Foundations of Quantum Mechanics
University College Dublin
Area of Study
Taught In English
Course Level Recommendations
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Recommended U.S. Semester Credits2.5 - 3
Recommended U.S. Quarter Units3.75 - 4.5
Hours & Credits
OverviewThis module introduces Quantum Mechanics in its modern mathematical setting. Several canonical, exactly-solvable
models are studied, including one-dimensional piecewise constant potentials, Dirac potentials, the harmonic
oscillator, and the Hydrogen atom. Three calculational techniques are introduced: time-independent perturbation
theory, variational methods, and numerical (spectral) methods. The postulates of Quantum Mechanics,
[Mathematical background] Complex vector spaces and scalar products, linear forms and duality, the natural scalar product derived from linear forms, Hilbert spaces, linear operators, commutation relations, expectation values, uncertainty, [Time evolution and the Schrodinger equation] Derivation of the Schrodinger equation for time-independent Hamiltonians, the position and momentum representations, the probability current, the free particle [Piecewise constant one-dimensional potentials] Bound and unbound states, wells and barriers, scattering, transmission coefficients, tunneling, [The harmonic oscillator] Solution by power series, Hermite polynomials, creation and annihilation operators, coherent states, [The Hydrogen atom] Solution by separation of variables, quantization of energy and angular momentum, general treatment of central potentials in terms of spherical harmonics, [Angular momentum] Motivation: angular momentum in the hydrogen atom, as derived from spherical harmonics, angular momentum in the abstract setting, intrinsic angular momentum, addition of angular momenta, Clebsch-Gordan coefficients,
[Approximation methods] Time-independent perturbation theory: the non-degenerate case, variational methods for
estimating the ground-state energy
Courses and course hours of instruction are subject to change.
Credits earned vary according to the policies of the students' home institutions. According to ISA policy and possible visa requirements, students must maintain full-time enrollment status, as determined by their home institutions, for the duration of the program.
ECTS (European Credit Transfer and Accumulation System) credits are converted to semester credits/quarter units differently among U.S. universities. Students should confirm the conversion scale used at their home university when determining credit transfer.
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