; uncertainty in momentum. Recipe for solving quantum mechanical problems: the propagator. Space-time propagator for a free particle in one dimension (Shankar p. 151-154).
Optional Lecture 8B. Simple Problems in One Dimension (cont’d). Time-evolution of the Gaussian wave packet. NONEXAMINABLE: The probability current. Wave packet incident on a potential step (1D scattering -- important for QMIII). (Shankar Chapter 5)
Lecture 9. [0.5] The Classical Limit and Simple Harmonic Oscillator in X-basis. (Revision of Fourier transforms. Midsemester exam up to here.) Ehrenfest’s Theorem (Shankar Chapter 6 or Susskind2 4.9, 4.10). Why is the motion of a particle in the quantum regime different to its motion in the classical regime? Under what conditions do the classical equations of motion hold? (Shankar Chapter 6) NONEXAMINABLE: Solution of the linear SHO in the X basis (Shankar Chapter 7).
Lecture 10. [1] SHO in the Energy Basis. Ladder operators: creation and annihilation operators. Number operator. (Shankar Chapter 7 or Susskind2 Chapter 10.)
Lecture 11. [0.5] Path Integral Approach. Simplistic introduction: calculating the propagator using Feynman's path integral approach. The space-time region of coherence. (Shankar Chapter 8 or Susskind2 9.8 for an elementary description.) NONEXAMINABLE: Equivalence to the Schroedinger equation. The propagator for systems with potential energy of a certain, useful general form is relatively easy to calculate using the path-integral approach. Why? (Shankar Chapter 8)
Optional Tutorial Lecture 11B. Path Integral Approach (cont’d). We complete optional topics not finished in Lecture 11.
Lecture 12. [0.75] Heisenberg Uncertainty Relation. A purely “mathematical” derivation. (Susskind2 5.3-5.7, 8.5) Another purely “mathematical” derivation that exposes conditions for minimum uncertainty. The minimum uncertainty wave packet is a Gaussian. Application to estimation of ground state energy of hydrogen atom. (Shankar Chapter 9) NONEXAMINABLE: The standard U.R. gives the wrong result for certain pairs of canonically conjugate observables. Why? Domain of unbounded operators revisited. Derivation of a more generally applicable U.R. (Chisolm, American Journal of Physics 2001) following a simple rule.
Lecture 13. [1] Systems with 2 or more identical particles. Pauli Exclusion Principle follows from a basic experimental fact. (Gottfried) Bosons, fermions. Symmetry or antisymmetry of the TOTAL wave function. Fermionic and bosonic spatial or spin wave functions. Normalisation of state vector. Interference. Combining quantum systems: direct product spaces. Quantization in 1, 2, 3 dimensions (separable partial differential equations). (Shankar Chapter 10)
Lectures 14. [1] Combining Quantum Systems, Entanglement, Correlation. (Susskind2 Chapters 6,7) We explore entanglement and correlations in a 2-qubit system. Density matrix (Shankar), reduced density matrix (Susskind2, Merzbacher, Gottfried).
Lectures 15. [1] Quantum teleportation. Local realism is dead: the CHSH (Clauser-Horne-Shimony-Holt) Inequality. Other examples of "spooky" quantum effects that have been confirmed by experiment. |
| | 履修条件
Course Prerequisites | | Calculus I; Calculus II; Linear Algebra I; Linear Algebra II; Mathematical Physics I; Mathematical Physics II; or Consent of Instructor.
You must be strong in physics and mathematics. |
| | 関連する科目
Related Courses | | Quantum Mechanics II; Physics Tutorial III; Statistical Physics III (next semester).
It is strongly advised that students concurrently enrol in Physics Tutorial III. |
| | 成績評価の方法と基準
Course Evaluation Method and Criteria | | | Attendance and class performance, attitude: 5%; Weekly quizzes or other written assessment: 30%; Midterm exam: 32.5%; Final Exam: 32.5% |
| | 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 the deadline in November. After that day, a letter grade will be awarded based on marks earned from all assessment during the semester.
If Quantum Mechanics II is NOT A COMPULSORY SUBJECT and the student plans never to take Quantum Mechanics II in the future, then a late withdrawal request will be considered. |
| | 参考書
Reference Book | | Reference Books/ Recommended Reading
1. Goldstein, H., Classical Mechanics, 2nd Edition.
2. Feynman, R.P., Leighton, R.B., Sands, M., 2011, Feynman Lectures on Physics (Volume 3), Basic Books. (Highly recommended introductory book on quantum mechanics.)
3. Merzbacher, E., Quantum Mechanics, 3rd Ed., Wiley, 1998. (A great teacher of QM.)
4. Feynman, R.P. and Hibbs, A. R., Quantum Mechanics and Path Integrals: Emended Edition, Dover Books on Physics, 2010.
5. Aharonov Y. and Rohrlich, D., Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley-VCH, 2005.
6. Gottfried, K. and Yan, T.-M., 2004, Quantum Mechanics: Fundamentals, Springer.
(Advanced reference. Excellent treatment of identical particles and PEP.)
7. Kreyszig, E., 1989, Introductory Functional Analysis with Applications, Wiley Classics.
(Clear introduction to infinite dimensional Hilbert space, inner product spaces, spectral theory of linear operators, self-adjoint linear operators, etc. Read this - particularly the latter chapters on unbounded operators - if you want to clear up some mathematical concepts encountered in Shankar.) |
| | 教科書・テキスト
Textbook | | 1. Shankar, R., 1994, Principles of Quantum Mechanics, 2nd ed., Kluwer Academic/Plenum.
2. Cohen-Tannoudji, C., Diu, B., Laloe, F., Quantum Mechanics, Wiley, 1991. Chapters 2 and 3 are required in the lectures. They complement, and at times supercede, the treatment in Shankar.
3. Susskind, L. and Hrabovsky, G., 2013, The Theoretical Minimum [Classical Mechanics], Basic Books.
4. Susskind, L. and Friedman, A., 2014, Quantum Mechanics: The Theoretical Minimum, Basic Books. |
| | 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.
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Advice
It is strongly advised that students concurrently enroll in Physics Tutorial IIIb.
No solutions are handed out in class. It pays to come prepared and pay attention during the tutorial.
Students must be willing to work hard if they wish to achieve a good, internationally competitive level. |
| | 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. |
| | 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 and ability in mathematics and physics. |
| | 他学科聴講の条件
Conditions for Other department student's attendance | | | Students from any department are welcome as long as they have the necessary grounding and ability in mathematics and physics. |
| | レベル
Level | | | | キーワード
Keyword | | | | 履修の際のアドバイス
Advice | | • It is strongly advised that students concurrently enrol in Physics Tutorial IIIb.
• Students must be willing to work hard if they wish to achieve a good, internationally competitive level.
• Alternatively, students who wish to just pass the unit may choose to work through the two books by Susskind, which cover the same topics (except for identical particles) in a far more elementary way, and submit a reasonable number of solved problems. The books by Susskind are written for people who have not previously learned physics. |
| | 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. For many G30 students, English is a 2nd or even 3rd language, so video recordings are an invaluable learning aid.
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. |
| | 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 to answer questions via Teams chat. |
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