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8.321 Quantum Theory I (MIT) 8.321 Quantum Theory I (MIT)

Description

8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement. 8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement.

Subjects

eigenstates | eigenstates | uncertainty relation | uncertainty relation | observables | observables | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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8.322 Quantum Theory II (MIT) 8.322 Quantum Theory II (MIT)

Description

8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation. 8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation.

Subjects

uncertainty relation | uncertainty relation | observables | observables | eigenstates | eigenstates | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | quantization | quantization | EM radiation field | EM radiation field | electromagnetic radiation field | electromagnetic radiation field | adiabatic theorem | adiabatic theorem | Berry?s phase | Berry?s phase | many-particle systems | many-particle systems | Dirac equation | Dirac equation | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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20.310J Molecular, Cellular, and Tissue Biomechanics (MIT) 20.310J Molecular, Cellular, and Tissue Biomechanics (MIT)

Description

This course develops and applies scaling laws and the methods of continuum and statistical mechanics to biomechanical phenomena over a range of length scales, from molecular to cellular to tissue or organ level. This course develops and applies scaling laws and the methods of continuum and statistical mechanics to biomechanical phenomena over a range of length scales, from molecular to cellular to tissue or organ level.

Subjects

biomechanics | biomechanics | molecular mechanics | molecular mechanics | cell mechanics | cell mechanics | Brownian motion | Brownian motion | Reynolds numbers | Reynolds numbers | mechanochemistry | mechanochemistry | Kramers' model | Kramers' model | Bell model | Bell model | viscoelasticity | viscoelasticity | poroelasticity | poroelasticity | optical tweezers | optical tweezers | extracellular matrix | extracellular matrix | collagen | collagen | proteoglycan | proteoglycan | cell membrane | cell membrane | cell motility | cell motility | mechanotransduction | mechanotransduction | cancer | cancer | biological systems | biological systems | molecular biology | molecular biology | cell biology | cell biology | cytoskeleton | cytoskeleton | cell | cell | biophysics | biophysics | cell migration | cell migration | biomembrane | biomembrane | tissue mechanics | tissue mechanics | rheology | rheology | polymer | polymer | length scale | length scale | muscle mechanics | muscle mechanics | experimental methods | experimental methods

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8.09 Classical Mechanics III (MIT) 8.09 Classical Mechanics III (MIT)

Description

This course covers Lagrangian and Hamiltonian mechanics, systems with constraints, rigid body dynamics, vibrations, central forces, Hamilton-Jacobi theory, action-angle variables, perturbation theory, and continuous systems. It provides an introduction to ideal and viscous fluid mechanics, including turbulence, as well as an introduction to nonlinear dynamics, including chaos. This course covers Lagrangian and Hamiltonian mechanics, systems with constraints, rigid body dynamics, vibrations, central forces, Hamilton-Jacobi theory, action-angle variables, perturbation theory, and continuous systems. It provides an introduction to ideal and viscous fluid mechanics, including turbulence, as well as an introduction to nonlinear dynamics, including chaos.

Subjects

Lagrangian mechanics | Lagrangian mechanics | Hamiltonian mechanics | Hamiltonian mechanics | systems with constraints | systems with constraints | rigid body dynamics | rigid body dynamics | vibrations | vibrations | central forces | central forces | Hamilton-Jacobi theory | Hamilton-Jacobi theory | action-angle variables | action-angle variables | perturbation theory | perturbation theory | continuous systems | continuous systems | ideal fluid mechanics | ideal fluid mechanics | viscous fluid mechanics | viscous fluid mechanics | turbulence | turbulence | nonlinear dynamics | nonlinear dynamics | chaos | chaos

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6.685 Electric Machines (MIT) 6.685 Electric Machines (MIT)

Description

6.685 explores concepts in electromechanics, using electric machinery as examples. It teaches an understanding of principles and analysis of electromechanical systems. By the end of the course, students are capable of doing electromechanical design of the major classes of rotating and linear electric machines, and have an understanding of the principles of the energy conversion parts of Mechatronics. In addition to design, students learn how to estimate the dynamic parameters of electric machines and understand what the implications of those parameters are on the performance of systems incorporating those machines. 6.685 explores concepts in electromechanics, using electric machinery as examples. It teaches an understanding of principles and analysis of electromechanical systems. By the end of the course, students are capable of doing electromechanical design of the major classes of rotating and linear electric machines, and have an understanding of the principles of the energy conversion parts of Mechatronics. In addition to design, students learn how to estimate the dynamic parameters of electric machines and understand what the implications of those parameters are on the performance of systems incorporating those machines.

Subjects

electric | electric | machine | machine | transformers | transformers | electromechanical | electromechanical | transducers | transducers | rotating | rotating | linear electric machines | linear electric machines | lumped parameter | lumped parameter | dc | dc | induction | induction | synchronous | synchronous | energy conversion | energy conversion | electromechanics | electromechanics | Mechatronics | Mechatronics | Electromechanical transducers | Electromechanical transducers | rotating electric machines | rotating electric machines | lumped-parameter elecromechanics | lumped-parameter elecromechanics | interaction electromechanics | interaction electromechanics | device characteristics | device characteristics | energy conversion density | energy conversion density | efficiency | efficiency | system interaction characteristics | system interaction characteristics | regulation | regulation | stability | stability | controllability | controllability | response | response | electric machines | electric machines | drive systems | drive systems | electric machinery | electric machinery | electromechanical systems | electromechanical systems | design | design | dynamic parameters | dynamic parameters | phenomena | phenomena | interactions | interactions | classical mechanics | classical mechanics

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2.797J Molecular, Cellular, and Tissue Biomechanics (MIT) 2.797J Molecular, Cellular, and Tissue Biomechanics (MIT)

Description

This course develops and applies scaling laws and the methods of continuum mechanics to biomechanical phenomena over a range of length scales. Topics include structure of tissues and the molecular basis for macroscopic properties; chemical and electrical effects on mechanical behavior; cell mechanics, motility and adhesion; biomembranes; biomolecular mechanics and molecular motors. The class also examines experimental methods for probing structures at the tissue, cellular, and molecular levels. This course develops and applies scaling laws and the methods of continuum mechanics to biomechanical phenomena over a range of length scales. Topics include structure of tissues and the molecular basis for macroscopic properties; chemical and electrical effects on mechanical behavior; cell mechanics, motility and adhesion; biomembranes; biomolecular mechanics and molecular motors. The class also examines experimental methods for probing structures at the tissue, cellular, and molecular levels.

Subjects

molecular mechanics | molecular mechanics | tissue mechanics | tissue mechanics | cell mechanics | cell mechanics | molecular electromechanics | molecular electromechanics | electromechanical and physiochemical properties of tissues | electromechanical and physiochemical properties of tissues | physical regulation | physical regulation | cellular metabolism | cellular metabolism | tissue-level deformation | tissue-level deformation | muscle constriction | muscle constriction

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5.61 Physical Chemistry (MIT) 5.61 Physical Chemistry (MIT)

Description

This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems -- the particle in a box, the harmonic oscillator, the rigid rotor and the hydrogen atom. The lectures continue with a discussion of atomic structure and the Periodic Table. The final lectures cover applications to chemical bonding including valence bond and molecular orbital theory, molecular structure, spectroscopy.AcknowledgementsThe material for 5.61 has evolved over a period of many years, and, accordingly, several faculty members have contributed to the development of the course contents. The original version of the lecture notes that are available on OCW was prepa This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems -- the particle in a box, the harmonic oscillator, the rigid rotor and the hydrogen atom. The lectures continue with a discussion of atomic structure and the Periodic Table. The final lectures cover applications to chemical bonding including valence bond and molecular orbital theory, molecular structure, spectroscopy.AcknowledgementsThe material for 5.61 has evolved over a period of many years, and, accordingly, several faculty members have contributed to the development of the course contents. The original version of the lecture notes that are available on OCW was prepa

Subjects

physical chemistry | physical chemistry | quantum mechanics | quantum mechanics | quantum chemistry | quantum chemistry | particles and waves; wave mechanics | particles and waves; wave mechanics | atomic structure | atomic structure | valence orbital | valence orbital | molecular orbital theory | molecular orbital theory | molecular structure | molecular structure | photochemistry | photochemistry | particles and waves | wave mechanics | particles and waves | wave mechanics

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18.307 Integral Equations (MIT) 18.307 Integral Equations (MIT)

Description

This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications. This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications.

Subjects

integral equations | integral equations | applied mathematics | applied mathematics | Volterra equation | Volterra equation | Fredholm equation | Fredholm equation | Fredholm theory | Fredholm theory | Hilbert-Schmidt theorem | Hilbert-Schmidt theorem | Wiener-Hopf Method | Wiener-Hopf Method | partial differential equations | partial differential equations | Hilbert Problem | Hilbert Problem | ingular integral equations | ingular integral equations | Cauchy type | Cauchy type | inverse scattering transform | inverse scattering transform | group theory | group theory | fluid mechanics | fluid mechanics | solid mechanics | solid mechanics | acoustics | acoustics | quantum mechanics | quantum mechanics

License

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8.05 Quantum Physics II (MIT) 8.05 Quantum Physics II (MIT)

Description

Together, this course and 8.06: Quantum Physics III cover quantum physics with applications drawn from modern physics. Topics covered in this course include the general formalism of quantum mechanics, harmonic oscillator, quantum mechanics in three-dimensions, angular momentum, spin, and addition of angular momentum. Together, this course and 8.06: Quantum Physics III cover quantum physics with applications drawn from modern physics. Topics covered in this course include the general formalism of quantum mechanics, harmonic oscillator, quantum mechanics in three-dimensions, angular momentum, spin, and addition of angular momentum.

Subjects

General formalism of quantum mechanics: states | General formalism of quantum mechanics: states | operators | operators | Dirac notation | Dirac notation | representations | representations | measurement theory | measurement theory | Harmonic oscillator: operator algebra | Harmonic oscillator: operator algebra | states | states | Quantum mechanics in three-dimensions: central potentials and the radial equation | Quantum mechanics in three-dimensions: central potentials and the radial equation | bound and scattering states | bound and scattering states | qualitative analysis of wavefunctions | qualitative analysis of wavefunctions | Angular momentum: operators | Angular momentum: operators | commutator algebra | commutator algebra | eigenvalues and eigenstates | eigenvalues and eigenstates | spherical harmonics | spherical harmonics | Spin: Stern-Gerlach devices and measurements | Spin: Stern-Gerlach devices and measurements | nuclear magnetic resonance | nuclear magnetic resonance | spin and statistics | spin and statistics | Addition of angular momentum: Clebsch-Gordan series and coefficients | Addition of angular momentum: Clebsch-Gordan series and coefficients | spin systems | spin systems | allotropic forms of hydrogen | allotropic forms of hydrogen | Angular momentum | Angular momentum | Harmonic oscillator | Harmonic oscillator | operator algebra | operator algebra | Spin | Spin | Stern-Gerlach devices and measurements | Stern-Gerlach devices and measurements | central potentials and the radial equation | central potentials and the radial equation | Clebsch-Gordan series and coefficients | Clebsch-Gordan series and coefficients | quantum physics | quantum physics | 8. Quantum mechanics in three-dimensions: central potentials and the radial equation | 8. Quantum mechanics in three-dimensions: central potentials and the radial equation | and allotropic forms of hydrogen | and allotropic forms of hydrogen

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20.410J Molecular, Cellular and Tissue Biomechanics (BE.410J) (MIT) 20.410J Molecular, Cellular and Tissue Biomechanics (BE.410J) (MIT)

Description

This course develops and applies scaling laws and the methods of continuum mechanics to biomechanical phenomena over a range of length scales. Topics include: structure of tissues and the molecular basis for macroscopic properties; chemical and electrical effects on mechanical behavior; cell mechanics, motility and adhesion; biomembranes; biomolecular mechanics and molecular motors. Experimental methods for probing structures at the tissue, cellular, and molecular levels will also be investigated. This course was originally co-developed by Professors Alan Grodzinsky, Roger Kamm, and L. Mahadevan. This course develops and applies scaling laws and the methods of continuum mechanics to biomechanical phenomena over a range of length scales. Topics include: structure of tissues and the molecular basis for macroscopic properties; chemical and electrical effects on mechanical behavior; cell mechanics, motility and adhesion; biomembranes; biomolecular mechanics and molecular motors. Experimental methods for probing structures at the tissue, cellular, and molecular levels will also be investigated. This course was originally co-developed by Professors Alan Grodzinsky, Roger Kamm, and L. Mahadevan.

Subjects

Scaling laws | Scaling laws | continuum mechanics | continuum mechanics | biomechanical phenomena | biomechanical phenomena | length scales | length scales | tissue structure | tissue structure | molecular basis for macroscopic properties | molecular basis for macroscopic properties | chemical and electrical effects on mechanical behavior | chemical and electrical effects on mechanical behavior | cell mechanics | motility and adhesion | cell mechanics | motility and adhesion | biomembranes | biomembranes | biomolecular mechanics and molecular motors | biomolecular mechanics and molecular motors | Experimental methods | Experimental methods | BE.410J | BE.410J | BE.410 | BE.410 | 2.798 | 2.798 | 6.524 | 6.524

License

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1.978 From Nano to Macro: Introduction to Atomistic Modeling Techniques (MIT) 1.978 From Nano to Macro: Introduction to Atomistic Modeling Techniques (MIT)

Description

The objective of this course is to introduce large-scale atomistic modeling techniques and highlight its importance for solving problems in modern engineering sciences. We demonstrate how atomistic modeling can be used to understand how materials fail under extreme loading, involving unfolding of proteins and propagation of cracks. This course was featured in an MIT Tech Talk article. The objective of this course is to introduce large-scale atomistic modeling techniques and highlight its importance for solving problems in modern engineering sciences. We demonstrate how atomistic modeling can be used to understand how materials fail under extreme loading, involving unfolding of proteins and propagation of cracks. This course was featured in an MIT Tech Talk article.

Subjects

large-scale atomistic | large-scale atomistic | large-scale atomistic modeling techniques | large-scale atomistic modeling techniques | modern engineering sciences | modern engineering sciences | atomistic modeling | atomistic modeling | extreme loading | extreme loading | ductile and brittle materials failure | ductile and brittle materials failure | molecular dynamics | molecular dynamics | simulations | simulations | Cauchy-Born rule | Cauchy-Born rule | biomechanics | biomechanics | biomaterials | biomaterials | copper nanocrystal | copper nanocrystal | nanomechanics | nanomechanics | material mechanics | material mechanics

License

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BE.410J Molecular, Cellular and Tissue Biomechanics (MIT) BE.410J Molecular, Cellular and Tissue Biomechanics (MIT)

Description

This course develops and applies scaling laws and the methods of continuum mechanics to biomechanical phenomena over a range of length scales. Topics include: structure of tissues and the molecular basis for macroscopic properties; chemical and electrical effects on mechanical behavior; cell mechanics, motility and adhesion; biomembranes; biomolecular mechanics and molecular motors. Experimental methods for probing structures at the tissue, cellular, and molecular levels will also be investigated.This course was originally co-developed by Professors Alan Grodzinsky, Roger Kamm, and L. Mahadevan. This course develops and applies scaling laws and the methods of continuum mechanics to biomechanical phenomena over a range of length scales. Topics include: structure of tissues and the molecular basis for macroscopic properties; chemical and electrical effects on mechanical behavior; cell mechanics, motility and adhesion; biomembranes; biomolecular mechanics and molecular motors. Experimental methods for probing structures at the tissue, cellular, and molecular levels will also be investigated.This course was originally co-developed by Professors Alan Grodzinsky, Roger Kamm, and L. Mahadevan.

Subjects

Scaling laws | Scaling laws | continuum mechanics | continuum mechanics | biomechanical phenomena | biomechanical phenomena | length scales | length scales | tissue structure | tissue structure | molecular basis for macroscopic properties | molecular basis for macroscopic properties | chemical and electrical effects on mechanical behavior | chemical and electrical effects on mechanical behavior | cell mechanics | motility and adhesion | cell mechanics | motility and adhesion | biomembranes | biomembranes | biomolecular mechanics and molecular motors | biomolecular mechanics and molecular motors | Experimental methods | Experimental methods | 2.798J | 2.798J | 6.524J | 6.524J | 10.537 | 10.537 | BE.410 | BE.410 | 2.798 | 2.798 | 6.524 | 6.524

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1.033 Mechanics of Material Systems: An Energy Approach (MIT) 1.033 Mechanics of Material Systems: An Energy Approach (MIT)

Description

1.033 provides an introduction to continuum mechanics and material modeling of engineering materials based on first energy principles: deformation and strain; momentum balance, stress and stress states; elasticity and elasticity bounds; plasticity and yield design. The overarching theme is a unified mechanistic language using thermodynamics, which allows understanding, modeling and design of a large range of engineering materials. This course is offered both to undergraduate (1.033) and graduate (1.57) students. 1.033 provides an introduction to continuum mechanics and material modeling of engineering materials based on first energy principles: deformation and strain; momentum balance, stress and stress states; elasticity and elasticity bounds; plasticity and yield design. The overarching theme is a unified mechanistic language using thermodynamics, which allows understanding, modeling and design of a large range of engineering materials. This course is offered both to undergraduate (1.033) and graduate (1.57) students.

Subjects

continuum mechanics | continuum mechanics | material modeling | material modeling | engineering materials | engineering materials | energy principles: deformation and strain | energy principles: deformation and strain | momentum balance | momentum balance | stress | stress | stress states | stress states | elasticity and elasticity bounds | elasticity and elasticity bounds | plasticity | plasticity | yield design | yield design | first energy principles | first energy principles | deformation | deformation | strain | strain | elasticity bounds | elasticity bounds | unified mechanistic language | unified mechanistic language | thermodynamics | thermodynamics | engineering structures | engineering structures | unified framework | unified framework | irreversible processes | irreversible processes | structural engineering | structural engineering | soil mechanics | soil mechanics | mechanical engineering | mechanical engineering | materials science | materials science | solids | solids | durability mechanics | durability mechanics

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12.620J Classical Mechanics: A Computational Approach (MIT) 12.620J Classical Mechanics: A Computational Approach (MIT)

Description

We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration. We will consider the following topics: the Lagrangian formulation; action, variational principles, and equations of motion; Hamilton's principle; conserved quantities; rigid bodies and tops; Hamiltonian formulation and canonical equations; surfaces of section; chaos; canonical transformations and generating functions; Liouville's theorem and Poincaré integral invariants; Poincaré-Birkhoff and KAM theorems; invariant curves and cantori; nonlinear resonances; resonance overl We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration. We will consider the following topics: the Lagrangian formulation; action, variational principles, and equations of motion; Hamilton's principle; conserved quantities; rigid bodies and tops; Hamiltonian formulation and canonical equations; surfaces of section; chaos; canonical transformations and generating functions; Liouville's theorem and Poincaré integral invariants; Poincaré-Birkhoff and KAM theorems; invariant curves and cantori; nonlinear resonances; resonance overl

Subjects

classical mechanics | classical mechanics | computational classical mechanics | computational classical mechanics | structure and interpretation of classical mechanics | structure and interpretation of classical mechanics | phase space | phase space | lagrangian | lagrangian | action | action | variational principles | variational principles | equation of motion | equation of motion | hamilton principle | hamilton principle | rigid bodies | rigid bodies | Hamiltonian | Hamiltonian | canonical equations | canonical equations | surfaces of section | surfaces of section | canonical transformations | canonical transformations | liouville | liouville | Poincare | Poincare | birkhoff | birkhoff | kam theorem | kam theorem | invariant curves | invariant curves | resonance | resonance | chaos | chaos

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8.321 Quantum Theory I (MIT)

Description

8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement.

Subjects

eigenstates | uncertainty relation | observables | eigenvalues | probabilities of the results of measurement | transformation theory | equations of motion | constants of motion | Symmetry in quantum mechanics | representations of symmetry groups | Variational and perturbation approximations | Systems of identical particles and applications | Time-dependent perturbation theory | Scattering theory: phase shifts | Born approximation | The quantum theory of radiation | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | probability | measurement | motion equations | motion constants | symmetry groups | quantum mechanics | variational approximations | perturbation approximations | identical particles | time-dependent perturbation theory | scattering theory | phase shifts | quantum theory of radiation | second quantization | many-body theory | relativistic quantum mechanics | one electron | Hilbert spaces | time evolution | Schrodinger picture | Heisenberg picture | interaction picture | classical mechanics | path integrals | EM fields | electromagnetic fields | angular momentum | density operators | quantum measurement | quantum statistics | quantum dynamics

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8.322 Quantum Theory II (MIT)

Description

8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation.

Subjects

uncertainty relation | observables | eigenstates | eigenvalues | probabilities of the results of measurement | transformation theory | equations of motion | constants of motion | Symmetry in quantum mechanics | representations of symmetry groups | Variational and perturbation approximations | Systems of identical particles and applications | Time-dependent perturbation theory | Scattering theory: phase shifts | Born approximation | The quantum theory of radiation | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | probability | measurement | motion equations | motion constants | symmetry groups | quantum mechanics | variational approximations | perturbation approximations | identical particles | time-dependent perturbation theory | scattering theory | phase shifts | quantum theory of radiation | second quantization | many-body theory | relativistic quantum mechanics | one electron | quantization | EM radiation field | electromagnetic radiation field | adiabatic theorem | Berry?s phase | many-particle systems | Dirac equation | Hilbert spaces | time evolution | Schrodinger picture | Heisenberg picture | interaction picture | classical mechanics | path integrals | EM fields | electromagnetic fields | angular momentum | density operators | quantum measurement | quantum statistics | quantum dynamics

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20.310J Molecular, Cellular, and Tissue Biomechanics (MIT)

Description

This course develops and applies scaling laws and the methods of continuum and statistical mechanics to biomechanical phenomena over a range of length scales, from molecular to cellular to tissue or organ level.

Subjects

biomechanics | molecular mechanics | cell mechanics | Brownian motion | Reynolds numbers | mechanochemistry | Kramers' model | Bell model | viscoelasticity | poroelasticity | optical tweezers | extracellular matrix | collagen | proteoglycan | cell membrane | cell motility | mechanotransduction | cancer | biological systems | molecular biology | cell biology | cytoskeleton | cell | biophysics | cell migration | biomembrane | tissue mechanics | rheology | polymer | length scale | muscle mechanics | experimental methods

License

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8.321 Quantum Theory I (MIT)

Description

8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement.

Subjects

eigenstates | uncertainty relation | observables | eigenvalues | probabilities of the results of measurement | transformation theory | equations of motion | constants of motion | Symmetry in quantum mechanics | representations of symmetry groups | Variational and perturbation approximations | Systems of identical particles and applications | Time-dependent perturbation theory | Scattering theory: phase shifts | Born approximation | The quantum theory of radiation | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | probability | measurement | motion equations | motion constants | symmetry groups | quantum mechanics | variational approximations | perturbation approximations | identical particles | time-dependent perturbation theory | scattering theory | phase shifts | quantum theory of radiation | second quantization | many-body theory | relativistic quantum mechanics | one electron | Hilbert spaces | time evolution | Schrodinger picture | Heisenberg picture | interaction picture | classical mechanics | path integrals | EM fields | electromagnetic fields | angular momentum | density operators | quantum measurement | quantum statistics | quantum dynamics

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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8.322 Quantum Theory II (MIT)

Description

8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation.

Subjects

uncertainty relation | observables | eigenstates | eigenvalues | probabilities of the results of measurement | transformation theory | equations of motion | constants of motion | Symmetry in quantum mechanics | representations of symmetry groups | Variational and perturbation approximations | Systems of identical particles and applications | Time-dependent perturbation theory | Scattering theory: phase shifts | Born approximation | The quantum theory of radiation | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | probability | measurement | motion equations | motion constants | symmetry groups | quantum mechanics | variational approximations | perturbation approximations | identical particles | time-dependent perturbation theory | scattering theory | phase shifts | quantum theory of radiation | second quantization | many-body theory | relativistic quantum mechanics | one electron | quantization | EM radiation field | electromagnetic radiation field | adiabatic theorem | Berry?s phase | many-particle systems | Dirac equation | Hilbert spaces | time evolution | Schrodinger picture | Heisenberg picture | interaction picture | classical mechanics | path integrals | EM fields | electromagnetic fields | angular momentum | density operators | quantum measurement | quantum statistics | quantum dynamics

License

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5.74 Introductory Quantum Mechanics II (MIT) 5.74 Introductory Quantum Mechanics II (MIT)

Description

5.74 explores time-dependent quantum mechanics and spectroscopy. Topics covered include: perturbation theory, two-level systems, light-matter interactions, relaxation in quantum systems, correlation functions and linear response theory, and nonlinear spectroscopy. The instructor would like to acknowledge Anne Hudson for assisting in preparation of the 5.74 notes. 5.74 explores time-dependent quantum mechanics and spectroscopy. Topics covered include: perturbation theory, two-level systems, light-matter interactions, relaxation in quantum systems, correlation functions and linear response theory, and nonlinear spectroscopy. The instructor would like to acknowledge Anne Hudson for assisting in preparation of the 5.74 notes.

Subjects

introductory quantum mechanics | introductory quantum mechanics | time-dependent quantum mechanics | time-dependent quantum mechanics | spectroscopy | spectroscopy | perturbation theory | perturbation theory | two-level systems | two-level systems | light-matter interactions | light-matter interactions | correlation functions | correlation functions | linear response theory | linear response theory | nonlinear spectroscopy | nonlinear spectroscopy

License

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5.74 Introductory Quantum Mechanics II (MIT) 5.74 Introductory Quantum Mechanics II (MIT)

Description

This course covers topics in time-dependent quantum mechanics, spectroscopy, and relaxation, with an emphasis on descriptions applicable to condensed phase problems and a statistical description of ensembles. This course covers topics in time-dependent quantum mechanics, spectroscopy, and relaxation, with an emphasis on descriptions applicable to condensed phase problems and a statistical description of ensembles.

Subjects

introductory quantum mechanics | introductory quantum mechanics | time-dependent quantum mechanics | time-dependent quantum mechanics | spectroscopy | spectroscopy | perturbation theory | perturbation theory | two-level systems | two-level systems | light-matter interactions | light-matter interactions | correlation functions | correlation functions | linear response theory | linear response theory | nonlinear spectroscopy | nonlinear spectroscopy

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.642 Continuum Electromechanics (MIT) 6.642 Continuum Electromechanics (MIT)

Description

Includes audio/video content: AV faculty introductions. This course focuses on laws, approximations and relations of continuum electromechanics. Topics include mechanical and electromechanical transfer relations, statics and dynamics of electromechanical systems having a static equilibrium, electromechanical flows, and field coupling with thermal and molecular diffusion. Also covered are electrokinetics, streaming interactions, application to materials processing, magnetohydrodynamic and electrohydrodynamic pumps and generators, ferrohydrodynamics, physiochemical systems, heat transfer, continuum feedback control, electron beam devices, and plasma dynamics. Acknowledgements The instructor would like to thank Xuancheng Shao and Anyang Hou for transcribing into LaTeX the problem set solution Includes audio/video content: AV faculty introductions. This course focuses on laws, approximations and relations of continuum electromechanics. Topics include mechanical and electromechanical transfer relations, statics and dynamics of electromechanical systems having a static equilibrium, electromechanical flows, and field coupling with thermal and molecular diffusion. Also covered are electrokinetics, streaming interactions, application to materials processing, magnetohydrodynamic and electrohydrodynamic pumps and generators, ferrohydrodynamics, physiochemical systems, heat transfer, continuum feedback control, electron beam devices, and plasma dynamics. Acknowledgements The instructor would like to thank Xuancheng Shao and Anyang Hou for transcribing into LaTeX the problem set solution

Subjects

continuum mechanics | continuum mechanics | electromechanics | electromechanics | mechanical and electromechanical transfer relations | mechanical and electromechanical transfer relations | statics | statics | dynamics | dynamics | electromechanical systems | electromechanical systems | static equililbrium | static equililbrium | electromechanical flows | electromechanical flows | field coupling | field coupling | thermal and molecular diffusion | thermal and molecular diffusion | electrokinetics | electrokinetics | streaming interactions | streaming interactions | materials processing | materials processing | magnetohydrodynamic and electrohydrodynamic pumps and generators | magnetohydrodynamic and electrohydrodynamic pumps and generators | ferrohydrodynamics | ferrohydrodynamics | physiochemical systems | physiochemical systems | heat transfer | heat transfer | continuum feedback control | continuum feedback control | electron beam devices | electron beam devices | plasma dynamics | plasma dynamics

License

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3.032 Mechanical Behavior of Materials (MIT) 3.032 Mechanical Behavior of Materials (MIT)

Description

Here we will learn about the mechanical behavior of structures and materials, from the continuum description of properties to the atomistic and molecular mechanisms that confer those properties to all materials. We will cover elastic and plastic deformation, creep, and fracture of materials including crystalline and amorphous metals, ceramics, and (bio)polymers, and will focus on the design and processing of materials from the atomic to the macroscale to achieve desired mechanical behavior. Integrated laboratories provide the opportunity to explore these concepts through hands-on experiments including instrumentation of pressure vessels, visualization of atomistic deformation in bubble rafts, nanoindentation, and uniaxial mechanical testing, as well as writing assignments to communicate th Here we will learn about the mechanical behavior of structures and materials, from the continuum description of properties to the atomistic and molecular mechanisms that confer those properties to all materials. We will cover elastic and plastic deformation, creep, and fracture of materials including crystalline and amorphous metals, ceramics, and (bio)polymers, and will focus on the design and processing of materials from the atomic to the macroscale to achieve desired mechanical behavior. Integrated laboratories provide the opportunity to explore these concepts through hands-on experiments including instrumentation of pressure vessels, visualization of atomistic deformation in bubble rafts, nanoindentation, and uniaxial mechanical testing, as well as writing assignments to communicate th

Subjects

Basic concepts of solid mechanics and mechanical behavior of materials | Basic concepts of solid mechanics and mechanical behavior of materials | stress-strain relationships | stress-strain relationships | stress transformation | stress transformation | elasticity | elasticity | plasticity and fracture. Case studies include materials selection for bicycle frames | plasticity and fracture. Case studies include materials selection for bicycle frames | stress shielding in biomedical implants; residual stresses in thin films; and ancient materials. Lab experiments and demonstrations give hands-on experience of the physical concepts at a variety of length scales. Use of facilities for measuring mechanical properties including standard mechanical tests | stress shielding in biomedical implants; residual stresses in thin films; and ancient materials. Lab experiments and demonstrations give hands-on experience of the physical concepts at a variety of length scales. Use of facilities for measuring mechanical properties including standard mechanical tests | bubble raft models | bubble raft models | atomic force microscopy and nanoindentation. | atomic force microscopy and nanoindentation. | plasticity and fracture | plasticity and fracture | Case studies | Case studies | materials selection | materials selection | bicycle frames | bicycle frames | stress shielding in biomedical implants | stress shielding in biomedical implants | residual stresses in thin films | residual stresses in thin films | ancient materials | ancient materials | standard mechanical tests | standard mechanical tests | solid mechanics | solid mechanics | mechanical behavior of materials | mechanical behavior of materials

License

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3.024 Electronic, Optical and Magnetic Properties of Materials (MIT) 3.024 Electronic, Optical and Magnetic Properties of Materials (MIT)

Description

This course describes how electronic, optical and magnetic properties of materials originate from their electronic and molecular structure and how these properties can be designed for particular applications. It offers experimental exploration of the electronic, optical and magnetic properties of materials through hands-on experimentation and practical materials examples. This course describes how electronic, optical and magnetic properties of materials originate from their electronic and molecular structure and how these properties can be designed for particular applications. It offers experimental exploration of the electronic, optical and magnetic properties of materials through hands-on experimentation and practical materials examples.

Subjects

electronic properites | electronic properites | optical properties | optical properties | magnetic properties | magnetic properties | materials | materials | Hamilton approach | Hamilton approach | Schrödinger’s Equation | Schrödinger’s Equation | mechanics | mechanics | quantum mechanics | quantum mechanics | spectral decomposition | spectral decomposition | symmetries | symmetries | angular momentum | angular momentum | periodic potentials | periodic potentials | band diagrams | band diagrams | Fermi | Fermi | Fermi-Dirac | Fermi-Dirac | p-n junction | p-n junction | light emitting diodes | light emitting diodes | wave optics | wave optics | electromagnetic waves | electromagnetic waves | magnetization | magnetization | semiconductor devices | semiconductor devices | Maxwell's equations | Maxwell's equations | photonic bands | photonic bands

License

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8.05 Quantum Physics II (MIT) 8.05 Quantum Physics II (MIT)

Description

This course, along with the next course in this sequence (8.06, Quantum Physics III) in a two-course sequence covering quantum physics with applications drawn from modern physics. General formalism of quantum mechanics: states, operators, Dirac notation, representations, measurement theory. Harmonic oscillator: operator algebra, states. Quantum mechanics in three-dimensions: central potentials and the radial equation, bound and scattering states, qualitative analysis of wavefunctions. Angular momentum: operators, commutator algebra, eigenvalues and eigenstates, spherical harmonics. Spin: Stern-Gerlach devices and measurements, nuclear magnetic resonance, spin and statistics. Addition of angular momentum: Clebsch-Gordan series and coefficients, spin systems, and allotropic forms of hydrogen This course, along with the next course in this sequence (8.06, Quantum Physics III) in a two-course sequence covering quantum physics with applications drawn from modern physics. General formalism of quantum mechanics: states, operators, Dirac notation, representations, measurement theory. Harmonic oscillator: operator algebra, states. Quantum mechanics in three-dimensions: central potentials and the radial equation, bound and scattering states, qualitative analysis of wavefunctions. Angular momentum: operators, commutator algebra, eigenvalues and eigenstates, spherical harmonics. Spin: Stern-Gerlach devices and measurements, nuclear magnetic resonance, spin and statistics. Addition of angular momentum: Clebsch-Gordan series and coefficients, spin systems, and allotropic forms of hydrogen

Subjects

General formalism of quantum mechanics: states | General formalism of quantum mechanics: states | operators | operators | Dirac notation | Dirac notation | representations | representations | measurement theory | measurement theory | Harmonic oscillator: operator algebra | Harmonic oscillator: operator algebra | states | states | Quantum mechanics in three-dimensions: central potentials and the radial equation | Quantum mechanics in three-dimensions: central potentials and the radial equation | bound and scattering states | bound and scattering states | qualitative analysis of wavefunctions | qualitative analysis of wavefunctions | Angular momentum: operators | Angular momentum: operators | commutator algebra | commutator algebra | eigenvalues and eigenstates | eigenvalues and eigenstates | spherical harmonics | spherical harmonics | Spin: Stern-Gerlach devices and measurements | Spin: Stern-Gerlach devices and measurements | nuclear magnetic resonance | nuclear magnetic resonance | spin and statistics | spin and statistics | Addition of angular momentum: Clebsch-Gordan series and coefficients | Addition of angular momentum: Clebsch-Gordan series and coefficients | spin systems | spin systems | allotropic forms of hydrogen | allotropic forms of hydrogen | Angular momentum | Angular momentum | Harmonic oscillator | Harmonic oscillator | operator algebra | operator algebra | Spin | Spin | Stern-Gerlach devices and measurements | Stern-Gerlach devices and measurements | central potentials and the radial equation | central potentials and the radial equation | Clebsch-Gordan series and coefficients | Clebsch-Gordan series and coefficients | quantum physics | quantum physics

License

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