 | Use this page to view syllabus information, learning objectives, required materials, and technical requirements for the module.
As a result of College adapting your modules to combine face-to-face on campus and online teaching and learning support, the breakdown of notional learning hours set out under the heading “Technical Requirements” below may not necessarily reflect how each module will be delivered this year. Further details relating to this will be made available by your department and will be updated as part of the student timetable. |
| PH 3210V - Quantum Theory |
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Associated Term:
2022/23 Academic Session
Learning Objectives:
Formal Aspects of Quantum Mechanics: Wavefunctions, principles of superposition, interference, state vectors and bra-ket notation (Dirac notation), delta function. Compatible observers, simultaneous measurement and commuting operators. Expansion postulate and complete sets of states. The generalised uncertainty relations. Matrix Representation of states and operators. Time dependent Schrödinger equation, expectation values, Hermitian operators, eigenstates, time evolution of operators. Periodic potential; Bloch theorem. Step-operator/ladder operator approach to the harmonic oscillator, derivation of energy eigenvalues and wavefunctions (explicit forms for n = 0, 1).
The Hydrogenic Atom: Solution of the non-relativistic Schrödinger equation for an electron in the field of a stationary nucleus in spherical polar coordinates, obtain normalised eigenfunctions. Energy levels, angular momentum quantum numbers and their allowed values.
Angular Momentum: Review of commutation relations, eigenvalues and eigenfunctions of angular momentum operators, generalised angular momentum, step operator techniques in angular momentum theory; spectrum of angular momentum eigenvalues. Rules for combining angular momenta in general. Representation of spin ½ operators by Pauli matrices, magnetic moments, Stern-Gerlach experiment.
Approximate Methods: Time independent perturbation theory for non-degenerate system to second order in the energy; to first-order for degenerate systems. Examples. Variational principle, He ground state example. Further examples of applications of quantum mechanics to atomic, nuclear and solid state physics: spin-dependent interactions, interaction of a hydrogen atom with a strong uniform external magnetic field, the stark effect, an harmonic oscillator.
Identical Particles: Exchange symmetry for a system with identical fermions or bosons; derivation of the Pauli principle. Independent particle model of He, singlet and triplet states, exchange interaction.
Simple time-dependent systems: Time dependent perturbation theory, interaction of a hydrogen atom with an oscillating electric field. Superposition of states of different energies. Electron in a magnetic field. Transition to a continuum; density of states, Fermi’s golden rule.
Learning Outcomes:
On completion of the course, students should be able to:
1. Understand and use the bra-ket (Dirac) notation for quantum states;
2. Understand and use the vector space and matrix representation of operator formalism, expansion of any states in terms of some complete set, the ladder operator approach to the harmonic oscillator;
3. Generalize the definition of angular momentum to include spin and solve the generalized angular momentum eigenvalue problem employing raising and lowering operator techniques;
4. Discuss the properties of spin-1/2 systems and use the Pauli matrices to solve simple problems; understand the concept and consequences of identical particles for fermions and bosons;
5. State the rules for the addition of angular momenta and to outline the underlying general, mathematical arguments, applying them in particular to two spin-1/2 particles;
6. Formulate first order and second order time-independent perturbation theory, and apply to some simple examples;
7. Formulate the variational method and apply to some simple systems;
8. Formulate first-order and second order time-dependent perturbation theory. Show, as an example, how it can lead to Fermi's Golden Rule.
Required Materials: Click here for the reading list system Technical Requirements: The total number of notional learning hours associated with course are 150. These will normally be broken down as follows: 35 hours of contact time including lectures, feedback sessions and revision lectures 115 hours spent learning material, answering coursework problems and revision. Formative Assessment: Students answer assessed problem sheets, which will then be discussed during feedback sessions. Summative Assessment: Problem Sets (20%) Quizzes: (10%) Exam: (70%) Deadlines: Normally within 2 weeks from issue of the problem sheets. |
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