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## Quantum Mechanics

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**Quantum Mechanics**The Realm Of The Bizarre, Freaky and Just Plain Weird OR Physics Walks the Planck**When It Comes To Understanding Matter, Remember There Is No**Spoon.**Some Famous Quotes**• I think I can safely say that nobody understands quantum mechanics. (Feynman) • I don’t like it and I wish I had never had anything to do with it. (Heisenberg) • Anyone who is not shocked by quantum theory has not understood it. (Bohr)**You Must Understand This**• Quantum mechanics, the theory that explains phenomena in the realm of the very small like atoms, is right. • Even so, most physicists find it so conceptually weird that they just are not comfortable with it.**It All Begins With A Little Problem**• Heat an object it an it glows. • Make it really hot and it glows orange, or yellow or even white. • In the late 1800’s scientists are heating all manner of things to measure there emission spectra. • And this guy, Max Planck (1901) finds a wee problem with the data.**Max And His Hot Body**• Our boy Max wants to examine not just any hot hunk of metal like iron, he wants to use platinum heated in an insulating cavity so that the light emitted comes out of a small hole. • Platinum is a “noble” metal so using it eliminates impurities and cleans up his data.**Max And His Hot Body**• Because it absorbs almost all light shined on it, it’s called a black body. • A perfect black body will absorb 100% of all radiation hitting it, thus it appears black. • There are no perfect black bodies, but we can get close enough for government work. • Go ahead and giggle now.**Max Has A Problem**• His data doesn’t agree with classical (meaning Newtonian) theory. • The issue is one of resonance. • Pluck one guitar string and others will vibrate slightly since they are similar in frequency. • You can try this by opening a piano lid, shouting loudly, and you’ll hear many strings humming. Each at a frequency associated with your complex voice induced sound wave.**A Little About Waves**This wave is a standing wave. It is captured by the cavity its in.**A Little More About Waves**• Heat something and it glows. Classical theory says it will emit a little light at all frequencies. • The problem is energy. • E αν • As frequency goes up so does energy. • As one wave induces another and another, more energy is released. • Also, each higher tone is a multiple of 1/n of the one before. Resonance occurs in discrete integer chunks . • This is HUGE!!!!!**The Ultraviolet Catastrophe**• Classical Prediction • All heated objects should • incinerate everything. • Clearly this does not happen, • unless you’re Joan of Arc.**Wanted: Quantum Mechanic**• Our boy Max (in 1900) applies a simple idea to the curve, integration by parts. • Maybe each part of the curve can be thought of as a discrete function unto its self using 1/n. • Like finding the area of a complex shape using a piece of graph paper. • More importantly, like a piano string maybe the energy comes in discrete integer based packets of 1/n. • This is also HUGE!!!!!**Max Planck Saves World, Film at 11.**• Each red bar represents a discrete frequency or wavelength of light. The energy defined by the area of each bar is related to the frequency.**Wanted: Algebra Student.**• Our boy Max applies the “Guitar Theory” E αν where ν (nu) is frequency f To get the units to work out Max follows a basic premise of proportionality – create a constant. E = nhν where h is now known as Planck’s Constan and n is an integer that describes the “string.” The implication of n is that strings can’t vibrate in half measures.**The Quantum Principle Is Born**• In nature certain quantities occur in discrete packets at discrete intervals. The distance between intervals is defined by h, Planck’s Constant.**Supermarket Door Openers?**• It had been known for a long time that if you shine light on certain metals they will emit electrons. • This is called the photoelectric effect first described by Heinrich Hertz and AleksandrStoletov in 1890. • Nobody knew why it worked. • It was thought to be the reverse of an incandescent bulb where electricity heats a metal filament which then emits light in a broad spectrum like a black body. • Except for yet another problem.**Photoelectric Effect**• Light strikes a metal such as zinc, selenium or silver, and electrons are ejected from it. • To determine the energy of the electrons a voltage or potential is placed across to plates which can be changed. • Electrons that make it across the gap have same potential V as the E field across the gap.**Photoelectric Effect**• Here red light won’t create electrons with enough energy to jump the gap.**Photoelectric Effect**• Here blue light, which has higher energy, does kick out electrons with enough energy to jump the gap.**Except For Another Small Problem**• When the voltage was set to ZERO, meaning no E field, there was a threshold below which light didn’t have enough umph to kick out electrons. • This violated the classical view. • Newton loses again.**Observational Summary**• Ordinary white light didn’t create a current no matter how bright it was. • Only above a certain frequency, say the UV, are electrons ejected. • Above this threshold the higher the frequency the more energetic the electrons. • Increase the brightness (intensity) and you made more electrons, not more energetic electrons. • There is no time delay between absorption and emission (This is Huge!!).**Classical Physics Dies, Long Live Physics.**• None of the observations made sense to classical physics. • Conservation of energy and momentum seem to be violated since intensity means more energy so you should make more energetic electrons. • Swing a hammer at a bowling ball and it moves. • Swing harder and it moves faster, this makes sense. • The observations said, “swing harder get 2 bowling balls, real hard and get 3.” • None of it made sense!!!!!!!!!!!**Big Al The Rescue**• All it took was Einstein to interpret the work of Planck and Hertz in one of his 1905 papers. • The slope of the graph is h, so nh is no longer used. • The units place holder constant h turns out to be real. How cools is that. Threshold energy**It Suddenly Makes Sense**• E = hν for black body emission • Eelectron = hν – Ethreshold • The threshold energy is also called the stopping voltage and since it takes energy to stop an electron, and we know that energy is work, we call Et the WORK FUNCTION Φ (phi) • Eelectron = hν – Φ**Einstein’s Nobel Winning 1905 Idea**• To explain the 5 observations that didn’t make sense… • Light comes in discrete quantized packets. • We now call them photons (remember QED?).**Einstein Performs Mathematical Miracle, Makes Massless Thing**Have Momentum. • To fit the emerging idea, Einstein invented photons to pass momentum around. But to meet the idea of relativity photons had to be massless, otherwise they can’t move at c. • c = λν • E = hν • And then a miracle happens • p = mv = hν/c = h/λ ……… E = pc • Which is a little ditty called E = mc2 (it’s still 1905)**The Hydrogen Emission Spectrum**• In 1913 Neils Bohr is trying to explain why the hydrogen emission spectrum has discrete lines. • Emission and absorption spectra lines had been known for many years. • In 1893 a Swiss school teacher, Johan Jacob Balmer, explained the pattern in a code like algorithm. • Building on Planck & Einstein, along with his Quantum Principle, Bohr speculates that electrons inhabit discrete energy shells.**Today’s Weather - Balmer**• Balmer cracks the code and finds…. R is the Ryberg constant and it makes the math works out.**Other Emission Series**• Many scientists are heating atoms and naming emission spectra series after themselves. • Now Enter Bohr.**Ogre’s Have Layers, Atoms Have Layers – Borh’s Big**Idea • The only way for light to be emitted in discrete frequencies is that that electrons must move in discrete ways. • He said, “the only allowed orbits are those where angular momentum of electrons mvr is quantized.” • His quantum leap of insight: The only time an electron radiates energy is when it jumps between discrete allowed orbits. The light emitted is then equal to the energy difference between the two orbits.**The Principle Of Elegance**• Bohr’s model explains the photoelectric effect, emission and absorption, and the Balmer series. • mvr = nh called h-bar • h = h/2π • It suddenly all makes sense. • Don’t worry, 2π will make sense in a minute.**Bohr Drops A Balmer On World**• Bohr expands the R in the Balmer series to account for the Atomic Number Z and writes it this way. Don’t forget, E = hν still applies • ν = 2π2meZ2e4 1 1 h3 k2 n2 • And It All Makes Sense And Is Still Elegant. { { Higher orbital integer Lower orbital integer Must be a value more than 2 But less than n.**Except For One Minor Problem**• When shooting x-rays at pure crystalline metals we find what we expect, wave diffraction patterns. • X-rays are generated by shooting high energy electrons at a metal target. • Hmm….what if we shoot electrons at our samples. • OOPS! We get a diffraction pattern! • Hang on, electrons are particles aren’t they?**Shooting Electrons At Crystals Causes Physicists Great**Diffraction**French Finally Explain Something, Rest Of World Astounded.**• Since electrons act like waves, perhaps the discrete orbits they inhabit must have the circumference that will hold one full wave sequence (remember the UV catastrophe problem?).**1923 - Louis DeBroglie Does The Wave**• de Broglie's picture of the Bohr model • The de Broglie relation • After a lot of arithmetic • Where k is called the wave number, a convenient way of reducing a relationship wave amplitude with respect to time. It merely makes the equation shorter. • The 2π thing? One rotation of a circle is 2π radians. • The full cycle of one wave takes places in 2π radians. • It All Still Makes Sense And Is Still Elegant.**Lambda λ Has An Implication**• λ = h/p implies a another huge idea…. • That all matter behaves like a wave at very small scales. • When in motion at great speed (mean p is large), particles exhibit wave like behavior because λ is insanely small and is related to h which 10-34 scale. • When p is low, like a bowling ball rolling in this room, λ is huge an so we perceive the object as physically solid. • More about this later.**Wave Particle Duality Causes Cats To Attempt Suicide, PETA**Accuses Schrödinger Of Feline Cruelty. Werner Heisenberg Not Certain Of Who’s Actually Responsible, “just don’t observe them” he says.**In A Nutshell**• Bohr’s Principle Of Complementarity: Waves and particles represent complementary aspects of the same phenomena. • Heisenberg’s Uncertainty Principle: It is not possible to measure the position and momentum of a particle with arbitrary precision. • De Broglie Relation: At very small scales matter behaves like waves. • Schrodinger develops a way to describe the probability of a particles location.**Gunfight At The Quantum Corral**• Like water waves, matter can be manipulated to a create a well in which waves of matter can be seen. • This is a recent advance, like within the last couple years. • But what’s the standing wave in the color image?**To Find Out Dig A Well**V -∞ +x • Take any length of wire and cut it like this picture, but before gluing it back together coat the ends with a super insulator. • In this way, whatever electron is placed into central trap or well will simply bounce back and forth since V on the ends approaches -∞ V -∞ V=0 L x=L x=0**A Very Deep Potential Well**• This diagram represents the physical artifact we just made before. • Because of the infinite potential outside of the well the potential inside can be described as: U=o for o < x < L U -∞ for x < 0 & x > L U(x) x=0 L**Allow An Electron To Resonate**The electron will now bounce back and forth like a ball. L = nλ/2 for n = 1,2,3… n is the principle quantum number The vertical displacement of the wave, called the amplitude A, can be described at any point along the wave as: yn(x) = A sin(x nπ/L) y x=0 L**Allow An Electron To Resonate**The question now is what are the odds of finding the electron at any given point along x? Although a standing wave is drawn imagine the electron is bouncing back and forth in wavelike manner. y x=0 L**Coulomb Example**• Near the well wall acceleration is huge so velocity is changing rapidly. • Near the center of the well the forces approach equilibrium and acceleration is near zero so the velocity approaches a constant value. e- -∞ -∞**Coulomb Example**• The odds of detecting e- near the well wall is very low. • Near the well middle it’s very high. • By this example, for n=1, the probability of detection is related to sinx. e- -∞ -∞