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Module CH2701: Physical Chemistry II Aims Quantum Mechanics I The aim of this course is to provide an introduction to quantum mechanics in general and the quantum mechanics of atoms. Georg Haehner Office 243, Purdie Literature: e.g. Atkins’ Physical Chemistry, OUP; ‘Quantum Mechanics for Chemists’, David O Hayward, RSC Publishing accessible online at http://pubs.rsc.org/en/Content/eBook/978-0-85404-607-2#!divbookcontent Objectives 1. To understand the concept of quantization and how it arises. 2. To know the form of the time-independent Schrödinger equation and the meaning of the quantities therein. 3. To follow the treatment of 'free electron' systems and understand how energy levels, quantum numbers, degeneracy and the effects of symmetry result from this. Objectives 4. To follow in broad outline the solution of the Schrödinger equation for the hydrogen atom and to know the general form of the energy levels and atomic orbitals and how the shapes of the latter arise. 5. To appreciate, in principle, how the electronic structure of 'many-electron‘ atoms can be constructed and understood. Quantum mechanics animations can be found at www.st-andrews.ac.uk/~qmanim/ and specifically for chemistry students at http://www.st-andrews.ac.uk/~qmanim/chemistry/ 1 1.2 Classical Mechanics 1.1 Why quantum mechanics? m r p According to classical mechanics an object and its path in space are characterized by its position in space: r and momentum: p mv (m: mass, v velocity ) Classical mechanics 1) predicts a precise trajectory for particles with precisely specified positions and momenta at each instant 2) allows the translational, rotational and vibrational modes of motion to be excited to any energy, simply by controlling the forces that are applied. The velocity , v, of a particle is the rate of change of its position: dr v dt The magnitude of the velocity is the speed, v. The linear momentum, p, of a particle of mass m is p mv An object and its path in space are characterized by: position in space r and momentum p m p r Conclusions are in agreement with everyday experience! 1.2.1 Linear Momentum 1.2.2 Newton‘s second law The linear momentum of a Newton’s second law of motion states that the rate of change of momentum is equal to the force acting on the particle: particle is a vector property and points in the direction of motion. The linear momentum of an object with mass m is defined as p mv v d r r dt Worked example: linear momentum of a running car dp F dt F dp d d d r m vm dt dt dt dt 2 m d 2r m r dt (if the mass does not change) The second derivative of the position with respect to time is the accelaration of the particle Worked examples: Free fall and accelerating car 2 1.3 The path of a particle 1.2.3 Potential (energy) and Force The force acting on a particle is determined by the slope of the potential energy at each point. The force points in the direction of lower potential energy. The force, F, experienced by a particle free to move in any direction is related to its potential energy, V, by F r V V ( , , ) x y z Worked example: climbing a slope 1.4 Energy E Fd r The path of a particle (its trajectory) can be determined from knowing its initial position and momentum and by integrating the two equations: dr p dt m dp V dt r m The particle will have a definite position and momentum (trajectory) at each instant! p r 1.4.1 Potential Energy and Total energy (Force opposing movement) Potential energy (depends often on the location of the object) r V (r ) F d r Kinetic energy of an object: Fd r Classical Mechanics describes the behavior of objects in terms of two equations. r0 dp d p dr dr dt dt dt dt F V V r The total energy is: Etotal Ekin E pot Etotal 1 2 mv V (r ) 2 Etotal p2 V (r ) 2m 2 p 1 p 1 2 dp mv m 2 m 2 Worked examples: energy required to jack up a vehicle (car-jack) and energy of a driving car Worked example: free fall 3