Why quantum mechanics?Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Slide 68Slide 69Slide 70Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Slide 78Slide 79Slide 80Slide 81Slide 82Slide 83Slide 84Slide 85Slide 86Slide 87Slide 88Slide 89Slide 90Slide 91Slide 92Slide 93Slide 94Slide 95Slide 96Slide 97Slide 98Slide 99Slide 100Slide 101Slide 102Slide 103Slide 104Slide 105Slide 106Slide 107Slide 108Slide 109Slide 110Slide 111Slide 112Slide 113Slide 114Slide 115Slide 116Slide 117Slide 118Slide 119Slide 120Slide 121Slide 122Slide 123Slide 124Slide 125Slide 126Slide 127Slide 128Slide 129Slide 130Slide 131Slide 132Slide 133Slide 134Slide 135Slide 136Slide 137Slide 138Slide 139Slide 140Slide 141Slide 142Slide 143Slide 144Slide 145Slide 146Slide 147Slide 148Slide 149Slide 150Slide 151Slide 152Slide 153Slide 154Slide 155Slide 156Slide 157Slide 158Slide 159Slide 160Slide 161Slide 162Slide 163Slide 164Slide 165Slide 166Slide 167Slide 168Slide 169Slide 170Slide 171Slide 172Slide 173Slide 174Slide 175Slide 176Slide 177Slide 178Slide 179Slide 180Slide 181Slide 182Slide 183Slide 184Slide 185Slide 186Slide 187Slide 188Slide 189Slide 190Slide 191Slide 192Slide 193Slide 194Slide 195Slide 196Slide 197Slide 198Slide 199Slide 200Slide 201Slide 202Slide 203Slide 204Slide 205Slide 206Slide 207Slide 208Slide 209Slide 210Slide 211Slide 212Slide 213Slide 214Slide 215Slide 216Slide 217Slide 218Slide 219Slide 220Slide 221Slide 222Slide 223Slide 224Slide 225Slide 226Slide 227Slide 228Slide 229Slide 230Slide 231Slide 232Slide 233Slide 234Slide 235Slide 236Determination of EigenenergiesSlide 238Slide 239Slide 240Slide 241Slide 242Slide 243Eigenenergies for 3D StructureDetermination of the EigenfunctionsFirst 4 EigenfunctionsNext 4 EigenfunctionsSlide 248Time Evolution (2162 meV)Time Evolution (cont’d)Slide 251Slide 252Slide 253Slide 254Slide 255Slide 2561 Quantum Mechanics forElectrical EngineersDennis M. Sullivan, Ph.D.Department of Electrical and Computer EngineeringUniversity of Idaho2Why quantum mechanics?At the beginning of the 20th century, various phenomena were being observed that could not be explained by classical mechanics.1. Energy is quantized2. Particles have a wave nature3Source of electrons4Source of electrons5Particles have wave properties.Conclusion:6Incident lightMaterialPhotoelectronsPhotoelectric EffectThe velocity of the escaping particles was dependent on the wavelength of the light, not the intensity as expected.70fFrequency fKineticEnergy TPhotoelectric EffectPlanck postulated in 1900 that thermal radiation is emitted from a heated surface in discrete packets called quanta.8Einstein postulated that the energy of each photonwas related to the wave frequency.E hfw= =h34156.625 104.135 10h J seV s--= � -= � -h is Plank’s constantThis is the first major result: energy is related to frequency9Second major result: Momentum is related to wavelengthhpl=In 1924, Louis deBroglie postulated the existence of matter waves. This lead to the famous wave-particle duality principle. Specifically thatthe momentum of a photon is given byOr the more familiar formp k=h10Everything is at the same time a particle and a wave.Bottom line:hpl=E hf=11Solve using only energy. Problem: Determine the velocity of the ball at the bottom of the slope.1 meter1 kgPhysicists formulate everything as an energy problem12While the ball is on the top of the hill, it has potential energy.1 meter1 kgSince the acceleration of gravity is29.8 /g m s=the potential energy is( ) ( )29.8 1 1 9.8PE g mass of the ball heightmkg m Js= � �� �= =� �� �13When it gets to the bottom of the hill, it no longer has potential energy, but it has kinetic energy. Since therehave been no other external forces, it must be the same1 meter219.82vT JM= =22219.8212 9.8 4.31T J Mvkg m mvs kg s= =� ��= � =� �� �14There are several methods of advanced mechanics that change everything into energy.1. Lagrangian mechanics2. Hamiltonian mechanics15Erwin Schrödinger was taking this approach and developed the following equation to incorporate these new ideas:This equation is 2nd order in time and 4th order is space.22 222 212Vt my y� ��=- � -� ��� �hh16Schrödinger realized that this was a completely intractable problem. (There were no computers in 1921.) However, he saw that by considering  to be a complex function, he could factor the above equation into two simpler equations, one of which is This is the Schrödinger equation.222i Vt my y y�=- � +�hh17The parameter in the Schrödinger equation, , is a state variable. It is not directly associated with any physical quantity itself, but all the information can be extracted from it. Also, remember that the Schrödinger equation was only half of the “real” equation from which it was derived. So to determine if a particle is located between a and b, calculate( ) ( )*( )baP a x b x x dxy y< < =�18( ) ( )*1dy y�- �=�r r rThis brings us to one of the basics requirementsof the state variable: it must be normalizedNote: The amplitude of the state verctor  does notrepresent the strength of the wave in the usual sense.It is chosen to achieve normalization19Computer simulations can show how the Schrödinger equation can model a particle like an electron propagating as a wave packet.The real part is blue, the imaginary part is red.202122232425In quantum mechanics, physical properties are related to operators, call observables.Two of the fundamental operators are:1. Momentum2. Kinetic energy26The momentum operator (in one dimension) is.pi x�=�hThis seems pretty strange until I rememberthat momentum is related to wavelength:.hp kl= =h27We think of a wavepacket as a superposition of plane waves( )pi x ti kx te eww� �-� �-� �=hWhen I apply the momentum operatorp E p Ei x t i x te pex� � � �- -� � � �� � � ��- =�h h h hh28Kinetic energy can be derived from momentum:222 22. .212 2pK Emm i x m x=� - �� �= =� ��