Module Specifications.
Current Academic Year 2024 - 2025
All Module information is indicative, and this portal is an interim interface pending the full upgrade of Coursebuilder and subsequent integration to the new DCU Student Information System (DCU Key).
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Date posted: September 2024
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Description To introduce the student to introductory topics in quantum mechanics. The formal rules of quantum mechanics are introduced. It is shown how observable quantities such as position, momentum and angular momentum are represented by operators. The properties of these operators are studied. The angular momentum operators are studied in detail leading to a series of applications of the wave equation to 1-, 2- and 3- dimensional physical systems. Emphasis is placed on the central potential and the energy and angular momentum properties of the Hydrogen atom are studied. Simple perturbation methods for solving non-trivial problems are introduced and applications of these methods to atomic and molecular systems are examined. | |||||||||||||||||||||||||||||||||||||||||||
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Learning Outcomes 1. Demonstrate an understanding of how quantum states are described by wave functions; 2. Solve one-dimensional problems involving transmission, reflection and tunnelling of quantum probability amplitudes; 3. Explain the significance of operators and eigenvalue problems in quantum mechanics; 4. Identify and construct the wavefunction for the hydrogen atom and explain the significance of angular momentum operators in atomic physics. 5. Solve the Schrödinger equation and describe the properties of a single particle in one, two and three dimensional potentials 6. Apply first and second order perturbation theory to simple systems | |||||||||||||||||||||||||||||||||||||||||||
All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml |
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Indicative Content and Learning Activities
Lecture Series: Formulation of QMPhysical states as abstract QM linear vector spaces; Observables as QM operators, Quantum Dynamics, the Schrödinger equation and Measurement. QM principles. Finite and infinite dimension QM systems. Statistical interpretation of QM; Compatible observables and commutators; Heisenberg's uncertainty relations; Eigenstates of an observable as a complete basis set. Superposition principle. Examples in low-dimensional systems (Spin and light polarization); Stern-Gerlach experiment.Lecture Series: Quantum DynamicsThe Schrödinger equation; connection with Hamilton's classical mechanics formulation; Quantum dynamics with a constant Hamiltonian, (conservation of energy); expansion in an Hamiltonian eigenbasis, the time-evolution operator, Ehrenfest theorem, characteristic time evolution of systems. Examples of low-dimensional systems (e.g. spin and light polarization systems)Wave mechanicsWavefunctions as states in a position basis; eigenvalue problem of the position operator; Normalization of improper states, box and delta function normalization. Momentum observable and eigenvalue problem; momentum states; Average and uncertainty of wavefunctions; bound and unbound motion. Particle in 1-D infinite and finite box; wavepacket propagation; momentum eigenstates; proper boundary conditions.Particle in a harmonic oscillator (HO) potentialThe classical HO; the QM formulation of dynamics in a harmonic oscillator potential. The HO eigenvalue problem. Spatial and momentum measurements, uncertainty relations, the Ehrenfest equations of HO-analogy with classical motion.Spin: a purely QM observableThe concept of Spin; electronic Spin; Stern-Gerlach experiment in more detail; Spin of a composite system; Spin dynamics in constant magnetic fields. Magnetic resonance; two-level systems and Pauli matrices.Particle in a Coulomb field (hydrogen)The time-independent Schrödinger equation; Angular momentum observable (conservation of angular momentum); Separation of variables and central-field eigenstates; Spherical harmonics and Laguerre polynomials; Classification of atomic states; Probability distribution; elements of atomic spectroscopyLearning ActivitiesTutorial problems and short projects. Physics: tutorial questions in subjects related with the module's content are worked out; additional questions are provided for homework. Maths: Linear algebra; vector and matrix calculus; eigenvalue matrix problem; ordinary differential equations with constant (and harmonic time-dependent) coefficients; Hermite, Spherical and Laguerre polynomials; delta function and Fourier expansion; spherical coordinate systems. | |||||||||||||||||||||||||||||||||||||||||||
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Indicative Reading List
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Other Resources None | |||||||||||||||||||||||||||||||||||||||||||