Quantum Mechanics for Materials Scientists

Abstract

Analysis and motivation for the necessity of a theory beyond classical mechanics to describe materials properties. The principles, terminology and concepts of quantum mechanics will be introduced and mathematically represented on the basis of simple problems.

Objective

Give reasons for the necessity of quantum mechanical description of matter and explain experimental observations leading to this description. Clarification of the term quantum object. Formulate and solve the Schrödinger equation for simple problems. Application of the operator formalism for the calculation of observables and the interpretation of physical processes. Interpretation of the wavefunction. Explain the solution of the hydrogen atom. Derivation of the approach to the solution in the application of symmetreis and angular momentum operators. Give reasons for the electron spin and calculate magnetic moments.

Content

Crisis of classical physics Planck's law of radiation (cavity radiation), photoelectric effect (Einstein's light quantum hypothesis), Bohr quantisation of the atom, De Broglie hypothesis Wave-particle dualism - wave mechanics, matter waves, double-slit experiment, comparison of classical mechanics and quantum mechanics Introduction of the wave function, de-Brogie relation, probability Postulates of quantum mechanics Introduction of the Schrödinger equation, normalisation of the wave function, stationary Schrödinger equation, location and momentum space, location representation of the momentum operator Wave packets (Gaussian bell curve), decay of wave packets, indeterminacy principle Wave mechanics with forces Piecewise constant potentials, particles in the potential well, potential step, probability current density, potential wall, tunnel effect, potential well Formalism of quantum mechanics Hilbert space, scalar product, vectors (basis), states, normalizability, completeness, eigenfunctions, notations, operators - general definitions and properties, Expectation values, spectrum (discrete, continuous), matrix representation, Ehrenfest theorem, measurement process and collapse of the wave function Central potential Eigenvalue problem in spherical coordinates, limiting cases, particles in a 3D pot, symmetries, rotation and angular momentum, angular momentum operator and spherical surface functions Hydrogen atom Coulomb potential, radial wave function, orbitals, atomic structure Charged particle in electric and magnetic field, magnetic moment, Stern-Gerlach experiment, spin, vector-valued wave function, free electron in magnetic field, spin resonance

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