Chapter 9 Microscopic systems and quantization Atomic structure the arrangement of electrons in atoms Principles of quantum theory Classical physics is based on three assumptions o A particle travels in a trajectory a path with a precise position and momentum at each instant o Any type of motion can be excited to a state of arbitrary energy o Waves and particles are distinct concepts 9 1 The emergence of the quantum theory Two conclusions of quantum theory o Energy can be transferred between systems only in discrete amounts o Light and particles have properties in common Electromagnetic radiation behaves like a stream of particles Electrons behave like waves a Atomic and molecular spectra Spectrum radiation that are absorbed or emitted by an atom or molecule a display of the frequencies or wavelengths of electromagnetic o Radiation is absorbed or emitted at a series of discrete frequencies The energy of the atoms or molecules is confined to discrete values Energy can be discarded or absorbed only in packets as the atom or molecule jumps between its allowed states The frequency of radiation is related to the energy difference between the initial and final states Bohr frequency condition energy between two states of an atom or molecule certain energies Quantization of energy Quantized the limitation of energies to discrete values the internal modes of atoms and molecules can possess only relates frequency of radiation to the difference in b Wave particle duality particles of electromagnetic radiation Photons Photoelectric effect exposed to ultraviolet radiation the ejection of electrons from metals when they are o No electrons are ejected regardless of the intensity of the radiation unless the frequency exceeds a threshold value characteristic of the metal Work function the energy required to remove the electron from the metal o The energy of the photon equals the sum of the kinetic energy and the Diffraction the interference between waves caused by an object in their pat o Results in a series of bright and dark fringes where the waves are work function detected Wave particle duality the joint wave particle character of matter and radiation 9 2 The Schrodinger equation A particle is spread through space like a wave There are regions where the particle is more likely to be found than others Wavefunction a concept in place of the trajectory to describe the distribution of a particle in space an equation for a single particle of mass m moving a The formulation of the equation Schrodinger equation with energy E in one dimension than just multiplying Hamiltonian an operator something that acts in a particular way rather The Schrodinger equation is a second order differential equation Three simple cases for which solution of the Schrodinger equation is important o The wavefunction for a freely moving particle is sin x o The wavefunction for the lowest energy state of a particle free to oscillate to and fro near a point is e x2 where x is the displacement from the point o The wavefunction for an electron in the lowest energy state of a hydrogen atom is e r where r is the distance from the nucleus An infinite number of possible solutions for the Schrodinger equation are allowed mathematically o Only some are acceptable physically Boundary conditions a set of constraints that a solution must satisfy o Energy is quantized b The interpretation of the wavefunction Born interpretation space of volume V is proportional to 2 V where is the value of the wavefunction in the region the probability of finding a particle in a small region of 2 is a probability density When 2 is large there is a high probability of finding the particle and vice versa Probabilistic interpretation predictions only about the probability of finding a particle somewhere an interpretation that accepts that we can make 9 3 The uncertainty principle Uncertainty principle it is impossible to specify simultaneously with arbitrary precision both the momentum and the position of a particle If a particle is at a definite position then its wavefunction must be nonzero there and zero everywhere else adding together the amplitudes of a large number of sine The uncertainty principle applies to location and momentum along the same Superposition functions axis Complementary not simultaneously specifiable Applications of quantum theory Three basic types of motion o Translation motion in a straight line o Rotation o Vibration 9 4 Translation a Motion in one dimension Normalization constant the walls finding the particle anywhere is 1 The only energy is kinetic energy Quantum number The potential energy of a particle is zero inside the box but rises to infinity at a constant that ensures that the total probability of an integer that labels the state of the system o Specifies certain physical properties of the system Nodes where the function passes through zero o Points at edges of the box are not nodes The number of nodes in a wavefunction is n 1 for a particle in a box Zero point energy Separation of energies decreases as the length of the box increases the lowest irremovable energy o Also decreases as the mass of the particle increases b Tunneling If the walls are thin and the particle is very light the wavefunction oscillates inside the box varies smoothly inside the region representing the wall and oscillates again on the other side of the wall outside the box leakage by penetration through classically forbidden zones Tunneling o A consequence of the wave character of matter the probability of tunneling Transmission probability Particles of low mass are more able to tunnel through barriers than heavy ones c Motion in two dimensions Separation of variables procedure product of wavefunctions for each direction different states with the same energy Degenerate the wavefunction can be expressed as a 9 5 Rotation 9 6 Vibration Harmonic oscillator Hooke s law of force Force constant Hooke s law when a particle is restrained by a spring that obeys the restoring force is proportional to the displacement the constant of proportionality o A stiff spring has a high force constant o A weak spring has a low force constant The potential energy of a harmonic oscillator has a shape of a parabola The wavefunctions must all go to zero for large displacements from x 0 but do not have to go all the way to zero at the parabola Hydrogenic atoms Hydrogenic atom Many electron atom a one electron atom or ion of general atomic number Z an atom or ion that has