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# GENERAL PHYSICS 3 PHYS 2130

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This 12 page Class Notes was uploaded by Mrs. Peter Toy on Friday October 30, 2015. The Class Notes belongs to PHYS 2130 at University of Colorado at Boulder taught by James Shepard in Fall. Since its upload, it has received 7 views. For similar materials see /class/232105/phys-2130-university-of-colorado-at-boulder in Physics 2 at University of Colorado at Boulder.

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Date Created: 10/30/15

Today s HW is the last graded HW I will post one more problem set and will later post solutions as well You need not turn this in though you will be responsible for that material for the final exam r Tie DIWij Lum usw away in WW ANA 4 44444 vaJ j 2 1m y w wwdjw 1 m JyLWLMx um u 5w aw u n sly u m4quot 4wa IWlt VZ 3 am 3de Aw7 Can Schrodinger make sense of the periodic table D Mela D semi em D Nonmelal Atomic weighl Schrodinger39s solution for multielectron atoms Need to account for all the interactions among the electrons Ax Amp l w WK 5 u hr sq ughV7 Must solve for all electrons at once W for q kqnucleus nlrm quqlrzrl kqaqlrm Schrodinger s sol n for mu lectron atoms What s different for these cases Potential energy V changes Now more protons AND other electrons IV for q kqnuclEUSqlrm quqlrzrl kqaqlrm I Need to account for all the interactions among the electrons Must solve for all electrons at once The 85 is now nonseparable N0 analytical solution tha can be solved exactly for ANY atom other than Hydrogen Probany NEVER will be done Helium numerical computer solutions to maybe 10 decimal places for ground and lowlying excited states best solutions ever done Lithium numerical solutions to 6 or 7 places Schrodinger39s solution for multielectron atoms What s different for these cases Potential energy V changes Now more protons AND other electrons IV for q kqnuclEUSqlrm quqlrzrl kqaqlrarl I Need to account for all the interactions among the electrons Must solve for all electrons at once Solutions change More protons higher Z easy to deal with Other electrons in the atom impossible to deal with in an exact way 9 Approximations needed This is o en what modern science deals with A problem biology weather etc is too complex to treat exactly One can hopefully still make great progress and gain a real understanding IPA lndeQendent Par le AQQroximation The potential Vr that any electron feels is assumed to be static and determined by the average probability densities of all other electrons Portion of electron that is far from nucleus Well screened by Portion of electron that is very close to nucleus gt at o I u E E a D4 Unscreened Other by other electrons electrons Screening of electron wavefunctions makes sense of the periodic table it tells us which electronic levels are lled first which last The last or highest energy electrons have the following character These valence electrons determine the chemistry metallicity magnetism etc s P W 39s v ElMehaI V Elsemvrnetal Z EINanmeLai 2 3d i r l A 3P l 1 l 1 3 y i l l l 439 4 52 9 1213 ay gt Poorly screened 35 states have a significant probability of electron density close to the nucleus where it is poorly screened by the other electrons 9Those electrons more tightly bound to nucleus gt ESs lt E 3p lt E 3d Ionization energies of Helium Z ke2 r E 292 136 eV E0n1 246 eV expt 1352136 eV 9 Zeff 135 What is 2nd ionization energy of Helium ram is w b ate1w 0 away A e 3 c xf 1 can any 4 m d is correct It is totally unscreened so Zeff Z 2 Today Review of SR Midterm 1 Oct 7 730900 PM in DUAN G1BZO Auxiliary date for people with a conflict on Oct 7th who contacted me earlier Oct 8th 215pm 345pm Room Reading Room Gamow Tower 11th floor People who have not contacted me until now with an excusable conflict on Oct 7th will not be allowed on the 8 amp exceptions You can bring 1 page Letter format written on both sides with whatever you think might help you during the exam But you cannot bring the textbook or lecture notes Bring your calculators but no PCs please No homework due for next week I will hold office hours on Monday 1pm 2pm F Review of SR The cast Yummy Chicken for Lunch What exactly did we do during the first few weeks Chapter 1 Spacetime Galileo transformation Classical relativity MichelsonMorley 9 39c39 is same in all inertial frames Einstein39s postulates Incompatible with Galilean relativity Consequences were 39time dilation39 and 39length contraction39 9 Lorentz transformation 9 Velocity transformation Spacetime interval Invariant under Lorentz transformation Chapter 2 Relativistic Mechanics Redefinition of important physical quantities to preserve conservation laws under LT Momentum Force Kinetic Energy Rest Energy Total Energy Let39s start with a few important concepts Einstein s Postulates of Relativity 1 If S is an inertial framel and if a second frame 8 moves with constant velocity relative to S then S is also an inertial frame 2 The speed of light is the same in all inertial frames of reference An inertial frame is a reference frame that is not accelerating Event Where something is depends on when you check on it and on the movement of your own reference frame Time and space are not independent quantities they are related by Lorentz Tr Definition An event is a measurement of where something occurs at what time xayazat Event are not invariant under Lorentz transformation In fact the LT converts the coordinates of an event from one frame to another such as from S xyzt to S39 x39y39z39t39 Proper time Proper time Time interval At t2 t1 between two events X1y1z1t1 and X2y222t2 measured in the frame in which the two events occur at the same spatial coordinates ie time interval that can be measured with one clock Same spatial coordinates means X1X2 Y1Y2 2122 Spacetlme Interval Say we have two events x1y1z1t1 and x2y222t2 Define the spacetime intenlal sort of the quotdistancequot between two events as As2 cmr PM PM Azrl With Ax x1 x2 Spacetime interval Ayy1y2 AZ2Z1Z2 Att1 t2 The spacetime interval has the same value in all reference frames ie As2 is invariant under Lorentz transformations This was part of HWi lu What does the Lorentz tr do The the Lorentz and Galileo transformation converts the coordinates Xyzt of an event xyzt in frame 8 to the corresponding coordinates x y z t of another frame 8 The way LT is presented here requires the following The frame 8 is moving along the Xaxes of the frame 8 with the velocity v measured relative to S and we assume that the origips of both frames overlap at the time t0 Proper length Proper length Length of object measured at rest object measured in the frame where it is at rest use a ruler Simultaneity time dilation amp length contraction As a consequence of Einstein s second postulate of relativity 39 The speed of light is the same in all inertial frames of reference we came to interesting conclusions Relativity of simultaneity Time dilation Length contraction All these effects are summarized in a set of equations The Lorentz transformation Transformations If S is moving with speed v in the positive x direction relative to S then the coordinates of the same event in the two frames are related by Galilean transformation Lorentz transformation classical relativistic x x vt x 7x vt y y y y Z Z z z t t v u39u v t 7T 7x c Remark This assumes 0000 is the same event in both frames Velocity transformation a consequence of the Lorentz transformation If S is moving with speed v in the positive x direction relative to S and the origin of S and S39 overlap at t0 then the coordinates of the same event in the two frames are related by orentz transtormation Velocity translormation relativistic relativistic 14X 7 v 7 4 7 x e 7x7vt x liuxvcz 39 u y i y u z39 z y Y ling02 14 quotll ungcl r V t ref 7 CZ X Remember this one Lucy X 70W 7 V s 72 rt 0 i 2 3 t ticizx v Ethel Lucy has a set of synchronized clocks in her reference frame 8 as shown Ethel is moving to the left past Lucy and has her own set of synchronized clo s Ethel passes Lucy at the events xx0tt393pm 39 Lucy39s frame checks the cl marked Compared to Lucy39s clocks all showing 3pm at that moment this one reads slightly after 3pm Quiz example A highspeed train is traveling at a velocity of v 05c The moment it passes over a bridge it launches a can on ball straight up as seen by the train conductor with avelocity of 04c What is thevelocity of the ball right after it launched as seen by an obsener standing on the bridge Situation seen by the onlooker on the bridge Cannon ball right after ring the cannon v Some applications of LT x39 x 7 Vt Ethel T e event has coordinates x3t3pm in Lucy39s frame Lucy sees Ethel travel to the left with vel ity v meaning that v is negative as seen c Therefore in Lucy39s frame the observer reads the clock a t39 yapme 7M 62 gt1 3y3pm 9 reads a time after 3pm Velocity transformation A highspeed train is traveling at avelocity ofv 05c The moment it passes over a bridge it launches a can on ball straight up as seen by the train conductor with a velocity of 04c What is the velocity of the ball right after it was launched as seen by an observer standing on the bridge Attach reference frame 8 to the t 39 39 Observer is in fr rain ame S39 traveling from right to left v is negative Now use the velocity translorm 14 7ux 2 leuxvc V 14 y17l1euvc2 u Velocity transformation u3905c y y 2 y 2 u39o34ecu luz Hui Velocity transf My 7 x7 7 liuxvcz u u y Y liuxvcz Length contraction Consequence of time dilation and vice versa Length of an object The length measured in the sticks rest frame is its proper length Length of an object 8 3 Another observer l l l v comes whizzing by at r r 7 speed v T 39 Is 839 I obsener measures f f f l f f f the length of the stick 0 and keeps track of time Event 1 Origin of 839 passes left end of stick Length of an object Event 1 Origin of 839 passes left end of stick Event 2 Origin of 839 passes right end of stick In frame 5 length of stick L39 this iswhatwe39re looking for time between measurements At39 this is proper time speed of frame 8 is v L39At39 In frame 5 length of stick L this is the proper length time between the two events At speed of frame 839 is v UAt Conclusion Speeds are the same both referto the relative speed And so y Length in moving frame Length in stick s rest frame proper length Length contraction is a consequence of time dilation and viceversa Now let39s do some relativistic mechanics Chapter 2 Definitions y 1 2 We redefined several physical quantities to 1 1L maintain the conservation laws for momentum c2 and energy in special relativity dr dr Relativistic momentum p m 7m 7mll dtproper d d Relativisticforce Fz pz ymu dt 11 Relativistic Energy E ymC2 K ch K kinetic energy Important relations These relations result from the previous definitions Rest energy of an object Em ch Energy momentum relation E2 pc2 mc22 Momentum of a massless particle p Ec Velocity of a massless particle 14 c How does nuclear power work The total rest energy of the particle equals the sum of the rest energy of all constituents plus the total internal energy such as binding energy EB M02 EmiCZ EB Application Nuclear Fusion 0 0 gt Deuterium Deuterium Helium Isotope mass Deuterium 201355321270 u MW 1 u 1661027 kg Helium 4 400260325415 u 9 1kg of Deuterium yields 0994 kg of Helium 4 Energy equivalent of 6 grams E0 me 0006 kg310lt 3mS2 541014 I Enough to power 20000 American households for 1 year Example Relativistic force A charged particle charge q with mass m is at rest at X 0 in a uniform electric field 8 Plot the velocity uof the particle as a function of time t assume the particle is released at f 0 Force acting on the particle F q8 Relativistic force F dymudt A Therefore q dymudt Or q dt dymu Integrating both sides fq dt dymu yields q t ymu Example Relativistic force cont Now Solve ymu qS t for the velocity u 1 Dividing by 7 yields mu qSt1u2CZ0395 Square both sides mZuz qZ Ztz uZcz Bring uto the left u2m2c2 1252 qzgIZCZ Divide by term in bracket and take squareroot LI cht th2 me2 Application Lorentz transformation 01 1 t397t7l2x D D C A B t yt izx C Two clocks one at A a d one at B are synch nized A third clock ies past A at a velocity v The moment it passes A all three clocks show the same time to 0 viewed by obseners i Aand B Seeleft image What timedoes the third clock shwv as seen by an obsener at B at the moment it passes the clock in B The clock in B is showing 1 is at that moment 9 Use Lorentz transformation A Y39 Il39IU B YII39IU1 W62 C mIo VZCZ D rtr0 I v E vatrum vxvc2gt Hint Use the following frames 0amp X39 139 t397tiLr HX 3 Q 02 A B A B nwi t C Two clocks one at A at d one at B are synchr unized A third clock ies past A at a velocity v The moment it passes A all three clocks shwv t e same time To 0 viewed by obseners in and B See left image What time does the third clock show as seen by an observer at B at the moment it passes the clock in B The clock in B is showing 11 is at that moment 9 Use Lorentz transformation A Y 39 IfIo B IfIoXl W62 C YI1Io1 VZCZ D II39IU Y E YItIo1 VX39lcz The moving clock shows the proper time intenal Atm n Aty Hint Use the following systems GV x 1 ID CX 2 I v t t 7 CM The clock travels from Ato B with speed v Assume A is at position x 0 then B is at position x vr rrlro Use this to substitute x in the Lorentz transformation v21 v2 L 705 730 17 9 We get exactly the expression of the time dilation Announcements Reading assignment for Monday TZT 58 59 HW6 due on Wed Oct 15 h Midterm solutions are posted on CUIearn There are still two people with unregistered clickers 19BC1QBC F46C28B If you don t register your clicker lwill not be able to give you credits for your class I 39 39 39 Last class I noticed that some of you were a bit confused with the unit electron volt eV Here it comes again Electron Volt A convenient unit for energy Joules are good for macroscopic energy conversions But when talking about energy of electrons or protons Joules are inconvenient way too big Define new energy unit The electronvolt eV 1eV kinetic energy gainedlost by an electron when accelerateddecelerated through 1 volt of potential difference 1eV 16 10quot9J ov 1V F E path You might remember this Current flowing through a wire I 39 I gt HHHHH 1 Ampere 1 CoulombSecond 1Watt1Ampere1Volt1Cs1V 1Joule1Watt1Second1C1V ie An electric potential of 1 V puts in one Joule of energy into a charge of 1 C 1 C kinetic energy increases by 1 Joule 1V An e39 has a charge of 1 1310490 3 therefore it ains a kinetic energy of 1eV 610quot9J V e Today39s class More about atoms and atomic spectra A semiclassical explanation of the atomic spectra Q1 Quiz on the reading The Rydberg formula describes a the colors of light emitted from all types of atom b the angular distribution of particles scattering off of atoms that Rutherford observed in his famous experimen c the colors of light emitted by hydrogen atoms d the energy distribution of cathode rays e the colors of light emitted by helium atoms The Rutherford experiment showed that the atomic core is a small hard positively charged thing inside a much lighter and bigger cloud of electrons We have seen that when atoms are struck by fast electrons they give off light concentrated in very narrow spectral lines 120V 0 J o J o 12 039 9 r f gt gt 0 Q l Cathode hot metal so electrons can come out Anode positive potential The Rutherford atom Tiny nucleus with protons and neutrons 999 of mass Surrounded by large diffuse cloud of low mass electrons What colors from white light What colors from neon What colors from hydrogen What from mercury What from sodium White light whole spectrum Each type of atom produces unique set of colors called its spectrum None of the atoms produces white light 6 i 39 600 Wavelength nanometers Hydrogen Hydrogen Hydrogen 700 J o o 500 Each type of atom produces unique set of colors w What does this imply about electrons in atoms lmplies that electrons only change between very specific energies Each time a photon is emitted an electron must be changing in energy by that amount reeasing energy Only way for individual atoms to give off energy is as light Atoms are lazy always want to do back to lowest energy state 2 Excited atom electron 3 Electron 1 Fast electron in atom goes to higher jumps back to Ms atom energy 1988 KE lowe ergy a 10ns IAN Higher energy Ground slate Energy level diagrams represent energy levels the electron can go to Different height different energy 5 5 Energy l For Hydrogen I I I transitions to No light emitted Ith ground state in colors in this regi n deep ultraviolet because no ener y levels spaced wit L 5 this energy Investigate energy levels in atoms 8 8 8 Dismargz Lamps 101 Help aaaaaaaaaa er way ll nr energy levels of electron stuck in atom energy of colliding electron G ground If the colliding electrons have an energy between that of level 2 and level 3 when they hit the atom how many different colors could be emitted bythe atom a no levels will be excited and so no light will come out b 1 colorof light will come out c 2 colors of light will come out d 3 colors of light will come out e 4 colors come out I ans d enough energy to excrte level 2 then get 2gt1 followed Q3 An electron bashes into an atom in a discharge lamp 2 eV 3 eV 6 eV If atom fixed at this point in tube 0 10ev list all the possible photon energies colors that you might see A 1eV 2eV 3eV 4eV 7eV 8eV B 4eV 7eV 8eV C 1eV 3eV 4eV D 4eV E Impossibleto tell by 125 but a so can do 2gtG Electron energy qAV eEd where E is the electric field battery Vtota distance D and d is the distance it goes before a collision Q4 A neon lamp emits a strong red line Sodium emits a strong yellow line What accounts for this difference aThe electrons in the discharge hit the neon atoms with more speed than the electrons hit the sodium atoms bThe electrons in the discharge hit the neon atoms with less speed than the electrons hit the sodium atoms cThe energy spacing between the electronic energy levels that are responsible forthese lines are smaller in the neon atom than in the sodium atom dThe energy spacing between the electronic energy levels that are responsible forthese lines is larger in the neon atom than in the sodium atom Summary of important Ideas 1 Electrons in atoms only found in specific energy levels 2 Different set of energy levels for different atoms 3 1 photon emitted per electron jump down between energy levels Photon color determined by energy difference 4 If electron not bound to an atom Can have any energy For instance free electrons in the PE effect Hydrogen Lithium Electron energy levels in 2 different atoms Levels have different spacing explains unique colors for each type of atom f l Energy A Atoms with more than one motto scale 00 electron lower levels filled Answers to the clicker questions Q1 C Q2 D see next slide 03 D Since the atom is approximately in the center of the tube the electrons gain only about 5eV of kinetic energy before they strike the atom Q4 C When the energy spacing is smaller the emitted photon has less energy Red photons longerwavelength lower frequency light have less energy than yellow photons Answer to 02 energy levels of electron stuck in atom G ground ans d enough energy to excite level 2 then get 2gt1 followed by 126 but also can go 226

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