## Quantum Mechanics II(2.0 credits) | |||

Code | : | 10224 | |

Course Type | : | Specialized Courses | |

Class Format | : | Lecture | |

Course Name | : | Fundamental and Applied Physics | |

Starts 1 | : | 3 Autumn Semester | |

Elective/Compulsory | : | Compulsory | |

Lecturer | : | John A. WOJDYLO Designated Professor |

•Course Purpose |

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: 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 are usually recorded and most of them are available on Youtube (private channel). see https://syllabus.sci.nagoya-u.ac.jp/detail/20190680430/ |

•Prerequisite Subjects |

Quantum Mechanics 1 or Consent of Instructor.
• Students must have passed Quantum Mechanics 1 to take Quantum Mechanics 2. |

•Course Topics |

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) see https://syllabus.sci.nagoya-u.ac.jp/detail/20190680430/ |

•Textbook |

1. Shankar, R., 1994, Principles of Quantum Mechanics, 2nd ed., Kluwer Academic/Plenum.
2. Susskind, L. and Hrabovsky, G., 2013, The Theoretical Minimum [Classical Mechanics], Basic Books. 3. Susskind, L. and Friedman, A., 2014, Quantum Mechanics: The Theoretical Minimum, Basic Books. 4. 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 supersede, the treatment in Shankar. |

•Additional 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. 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.) |

•Grade Assessment |

Attendance: 5%; Weekly quizzes or other written assessment: 30%; Mid-semester exam: 32.5%; Final Exam: 32.5% |

•Notes |

Quantum Mechanics 1 or Consent of Instructor.
• Students must have passed Quantum Mechanics 1 to take Quantum Mechanics 2. *See "Course List and Graduation Requirements" for your program for your enrollment year. • Students must be willing to work hard 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 more or less 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. |

•Contacting Faculty |

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