Undergraduate Course: Principles of Quantum Mechanics (PHYS10094)
|School||School of Physics and Astronomy
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 3 Undergraduate)
||Availability||Available to all students
|Summary||It provides a good starting point for SH quantum courses, in particular Symmetries of Quantum Mechanics, Quantum Theory, and QFT. The second half of the course, Quantum Physics, will continue to be available as a level 10 10pt S2 SH course for Physics, Astrophysics etc: the syllabus is essentially unchanged from the current syllabus of this course, and contains, as at present, some introductory lectures to bring together the cohorts.
Semester 1: Mathematical foundations of quantum mechanics
- Introduction: Particles vs Waves: photoelectric effect, double slit diffraction, linear superposition, the need for a probabilistic interpretation. 
- States of a quantum system as vectors in a Hilbert space. Dirac notation. Inner product of state vectors, probability. 
- Operators and observables, properties of Hermitian operators (orthogonality and completeness). Commutators and compatible observables. Complete Sets of Commuting Observables. Degeneracy. 
- The Hamiltonian as the generator of time translations: the generic Schrodinger equation. Energy and Hamiltonian. Two-state systems, examples of finite-dimensional Hilbert spaces. 
- Completeness and orthogonality relations for the eigenfunctions of a continuum spectrum. Dirac's delta and distributions. Position and momentum space representations. The commutation relations for x and p: the Heisenberg uncertainty relation. 
- The Schrodinger equation for a point particle. One-dimensional square wells, tunneling. 
- Solution of the one- dimensional harmonic oscillator using both Hermite polynomials, and creation and annihilation operators. Spectrum of the Hamiltonian. Explicit construction of the eigenfunctions of the Hamiltonian. 
- Symmetries of the Hamiltonian and eigenfunctions. Degeneracy and symmetries of the Hamiltonian. Rotational symmetry: hydrogen atom as an example. 
- Angular momentum and the generators of three-dimensional rotations. Algebra of the generators. Simultaneous eigenstates of L^2 and L_z. 
- Angular momentum using raising and lowering operators. Derivation of the spectrum, quantization of the eigenvalues. Normalization of the eigenstates. Explicit form of the eigenfunctions in terms of spherical harmonics. 
- Derivation of the spectrum of the hydrogen atom (both differential and algebraic). 
- Spin as an intrinsic property of a quantum system. More on representations of the group of rotations. Arbitrary spin and bases for the space of physical states. 
- Tensor product and addition of angular momentum. 
- Identical particles, link with the group of permutations. Two-electron wave function, combining spin and spatial wave functions. Pauli exclusion principle. Spin and statistics: bosons and fermions. Atomic structure. 
Semester 2: Quantum Physics
- Revision: state vectors and wavefunctions, observables and operators, repeated and successive measurements, uncertainty, compatible observables, Hilbert space. 
- Degeneracy and measurement. Commuting sets of operators. Good quantum
numbers and maximal measurements. 
- The time-dependent Schrodinger equation, the Hamiltonian operator, stationary states and the time-independent Schrodinger equation. Constants of motion. 
- Non-degenerate time-independent perturbation theory: the first-order formulae for energy shifts and wavefunction mixing. Higher-orders. 
- Perturbing degenerate systems: the first-order calculation of energy shifts. Lifting of degeneracy by perturbations. Special cases. 
- Hydrogen fine structure. Kinetic energy correction, spin-orbit correction and Darwin correction. 
- The Helium atom. Two-electron wavefunctions and their symmetries. First-order perturbative treatment of the inter-electron repulsion in the ground state. The first excited states of Helium: singlet-triplet splitting and exchange interaction. 
- Brief outline of multi-electron atoms and their treatment in the central field approximation. Slater determinants. 
- The Rayleigh-Ritz variational method. Ground state of Hydrogen as an example. Result for Helium atom. Variational bounds for excited states. 
-Hidden variables, EPR Paradox, and Bell's inequality. 
-Characterising entanglement - qubit, no-cloning theorem, quantum communication - teleportation.
-Quantum secure communication, superdense coding. 
-Quantum computing basics, gates and registers.
-Quantum computing algorithms - Deutsch, Grover.
-Role of information theory in quantum entanglement - Shannon entropy, density Matrix, von Neumann entropy, entanglement entropy.
- Time-dependent Hamiltonians, the Heisenberg picture, and Heisenberg equation of motion. Equivalence of Heisenberg and Schrodinger representations. 
- Time-dependent perturbation theory, the Dirac picture. Transition probabilities. Special case: time-independent perturbations. 
- Transitions to a group of states induced by a constant perturbation: Fermi's Golden Rule. Harmonic perturbations and transitions to a group of states. 
- Interaction of radiation with quantum systems. Electromagnetic radiation. Interaction with a 1-electron atom. The dipole approximation. Absorption and stimulated emission of radiation. Laser action. 
- Spontaneous emission of radiation. Einstein A and B coefficients. Electric dipole selection rules. Parity selection rules. 
- Quantum scattering theory. Differential and total cross-sections. The Born approximation via Fermi's Golden Rule: density of states, incident flux, scattered flux. Differential cross-section for elastic scattering. 
- Scattering by central potentials. The screened Coulomb potential. Coulomb scattering and the Rutherford cross-section. Two-body scattering and the CM frame. 
- The H+2 ion & molecular bonding. Born-Oppenheimer approximation. Variational estimates of the ground-state energy. Brief discussion of rotational and vibrational modes. 
Information for Visiting Students
|High Demand Course?
Course Delivery Information
|Academic year 2023/24, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 44,
Seminar/Tutorial Hours 44,
Summative Assessment Hours 3,
Revision Session Hours 1,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Feedback is provided through written comments on hand-ins, Q&A sessions during tutorial. One on one meetings with the lecturer, or the teaching assistants.
||Hours & Minutes
|Main Exam Diet S2 (April/May)||3:00|
On completion of this course, the student will be able to:
- Understand the foundational principles of quantum mechanics, the underlying mathematical concepts, and how they relate to broader physical principles. - Understand in detail the Hilbert space structure underlying the foundations of quantum mechanics. - Understanding the difference and relation between algebraic and differential formulations and solutions of quantum mechanical problems. - Be able to derive many of the wide range of phenomena in quantum mechanics from basic principles, including bound states and scattering. - Derive and understand the use of quantum mechanics in atomic physics, and the wider implications of the probabilistic interpretation, spin, the Pauli principle, so on and also understand quantum entanglement with applications to quantum teleportation, computing and information. - Devise and implement a systematic strategy for solving a complex problem by breaking it down into its constituent parts. - Use the experience, intuition and mathematical tools learned from solving physics problems to solve a wider range of unseen problems. - Resolve conceptual and technical difficulties by locating and integrating relevant information from a diverse range of sources.
- Understanding the difference and relation between algebraic and differential formulations and solutions of quantum mechanical problems.
- Be able to derive many of the wide range of phenomena in quantum mechanics from basic principles, including the use of quantum mechanics in atomic physics, and the wider implications of the probabilistic interpretation, spin, the Pauli principle, bound states and scattering.
- Devise and implement a systematic strategy for solving a complex problem by breaking it down into its constituent parts. Use the experience, intuition and mathematical tools learned from solving physics problems to solve a wider range of unseen problems. Resolve conceptual and technical difficulties by locating and integrating relevant information from a diverse range of sources.
|None of these texts is compulsory for the course. However you may find it interesting to compare the material in the lectures with other presentations. There are many good textbooks on QM. Browse until you find something that you like. |
PAM Dirac "Principles of Quantum Mechanics"
J Sakurai "Modern Quantum Mechanics"
F Mandl "Quantum Mechanics"
DJ Griffiths "Introduction to Quantum Mechanics"
S Gasiorowicz "Quantum Physics"
|Graduate Attributes and Skills
|Course organiser||Prof Luigi Del Debbio
Tel: (0131 6)50 5212
|Course secretary||Ms Lucy Davis-Jenkins