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# PHYS 2213 - 29 Mar - 4 Feb Lecture and Book Notes PHYS 2213 A

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This 11 page Class Notes was uploaded by Connor Hawley on Friday January 15, 2016. The Class Notes belongs to PHYS 2213 A at Georgia Institute of Technology - Main Campus taught by Dr. Joseph Conrad in Spring 2016. Since its upload, it has received 15 views. For similar materials see Intro to Modern Physics in Physics 2 at Georgia Institute of Technology - Main Campus.

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Date Created: 01/15/16

Ch. 38 Sc. 6 Sunday, February 28, 2016 10:55 PM Chapter 38 - Quantization Section 6 - The Bohr Hydrogen Atom - Bohr's model of the hydrogen atom treats the electrons orbiting the nucleus as both waves and particles ○ They rotate circularly about the center, like a particle ○ However,the electron also acts as a circular standing wave such that - Now, the electrons orbit is subject to both a constraint on the velocity and on the radius ○ ○ From , we obtain - When , we call the radius the Bohr radius ○ ○ Now, - Also, using classical analysis, the energy of the orbit is found to be - Binding energy - the of the electron in stationarystate ○ Because nearly all atoms are in their ground state, the binding energy of the ground state is called the ionization energy of an atom - Rememberangular momentum: ○ It is quantized in atoms as L: M Monday, February 29, 2016 10:21 AM When is quantum behavior indistinguishable from classical behavior? - When the difference in energy between energy levels is very small Bohr Correspondence theorem - Things described by quantum mechanics should converge to classical theory at large scales Moseley'sx-ray experiment - was shown by Barkla ○ The periodic table used to be ordered by atomicmass, not proton count ○ Using the x-ray experiment,Moseley predicted a whole bunch of new elements - With only a few exceptions, Moseley found that the x-ray emission line frequency from all known elements could be fit to ○ ○ Where and are constants for each line in his plots ○ It was plotted on a square root of the micro-frequencyversus atomic number K-shell spectrum - The K-shell is the first energy level - The spectrum is essentially the Lyman series - ○ This is what the Bohr model predicted ○ However,the real result is Auger Electron Spectroscopy - X-ray photoemission - very hard (unlikely to happen) to get a photon out of the metal - Auger spectroscopy - uses high energy x-rays to emit electrons Limitations of the Bohr model - Could not explain why all electrons were not in the level - Could not explain transition rates The Compton Effect - Put an end to the deniers who said light is not a particle - The Compton effect is observedwhen a photon collides with an electron, igniting an elastic collision - Conservationof momentumgives ○ - Conservationof energy gives ○ - This results in the simple equation ○ This change in wavelength of the scattered photon is known as the Compton effect ○ Detectedusing diffraction patterns Ch. 38 Sc. 7 Sunday, February 28, 2016 11:32 PM Chapter 38 - Quantization Section 7 - The Hydrogen Spectrum - Ionization limit - the top rung of the energy-leveldiagram corresponding to the energy level at which ○ An emitted photon in an energy-leveltransition will have a frequency - Now, the wavelength of the photon will be ○ ○ We now define ○ - Hydrogen-like ion - an ion with a single electron orbiting some number of protons ○ From here, we can adjust our analysis to find ○ ○ ○ - Bohr's model is semi-classical;the missing link ○ However,it cannot explain anything but hydrogen-like atoms L: W Wednesday, March 2, 2016 10:06 AM Wave functions and uncertainty - The double slit experiment ○ A single wave hits two slits and leaves causing an intensity pattern on the screen ○ The wave has an amplitude but the intensity (proportionallyto the amplitude squared) that we see ○ Photons must give the same pattern! - A wave analysis ○ There is an equation for amplitude - Probabilities ○ ○ is the probability of the dart landing iand is the total number that landed in the area desired - If you circumscribe a circle in a square, the probability that a rock lands in the circle is ○ ○ You can get this pretty accurately with just a small number of rocks - Back to photon-slit problem ○ The probability the photon ends up in the little window by H is; is the number of photons in a standard narrow strip when it is at position x ○ Rememberthat the light intensity is . So the energy per unit time is ○ Number of photons per unit time is ○ The probability that a photon ends up in the little window is ○ The probability is proportional to the intensity ○ - Probability density ○ ○ Where is the probability density The function is not a probability you must multiply it by a length to find an actual probability Normalization condition for the probability density □ ○ This was Schrödinger's idea - What about matter waves? ○ Schrödinger establishes this convention for matterwaves Psi is called the wave function of the particle at a particular position in space and must be properly normalized so the particle exists somewhere must be properly normalized so the particle exists somewhere ○ Probability of the particle being between two positions is given by - Particle in a box with infinite potential walls ○ A particle (wave)with mass m is in a one-dimensional box of width ○ The box puts boundary conditions on the wave - Probability of the particle vs. position ○ is not a possible energy level ○ You can tell the probability of where the particle is, not the position - When I put two neutrons in this box, where are they? ○ It is only known what energy level is occupied - Which slit do photons go through? ○ Dimming the light in Young's experiment results in single photons on the screen ○ At low intensities, their distribution should become the single-slit patterns ○ Still, you get the double slit experiment pattern ○ The photon is going through two slits at once - The experiment is replicable using electrons ○ However,when you detect which slit the electron goes through, it becomesa particle ○ Literally, it is observed using a microscope The photons change what is happening ○ "Nobody knows what the hell Psi is" Ch. 39 Sc. 1 - 4 Thursday, March 3, 2016 11:20 AM Chapter 39 - Wave Functions and Uncertainty Section 1 - Waves, Particles,and the Double-Slit Experiment - Why not just jump right into quantum mechanics? ○ Quantum mechanics explains microscopicphenomena that we cannot directly sense or experience, so it was important to begin by learning how light and atoms behave ○ The concepts we'll need in quantum mechanics are rather abstract, so before launching into the mathematics,we need to establish a connection between theory and experiment The connection is the double-slit interferenceexperiment - The amplitude of the superposition of two waves will be ○ ○ Because , Where is the proportionality constant The intensity of the wave is the experimentalreality that you measure - The probability that any throw lands in area A of the dart board is ○ - Probability also possessesa linearity property: - Also, for some number of darts thrown at the board, ○ The expected value is your best prediction of the outcomeof an experiment - For the double-slit experiment, suppose photons are fired at the slits ○ The probability that any one photon ends up in the strip at position is ○ ○ - The point is, we cannot predict what any individual photon will do, but we can predict the fraction of the photons that should land in this little region of space Section 2 - Connecting the Wave and Photon Views - There is a clear correlationbetween the intensity of the wave and the probability of detecting photons ○ - Because photons are quantized we write, ○ ○ This is a critical link between the wave model and the photon model ○ This is a critical link between the wave model and the photon model - Finally, we write ○ ○ In other words, the probability of detecting a photon at a particular point is directly proportional to the square of the light wave amplitude function at that point - Now, we change the amplitude squared to the probability density ○ ○ This is analogous to the linear mass density ○ It says that for any experimentin which we detect photons, the probability density for detection a photon is directly proportional to the square of the amplitude function of the corresponding electromagneticwave Section 3 - The Wave Function - For electrons and other matter, there is no amplitude function, unlike electromagneticwaves - We assume there is somecontinuous, wave-likefunction for matter that plays the same role as for electromagneticwaves ○ This is called the wave function ○ ○ Thus, - Note that it is only a wave-like function - Back to theory: a theory needs two basic ingredients ○ A descriptor - a mathematicalquantity used to describe our knowledge of a physical object is our descriptor ○ One or more laws that govern the behavior of the descriptor - The difficulty facing physicists early in the 20th century was the astounding discoverythat the position of an atom-sized particle is not well-defined Section 4 - Normalization - Normalizationcondition for the probability density ○ ○ Probability of the particle being between two positions is given by L: F Friday, March 4, 2016 10:04 AM - Momentumuncertainty principle ○ ○ For a wave packet the uncertainty in x and k are related ○ So it's impossible to measure simultaneouslyboth x and k ○ Now the momentumcan be written in terms of k ○ (multiplying by the first condition) - Energy uncertainty ○ ○ ○ ○ It means if we can tell the energy, we can't do a good job of telling when it has that energy - How to think about Uncertainty ○ The act of making one measurementon either momentumor position perturbs the other ○ Preciselymeasuring the time disturbs the energy ○ Preciselymeasuring the position disturbs the momentum - Wave-particle-duality solution ○ Bohr's principle of complementarity - it's not possible to describe physical observables simultaneouslyin terms of both particles and waves ○ When we're making a measurement,use the particle description, but when we're not, use the wave description - Wave packets are the result of a Fourier series? ○ Then how are there many and only one frequencies? - Copenhagen Interpretation ○ Three principles Heisenberg's uncertainty principle Bohr's complementarityprinciple Born's statistical interpretation, based on probabilities determined by the wave function ○ Together these three concepts form a logical interpretation of the physical meaning of quantum theory Physics describes only the results of these measurements - What is the minimum energy state for a particle in a box? ○ The uncertainty in position is roughly , so ○ We'll calculate the standard deviation of the momentum ○ ○ The average energy predicted by the uncertainty principle is greater than zero "If it's zero, it doesn't exist" ○ "If it's zero, it doesn't exist" - What is the minimum energy of a harmonic oscillator (a photon included) ○ From Newtonian mechanics, the energy of a 1D harmonic oscillator is ○ To find the minimum of the potential well, we take the partial derivative , from the derivative , plugging in the value for xmin - In other words, even in a vacuum, there are still photons around ○ The CasimirForce is a result of this zero point energy. Two uncharged plates will repel each other because zero point photons must be squeezed out by the boundary condition - Also, there's a well of antimatter in the universe? ○ It's like the electrons being bound to the atom: they can be emitted using the photoelectric effect ○ It takes about a billion volts to overcomethe universe's work function and to pull antimatter out of space

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