| CODE | PHY2170 | ||||||
| TITLE | Waves and Quantum Mechanics | ||||||
| UM LEVEL | 02 - Years 2, 3 in Modular Undergraduate Course | ||||||
| MQF LEVEL | 5 | ||||||
| ECTS CREDITS | 4 | ||||||
| DEPARTMENT | Physics | ||||||
| DESCRIPTION | This study-unit constitutes an advanced study-unit in waves leading to an introductory unit in quantum mechanics. Study-Unit Aims: The study of waves will provide: - a review of forced and damped simple harmonic motion; - a review of the concept of dispersion; - a review and extension of the concept of superposition of oscillations; - an illustration of the analogy between electric circuits and mechanical systems; - an analysis of non-linear vibrations, including the mathematical formulation of the analysis of the equilibrium points for autonomous systems; - a heuristic discussion of chaotic systems; - an analysis of coupled oscillators leading to the derivation of the wave equation for an oscillating string; - a detailed explanation of how to split a complex oscillatory pattern into its harmonics using the Fourier series and its complex analogue; - a detailed explanation how to derive the Fourier Transform for a non-periodic function; - an analysis of the properties of the Fourier Transform; - a detailed explanation of how to obtain the discrete and continuous Fourier spectrum; - an overview of the practical applications of the Fourier series and Transform. The study of quantum mechanics will provide: - an overview of basic concepts in probability and statistics; - an introduction to vector algebra; - a review of the experimental theoretical results that lead to the breakdown of classical mechanics and the introduction of quantum mechanics; - a discussion on the basic postulates of quantum mechanics; - an introduction to the Dirac notation and its use; - the derivation of general properties of wavefunctions and operators; - the definition of the commutator and its physical implication; - the derivation of the general uncertainty principle; - simple examples on how to solve Schrödinger equation in one dimension when the potential is a constant; - an introduction to the notion of density of states; - an introduction to the notion of parity. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - explain the concept of dispersion; - explain how electric circuits are related to mechanical systems; - explain how chaos arises in a system; - explain the practical importance of the Fourier series and Transform; - explain how amplitude modulation works; - distinguish between the mean, expectation value and mode; - state the conditions for a function to be used as a probability distribution; - be able to describe at least three experiments that lead to the breakdown of classical mechanics and the introduction of quantum mechanics; - describe the Dirac notation; - state the definition of Hermitian operator; - explain why operators corresponding to observables are Hermitian; - explain when the time part of a wavefunction can be separated from the space part; - explain why the general wavefunction is obtained by a linear combination of all possible states; - interpret a general wavefunction in terms of the probability the system is found in a particular state; - give the definition of the commutator; - explain the physical significance when two operators have a non-zero commutator; - explain the physical significance when two operators have a non-zero uncertainty; - explain what is meant by density of states; - explain what is meant by parity. 2. Skills: By the end of the study-unit the student will be able to: - work out hard problems involving forced and damped simple harmonic motion including the derivation of the equations of motion for a given system and the determination of their solution; - work out the resultant of multiple waves that are superposed; - make use of the analogy between electric circuits and mechanical systems so that given a set of describing equations for an electric circuit they can determine how the system will behave; - derive the relation between the velocity and position of a non-linear system; - determine the nature of equilibrium points for nonlinear systems and use this information to draw the corresponding phase plot; - derive the equation of the curve in phase space that separates stable and unstable equilibrium points; - derive the equation of motion of coupled oscillators and solve them; - derive the Fourier series and its complex analogue for a given oscillatory function; - derive the Fourier Transform for a non-periodic function; - draw the discrete and continuous Fourier spectrum; - derive the properties of the Fourier Transform and use them to solve simple problems; - derive the mean, mode and the expectation value of a function for a giving probability distribution; - be able to apply the Dirac notation to solve problems; - determine if an operator is Hermitian or not; - derive the time dependant wavefunction from the initial state; - obtain the general wavefunction from a list of all possible states; - determine the probability that in a measurement the system is found in a particular state; - determine the expectation value of an operator for a given wavefunction; - derive the commutator of two given operators; - given two operators determine the uncertainty associated with the successive measurements of the corresponding observables; - solve the Schrödinger equation in one dimension for the infinite square well and obtain the general solution. Main Text/s and any supplementary readings: Waves: Recommended textbooks: French, A.P., Vibrations and Waves, Van Nostrand Reinhold or Ingard, K.U., Fundamentals of waves and oscillations, Cambridge press or Pain, H.J., The Physics of Vibrations & Waves, John Wiley & Sons. Bibliography: Tongue, B.H., Principles of Vibration, Second edition, Oxford University Press Quantum Mechanics: Recommended textbooks: Zettili, N., Quantum Mechanics, concepts and applications, John Wiley and Sons or Griffiths, D. J., Introduction to Quantum Mechanics, Prentice Hall. Bibliography: Liboff, R. L., Introductory Quantum Mechanics, Addison-Wesley. Cassels, J. M., Basic Quantum Mechanics, Macmillan. Pebbles, P. J. E., Quantum Mechanics, Princeton University Press. Goswani, A., Quantum Mechanics, Wm C Brown Publishers. Eisberg R. and Resnick, R., Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, John Wiley and Sons. Park, D., Introduction to the Quantum Theory, McGraw-Hill. Additional reading: Al Khalili, J., Quantum: A Guide for the Perplexed, Weidenfeld Nicolson. The use of other books with similar content would be equivalently good. |
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| ADDITIONAL NOTES | Pre-Requisite qualifications: This unit follows from PHY1160 Waves and Optics or any basic course on vibrations and waves. Knowledge of advanced calculus, matrices and geometry at the level of MAT1091 Mathematical Methods and MAT1511 Analytical Geometry or equivalent is expected. | ||||||
| STUDY-UNIT TYPE | Lecture | ||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Pierre Sandre Farrugia |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2023/4. It may be subject to change in subsequent years. |
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