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 | | 量子力学２ | | 科目名 【英語】
Course Title | | 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 | | | Check the graduation requirement for your program. |
| |
授業の目的 【日本語】
Goals of the Course(JPN) | | | | 授業の目的 【英語】
Goals of the Course | | The main goal of this course is to make the Hilbert Space approach to quantum mechanics seem natural to the student; then to enable the student to solve basic problems in quantum mechanics involving one-dimensional problems, entanglement (essential in quantum computation) and identical particles.
This 2nd course in quantum mechanics is the first half of a full-year course. The goal is to enable students to attain a solid grasp of basic concepts. Underlying the teaching approach is the philosophy that in order to learn well, learners must make it a habit to produce many simple calculations, including working through each line in the textbook: in this way the mathematical language becomes second nature and students learn not to be overwhelmed by mathematical symbols, and to discern the simple physical principles expressed by them: students learn to express with ease their physical intuition using mathematical language. 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.
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 for interest. The intention is to help students grasp the abstract content in the textbook, much of which strikingly contradicts classical intuition; and to see the amazing quantum reality.
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. In this way, non-physics majors such as Biology students can learn concepts at the forefront of physics, such as the path integral, which is useful for computational treatments of the protein-folding problem, opening the possibility of their entry into physics labs such as the computational biology lab in the Department of Physics.
Lectures will be recorded and made available on MS Teams immediately after each lecture.
The first three lectures (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 III) 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.
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. |
| | 到達目標 【日本語】
Objectives of the Course(JPN)) | | | | 到達目標 【英語】
Objectives of the Course | | | In the first three lectures (based on Susskind1), students will receive 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 treated in more depth 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 entanglement: 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. |
| | 授業の内容や構成
Course Content / Plan | | Course contents:
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, 15. [2] 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 (Merzbacher, Gottfried). |
| | 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 |
| | 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 | | 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. Gottfried, K. and Yan, T.-M., 2004, Quantum Mechanics: Fundamentals, Springer.
(Advanced reference. Excellent treatment of identical particles and PEP.)
5. 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.
Alternative, simple treatment of a subset of topics, for students who want to just pass:
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. |
| | 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 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 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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