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Quantum Mechanics • Bohr Theory worked pretty well for one- electron atoms (H, He+, Li++, etc. • Failed totally for He with two electrons • Close examination of spectra of one electron atoms showed that some spectral lines were actually multiple lines • Couldn’t predict intensity of lines • Couldn’t explain chemical bonding Quantum Mechanics • A new approach was needed and appeared in two different forms • The first was called wave mechanics and was developed by Erwin Schrodinger • The second was called matrix mechanics and was developed by Werner Heisenberg • Eventually shown to be the same Quantum Mechanics • Deals with the microscopic world of atoms and light (photons) • Blends smoothly with classical mechanics as we approach the macroscopic world • This is the correspondence principle • Mathematics of quantum mechanics involve matrices and partial differential equations • We’ll just have to look at results Wave Mechanics • EM Waves have frequency, wavelength, energy • Relate wave to particle properties by using E = hf • Amplitude of an EM wave is the strength of the electric or magnetic field and is related to the intensity of the wave Wave Mechanics • Particles have wave properties as well • Wavelength is h/mv • We say this is a matter wave • What is the amplitude of a matter wave? • Schrodinger defined a wave function • is a function of position and time Wave Mechanics • can be compared to the electric field • For light the intensity is proportional to the square of the electric field strength • Light can be considered a stream of particles • Then the intensity depends on the number of photons Wave Mechanics So, to say how much light we have, we talk about the intensity. We can either talk about the square of the field 2 I E N strength or the number of photons. The two quantities are proportional to each other. 2 N E We said that is comparable to E, so 2 must be related to numbers of particles or something similar. Wave Mechanics • The great leap is that we might consider the probability of finding photons somewhere in space to be proportional to E2 • Now, we make the same leap for particles • Consider the probability of finding particles somewhere in space to be proportional to 2 Wave Mechanics Direct photons or particles at a pair of slits. E2 gives the probability of finding a photon at the viewing screen. 2 gives the probability of finding a particle at the viewing screen. Now send photons or particles one at a time. What happens? Place a piece of film at screen. Wave Mechanics We will see a dot on the film when a photon or particle hits the screen. If we keep sending photons or particles and keep watching, eventually the interference patterns appear. We can’t predict what any single particle or photon will do, but we can predict what a lot of them will do! Wave Mechanics If we cover one slit for a while and then the other for a while, no interference pattern is seen! So, a single photon or particle must somehow pass through both slits in order to interfere! Says our macroscopic view of waves and particles cannot be extended to the microscopic! Wave Mechanics What we do know is that E2 and 2 give us the probability of finding the photon or particle at a point in space and time! We can treat a wad of particles as a wave, but we treat individual particles by probabilities! Uncertainty Principle • We assume that if we measure something we will have some small errors • You know this from your lab work • With better instruments and techniques you can reduce these errors • Heisenberg showed that there is a limit to how small you can make the error! Uncertainty Principle • There are two factors involved • One is wave-particle duality • The other is that to measure something you have to disturb it • Place a ruler to measure length • You must use pressure to line up the end of the ruler and the end of the object • You have to apply pressure to make the alightment • This changes the length of the object! Uncertainty Principle • Example in the text about finding a ping- pong ball in a completely darkened room • You grope around, moving your hand • You touch the ball during the movement • You know where the ball was, but you don’t know where it is after the touch • You can’t predict the exact future position • You gave the ball some momentum! Uncertainty Principle Recall the diffraction limit of light. We can measure position to about a wavelength of the light we use. To get more accurate position, shorten the wavelength which ups the frequency. But E=hf, so the photon has higher energy. Whacks the electron harder and you don’t know where it goes or how fast it is moving. Lower the frequency and you get more uncertainty in position. Uncertainty Principle x p h / xp h xp h /2 Uncertainty Principle Particles have uncertainty in position of x=. We try to detect with a photon that has speed c and takes a time t= x/c= /c to pass through the uncertainty distance. So the time of measurement is uncertain by t= /c. Now the photon can transfer some or all of its energy to the particle. The energy of the photon is E=hf=hc/. So, the uncertainty in the particle’s energy after the photon hits it is just this same amount. So the product of the two undertainties in time and energy is Et= (hc/)(/c)=h. Heisenberg’s exact calculation gives Et h/2 Uncertainty Principle • So what does all this mean? • When dealing with the microscopic world we cannot simply take our macroscopic picture of particles and waves to a smaller level • We can’t describe in words what a photon or an electron is • These objects have both wave and particle properties and we have to consider both to gain understanding of microscopic phenomena Uncertainty Principle • These issues force us to deal with probabilities rather than certainties when we discuss energies, times, positions, momenta • Can we violate energy of momentum conservation? • Sure, if we do it for a short enough time interval or a short enough position space since we can’t measure the failure!!! • Is this real? YES, as we shall see! Atoms • So what are atoms? • Well, Rutherford’s experiments established the notion of a small very dense nucleus • We know atoms contain electrons • But the classical orbit picture is garbage! • We only know what we can measure!!! Atoms • What we can measure is that atoms have definite precise energies • When electrons change energy levels, photons with precise energies emerge • The electrons have highest probability of being in some well defined region of space with fuzzy edges • These probability regions are not necessarily spherical as the classical orbits might imply Atoms • Atoms have other precise characteristics besides energies • We will take up these ideas next time when we examine the notion of quantum numbers • For now, we can simply say that just as a guitar string can generate musical tones as a series of harmonics, so the electrons in atoms can take on a precise series of characteristics • These properties are inherent in nature

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posted: | 10/8/2012 |

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