This course addresses quantum mechanics, the foundation of modern engineering. The instructor will explain that particles such as electrons behave as waves in microscopic systems and lead to the Schrödinger equation, which mathematically describes the wave-like behavior of matter. The instructor will also discuss subjects such as the properties of wave functions, operations for quantization, and the uncertainty principle. The Schrödinger equation will be solved to investigate bound states in quantum wells and quantum tunneling in potential barriers. Further the instructor will discuss harmonic oscillators and central force fields, leading to concepts and mathematics specific to quantum mechanics, such as state vectors and angular momentum. The instructor will reveal the structure of hydrogen atoms and explain the electron configuration of many-electron systems. Also, perturbation theory will be discussed as a method for approximating the Schrödinger equation.
Quantum mechanics is necessary in the field of chemistry and materials engineering related to such things as semiconductors to analyze the behavior of electrons and holes. Quantum computers and other quantum information processing also require knowledge of quantum mechanics. On the other hand, quantum mechanics is said to be difficult to grasp for introductory students. This course provides explanations for an efficient mastery of quantum mechanics, and students will develop a foundation for utilizing and applying it in various fields by completing numerous exercise problems.
By taking this course, you are able to
(1) acquire the innovative view of nature being created by quantum mechanics,
(2) understand the concepts specific to quantum mechanics of the wave probability, state vectors, the uncertainty principle, the tunnel effect, and spin,
(3) solve the Schrödinger equation for square well potential, the harmonic oscillator, and the central force field,
(4) apply the knowledge of linear algebra including matrices and eigenvalues, and special functions,
(5) understand the algebra of angular momenta and visualize the structure of hydrogen atom, and
(6) acquire an approximate means for problems that cannot be exactly solved.
Schrödinger equation, quantum well, tunnel effect, harmonic oscillator, central force field, perturbation theory
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||✔ Practical and/or problem-solving skills|
Read the sections of the textbook corresponding to each class in advance. In each class, basic points are explained and exercises are done. Do the end-of-section problems after class.
|Course schedule||Required learning|
|Class 1||The Planck's Quantum Hypothesis, Einstein's Light Quanta and the Photoelectric Effect||Read the collapse of classical mechanics and the rise of quantum mechanics.|
|Class 2||The Wave-Particle Duality||Learn de Broglie waves.|
|Class 3||Wave Packets and Wave Functions, The Schrödinger Equation||Learn the wave probability, and derive the Schrödinger equation.|
|Class 4||Observables and Operators, The Heisenberg Uncertainty Principle||Learn the relationship between observables and Hermitian operators and how to calculate expectation values, and derive the uncertainty relation.|
|Class 5||Eigenvalues and Eigenfunctions, Free Particles||Derive the energy and momentum eigenstates from the time-independent Schrödinger equation.|
|Class 6||Infinite Square Well Potential||Solve the one-dimensional infinite square well potential problem, and learn the characteristics of eigenfunctions including the complete system and the basis.|
|Class 7||Finite Square Well Potential||Solve the finite square well potential problem, and learn Parity.|
|Class 8||The Comprehension test||Confirm your understandings of Class 1–7.|
|Class 9||The Tunnel Effect||Learn tunneling through a thin potential barrier and the resonant transmission.|
|Class 10||The Harmonic Oscillator||Solve the energy eigenvalues of the harmonic oscillator and express the eigenfunctions with the Hermite polynomial, and derive the zero-point oscillation.|
|Class 11||Creation and Annihilation Operators, State Vectors||Solve the harmonic oscillator problem with the operator method, and acquire the ket notation of state vectors and the matrix notation of operators.|
|Class 12||The Central Force Field||Express the Schrödinger equation with spherical coordinates, and learn angular momentum operators and spherical harmonics.|
|Class 13||The Hydrogen Atom||Learn the directional quantization, and express the eigenfunctions of the hydrogen atom with the Laguerre polynomial.|
|Class 14||Spin and The Pauli Exclusion Principle||Derive spin angular momenta with commutation relations and ladder operators, and learn the Pauli exclusion principle.|
|Class 15||Perturbation Theory||Solve the Schrödinger equation under a perturbation.|
Haruhiko Ito, Quantum Mechanics for Students of Science and Engineering, e-book edition, Kodansha Scientific (Japanese).
The Feynman Lectures on Physics, Vol. III: Quantum Mechanics.
You are assessed by the comprehension of the concepts unique to quantum mechanics and the solution of the Schrödinger equation.
The point allocation is 80 % for midterm and final exams, and 20 % for exercises.
Calculus, Linear Algebra, and Mechanics are needed for this course.