5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Andrei Tokmakoff SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mathematical formulation of the dynamics of a quantum system is not unique. Introduction to matrix methods in structural mechanics ... Stiffness matrices are often in the literature derived on basis of energy- and variational methods, which qualitatively speaking let work done by inner force (strain energy) balance work done by . T. F. Jordan, Quantum Mechanics in Simple Matrix Form ~Wiley, New York, 1986!. Introduction Nowadays a classical mechanics course devotes a lot of time to various formulations of classical mechanics (Newtonian, Lagrangian, and Hamiltonian). 2. 1926 (Schrodinger): Erwin Schrodinger formulated a version of quantum mechanics that was based on waves. (1.5) we see that ˆ =∑A mnρnm ()t nm At(), (1.7) =Tr ⎡A ρ t ⎤ ˆ () In practice this makes evaluating expectation values as simple as tracing over a product of p. 2 We see that ρnm , the density matrix elements, are made up of the time-evolving expansion coefficients. • The commutator measures the difference between the two composite operations. For the complex square to be meaningful statistically, we need the probabilities to sum to 1. For example, it has the property of hardness and can be either hard or soft. 1. Lecture Notes Introduction to Strength of Materials Prof. Dr. N.H. Østergaard pp. The founding papers are 3. equivalence between matrix mechanics and wave mechanics at a level accessible to physics students, teachers and researchers. Parallels between matrix mechanics and ordinary vectors and matrices are extensively utilized. Substituting into eq. History. function (PDF) of a quantum system. Ultimately we are interested in observables (probability amplitudes)—we can’t measure a wavefunction. Green, Matrix Mechanics ~P. made use of matrix mechanics. Commutators • In general, two operators do not commute in quantum mechanics. Matrix mechanics was the first formulation of quantum mechanics to be discovered. We won’t deal with this matrix formulation (it’s rather di–cult), but instead with the following wave formulation due to Schrodinger (this is a waves book, after all). Noordhoff, Ltd., Groningen, The Netherlands, 1965!. • Example: Canonical Commutation Relation. Basic Matrix Mechanics Frank Rioux A quon (an entity that exhibits both wave and particle aspects in the peculiar quantum manner - Nick Herbert, Quantum Reality, page 64) has a variety of properties each of which can take on two values. 4 outer forces (external loads). This chapter introduces the matrix-mechanics formulation of quantum mechanics, emphasizing both calculational techniques and conceptual understanding.
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