# Humboldt-Universität zu Berlin - Faculty of Mathematics and Natural Sciences - Department of Chemistry

Introduction to Quantum Theory

## Introduction to Quantum Theory

Classical mechanics: total energy, phase space. Harmonic oscillator, double-minimum potential. Movement of a particle, of an ensemble. Evolving distributions, diffusion as operator, stationary distribution. Statistics: distribution of an observable, normalization, expectation value. Representation of distributions: diffusion on a ring, Fourier-expansion of a spatial distribution, basis functions, expansion coefficients. Action of operators on basis functions. Quantisation: axioms of quantum mechanics. 1d particle in a box: classical H, quantum-mechanical H, boundary conditions, eigenfunctions and -energies. Harmonic oscillator: potential energy, zero-point amplitude and dimensionless coordinate. Classical H(p,q) and q.m. Hamiltonian. Eigenfunctions and -energies. Expectation values for q, q2, p, p2. Quantisation as h in phase space. Rigid 1d rotor: classical H(p,q) in carthesian coordinates, trafo to polar coordinates, moment of inertia. Operator for kinetic energy in polar coordinates. Ansatz and solutions. Particle on a sphere: classical H, coordinate trafo, corresponding trafo of Laplace operator, separation ansatz for spherical harmonics Y(q,j), solutions as Legendre-polynomials (cos(q)) or exp(imj). Eigenfunctions for L2 or kinetic energy and simultaneously for Lz. Electron in Coulomb potential: classical H, quantisation, trafo to polar coordinates, separation ansatz R(r) Y(q,j) and Radial Schrödinger Equation for P(r) = r R(r). Effective potential from Coulomb and centrifugal terms. Structure of solutions rL Laguerepolynom(r) Exp(-r/2). Atomic Orbitals and their qualitative discussion.

P.W. Atkins: "Physical Chemistry", Oxford University Press, chapters 0, 13, 14;
P. W. Atkins: "Molecular Quantum Mechanics", Oxford University Press, 2nd ed., 1994;
J. Reinhold: "Quantentheorie der Moleküle", chapters 1,2.