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Fund Quantum Mechanics Lect & HW Solutions 118

Course: PHY 3604, Fall 2011
School: UNF
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9. 100 CHAPTER NUMERICAL PROCEDURES 9.2.1 The Hamiltonian 9.2.2 The basic Born-Oppenheimer approximation 9.2.3 Going one better 9.3 The Approximation 9.3.1 Wave Hartree-Fock function approximation 9.3.2 The Hamiltonian 9.3.3 The expectation value of energy

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9. 100 CHAPTER NUMERICAL PROCEDURES 9.2.1 The Hamiltonian 9.2.2 The basic Born-Oppenheimer approximation 9.2.3 Going one better 9.3 The Approximation 9.3.1 Wave Hartree-Fock function approximation 9.3.2 The Hamiltonian 9.3.3 The expectation value of energy
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UNF - PHY - 3604
9.3. THE HARTREE-FOCK APPROXIMATION9.3.4The canonical Hartree-Fock equations9.3.5Additional points9.3.5.1Meaning of the orbital energies9.3.5.2Asymptotic behavior9.3.5.3Hartree-Fock limit9.3.5.4Conguration interaction101
UNF - PHY - 3604
102CHAPTER 9. NUMERICAL PROCEDURES
UNF - PHY - 3604
Chapter 10Solids10.1Molecular Solids10.2Ionic Solids10.3Metals10.3.1Lithium10.3.2One-dimensional crystals103
UNF - PHY - 3604
104CHAPTER 10. SOLIDS10.3.3Wave functions of one-dimensional crystals10.3.4Analysis of the wave functions10.3.5Floquet (Bloch) theory10.3.6Fourier analysis10.3.7The reciprocal lattice10.3.8The energy levels10.3.9Merging and splitting bands
UNF - PHY - 3604
10.4. COVALENT MATERIALS10.4Covalent Materials10.5Free-Electron Gas10.5.1Lattice for the free electrons10.5.2Occupied states and Brillouin zones10.6Nearly-Free Electrons10.6.1Energy changes due to a weak lattice potential10.6.2Discussion of
UNF - PHY - 3604
106CHAPTER 10. SOLIDS10.7Additional Points10.7.1About ferromagnetism10.7.2X-ray diraction
UNF - PHY - 3604
Chapter 11Basic and Quantum Thermodynamics11.1Temperature11.2Single-Particle versus System States11.3How Many System Eigenfunctions?11.4Particle-Energy Distribution Functions107
UNF - PHY - 3604
108CHAPTER 11. BASIC AND QUANTUM THERMODYNAMICS11.5The Canonical Probability Distribution11.6Low Temperature Behavior11.7The Basic Thermodynamic Variables11.8Intro to the Second Law11.9The Reversible Ideal11.10Entropy11.11The Big Lie of Dis
UNF - PHY - 3604
11.12. THE NEW VARIABLES11.12The New Variables11.13Microscopic Meaning of the Variables11.14Application to Particles in a Box11.14.1Bose-Einstein condensation11.14.2Fermions at low temperatures11.14.3A generalized ideal gas law11.14.4The ide
UNF - PHY - 3604
110CHAPTER 11. BASIC AND QUANTUM THERMODYNAMICS11.14.5Blackbody radiation11.14.6The Debye model11.15Specic Heats
UNF - PHY - 3604
Chapter 12Angular momentum12.1Introduction12.2The fundamental commutation relations12.3Ladders12.4Possible values of angular momentum111
UNF - PHY - 3604
112CHAPTER 12. ANGULAR MOMENTUM12.5A warning about angular momentum12.6Triplet and singlet states12.7Clebsch-Gordan coecients12.8Some important results12.9Momentum of partially lled shells12.10Pauli spin matrices12.11General spin matrices
UNF - PHY - 3604
12.12. THE RELATIVISTIC DIRAC EQUATION12.12The Relativistic Dirac Equation113
UNF - PHY - 3604
114CHAPTER 12. ANGULAR MOMENTUM
UNF - PHY - 3604
Chapter 13Electromagnetism13.1The Electromagnetic Hamiltonian13.2Maxwells Equations13.3Example Static Electromagnetic Fields13.3.1Point charge at the origin13.3.2Dipoles115
UNF - PHY - 3604
116CHAPTER 13. ELECTROMAGNETISM13.3.3Arbitrary charge distributions13.3.4Solution of the Poisson equation13.3.5Currents13.3.6Principle of the electric motor13.4Particles in Magnetic Fields13.5Stern-Gerlach Apparatus13.6Nuclear Magnetic Reso
UNF - PHY - 3604
13.6. NUCLEAR MAGNETIC RESONANCE13.6.1Description of the method13.6.2The Hamiltonian13.6.3The unperturbed system13.6.4Eect of the perturbation117
UNF - PHY - 3604
118CHAPTER 13. ELECTROMAGNETISM
UNF - PHY - 3604
Chapter 14Nuclei [Unnished Draft]14.1Fundamental Concepts14.2The Simplest Nuclei14.2.1The proton14.2.2The neutron14.2.3The deuteron119
UNF - PHY - 3604
120CHAPTER 14. NUCLEI [UNFINISHED DRAFT]14.2.4Property summary14.3Modeling the Deuteron14.4Overview of Nuclei14.5Magic numbers14.6Radioactivity14.6.1Decay rate14.6.2Other denitions
UNF - PHY - 3604
14.7. MASS AND ENERGY14.7Mass and energy14.8Binding energy14.9Nucleon separation energies14.10Liquid drop model14.10.1Nuclear radius14.10.2von Weizscker formulaa14.10.3Explanation of the formula121
UNF - PHY - 3604
122CHAPTER 14. NUCLEI [UNFINISHED DRAFT]14.10.4Accuracy of the formula14.11Alpha Decay14.11.1Decay mechanism14.11.2Comparison with data14.11.3Forbidden decays14.11.4Why alpha decay?14.12Shell model
UNF - PHY - 3604
14.13. COLLECTIVE STRUCTURE14.12.1Average potential14.12.2Spin-orbit interaction14.12.3Example occupation levels14.12.4Shell model with pairing14.12.5Conguration mixing14.12.6Shell model failures14.13Collective Structure123
UNF - PHY - 3604
124CHAPTER 14. NUCLEI [UNFINISHED DRAFT]14.13.1Classical liquid drop14.13.2Nuclear vibrations14.13.3Nonspherical nuclei14.13.4Rotational bands14.13.4.1Basic notions in nuclear rotation14.13.4.2Basic rotational bands14.13.4.3Bands with intri
UNF - PHY - 3604
14.14. FISSION14.13.4.5Even-even nuclei14.13.4.6Non-axial nuclei14.14Fission14.14.1Basic concepts14.14.2Some basic features14.15Spin Data14.15.1Even-even nuclei125
UNF - PHY - 3604
126CHAPTER 14. NUCLEI [UNFINISHED DRAFT]14.15.2Odd mass number nuclei14.15.3Odd-odd nuclei14.16Parity Data14.16.1Even-even nuclei14.16.2Odd mass number nuclei14.16.3Odd-odd nuclei14.16.4Parity Summary
UNF - PHY - 3604
14.17. ELECTROMAGNETIC MOMENTS14.17Electromagnetic Moments14.17.1Classical description14.17.1.1Magnetic dipole moment14.17.1.2Electric quadrupole moment14.17.2Quantum description14.17.2.1Magnetic dipole moment14.17.2.2Electric quadrupole mom
UNF - PHY - 3604
128CHAPTER 14. NUCLEI [UNFINISHED DRAFT]14.17.2.4Values for deformed nuclei14.17.3Magnetic moment data14.17.4Quadrupole moment data14.18Isospin14.18.1Basic ideas14.18.2Heavier nuclei14.18.3Additional points
UNF - PHY - 3604
14.19. BETA DECAY14.18.4Why does this work?14.19Beta decay14.19.1Energetics Data14.19.2Von Weizscker approximationa14.19.3Kinetic Energies14.19.4Forbidden decays14.19.4.1Allowed decays14.19.4.2Forbidden decays allowed129
UNF - PHY - 3604
130CHAPTER 14. NUCLEI [UNFINISHED DRAFT]14.19.4.3The energy eect14.19.5Data and Fermi theory14.19.6Parity violation14.20Gamma Decay14.20.1Energetics14.20.2Forbidden decays14.20.3Isomers
UNF - PHY - 3604
14.20. GAMMA DECAY14.20.4Weisskopf estimates14.20.5Comparison with data14.20.6Cage-of-Faraday proposal14.20.7Internal conversion131
UNF - PHY - 3604
132CHAPTER 14. NUCLEI [UNFINISHED DRAFT]
UNF - PHY - 3604
Appendix AAddendaA.1Lagrangian mechanicsA.1.1IntroductionA.1.2Generalized coordinatesA.1.3Lagrangian equations of motionA.1.4Hamiltonian dynamics133
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Ashley StephensIST 444Laurie JervaLogo AssignmentWhen designing the logo for my groups company, Syracuse Miracle Works, there were afew factors that attributed to the overall design. Being a Hair Salon and Spa with the ultimategoal of providing rela
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Pho synthe istos11/07/11Quick Revision :Where does photosynthesis occur Leaf X-section :PalisademesophyllChloroplast under electron microscopeChloroplast 11/07/11Warm-up QuestionsWarm-upThe following microphotograph is atransverse section
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LeafStructureandPhotosynthesisStandardGradeBiologyPhotosynthesis Greenplants(producers)canuselight energytomaketheirownfood Thisprocessiscalledphotosynthesis Greenplantsaregreenbecausetheycontainachemicalcalledchlorophyll.Thischemicalisusedtotrapli
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HydrogeLightenergy nOxygenChlorophyllChemicalATPenergywaterADP + PiHydrogenCarbonDioxideATPADP and PiGlucoseThis powerpoint was kindly donated towww.worldofteaching.comhttp:/www.worldofteaching.com is hometo over a thousand powerpoints
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Boise State - BIOL - 191
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UNF - PHYSICS - 4360C
(b) T =T=1211 mg 1 m 2 g2mv = mvt = m =222 c2 2 .22 D 2(.145kg )2m 9.8 2 skg( 2 ) (.22 ) ( 2 ) (.0366 )m2= 87 J3 t 23 Fdx = cv dx = c v dt = c vt tanh dt t t t 31= cvt tanh 2 + tanh d 2t t 31= cvt tanh 2 + ln cosh
UNF - PHYSICS - 4360C
F2F1 2F2t1 + t1 ( t t1 ) +( t t1 )m2m2mF 2 F 2 F 2 5F 2At t = 2t1 : x =t1 + t1 + t1 =t1mm2m2mdv dv dxdvc3a== = v = v2dt dx dtdxm1cv 2 dv = dxm1vxmaxcv v 2 dv = 0 mdx1c2v 2 = xmaxmx=2.111xmax2.122mv 2=cGoing up:
UNF - PHYSICS - 4360C
v=vt v(vt2.142+v122), vt =gmg=kc2Going up: Fx = mg c2 v 2dvc22a = v = g kv , k =dxmvxvdvv g kv 2 = 0 dxv1ln ( g kv 2 ) = xv2kg + kv 2= e 2 kx2g + kvgg2v 2 = + v e 2 kx kk2Going down: Fx = mg + c2 vdvv = g + kv
UNF - PHYSICS - 4360C
22c2 v c1 c1 + 4mgc2t1=ln22mc1 + 4mgc2 2c2 v c1 + c1 + 4mgc2(t2c1 + 4mgc2m)12( 2c v + c += ln( 2c v + c 2v0) (c + 4mgc ) ( c +21c1 + 4mgc221c122)+ 4mgc )112c1 + 4mgc2c1222as t , 2c2 vt + c1 c1 + 4mgc2 = 0c1 c1
UNF - PHYSICS - 4360C
2.17dvdv= mv = f ( x )i g ( v )dtdxmvdv= f ( x ) dxg (v)mBy integration, get v = v ( x ) =If F ( x, t ) = f ( x )i g ( t ) :dxdtd 2xd dx = m = f ( x )i g ( t )2dtdt dt This cannot, in general, be solved by integration.If F ( v , t ) =
UNF - PHYSICS - 4360C
Integrating11A = tuu mand substituting e v = uAln 1 + e v t m(b) t = T @ v = 0A v = ln 1 + e v T mA vm1 e v e v = 1 + e TT=AmdvAvdvA= dx(c) v = v = e vvdxmem(a) v = v 1again, let u = e vdu = udvordv =duuv=1ln u1 d
UNF - PHYSICS - 4360C
3r 2= 1gr4Using Mathcad, solve the above non-linear d.e. lettingwhich reduces to: r +1 103 and R 0.01mm (small raindrop). Graphsshow thatv r t and r t 210
UNF - PHYSICS - 4360C
CHAPTER 3OSCILLATIONS3.1x = 0.002 sin 2 ( 512 s 1 ) t [ m ]mmxmax = ( 0.002 ) ( 2 ) ( 512 ) = 6.43 ss22 mmxmax = ( 0.002 ) ( 2 ) ( 512 ) 2 = 2.07 104 2 s s 3.2x = 0.1sin t [m]When t = 0, x = 0and = 5 s 13.3mx = 0.1 cos t smx = 0.5
UNF - PHYSICS - 4360C
3.63.711l=T =s 2.5 s9.82g6For springs tied in parallel:Fs ( x ) = k1 x k2 x = ( k1 + k2 ) x1(k + k ) 2= 1 2 mFor springs tied in series:The upward force m is keq x .Therefore, the downward force on spring k2 is keq x .The upward force
UNF - PHYSICS - 4360C
Fr = mg +mkmkdkcosxm = mg +tM +mM +mM +mFor the block to just begin to leave the bottom of the box atthe top of the vertical oscillations, Fr = 0 at xm = d :mkd0 = mg M +mg ( M + m)d=k3.9x = e t A cos ( d t )dx= e t A d sin ( d t ) e
UNF - PHYSICS - 4360C
3.1117 2 mx = 02317 = and 2 = 2222222 r = 2 = 4 mx + 3 mx +(a)Amax =(b)=2d = 2 2 =25 24 d =522A15 2e Td =3.12F2m d r = 2 =121ln 2 = f d ln 2Td1 d = ( 2 2 ) 2(a)122So, = ( + )2d1111 2 2 2 ln 2 2 2f = fd
UNF - PHYSICS - 4360C
21Td d 1 2=== 1 + 2 2 2 d 4 n TFor large n,Td1 1+ 2 28 nT13.14((a) r = 22122)22 = 2 2 = 0.70721 1 22 1 4(b) Q = d === 0.86622 2 2 2 ( 2 )222(c) tan = 2= 2 2 =2 43122() 2 = tan 1 = 146.331222
UNF - PHYSICS - 4360C
3.16(b) Q =2 2d=222 =1,LC=R2L2 1 R 2 LC 4 L L 1Q== 2 R R C 42 2L L R(c) Q = = C =RR 2 3.17Fext = F sin t = Im F ei t and x ( t ) is the imaginary part of the solution to:mx + cx + kx = F ei ti.e.i t x ( t ) = Im Ae
UNF - PHYSICS - 4360C
= tan 1 ( c 2m )m ( 2 2 ) c + kUsing sin 2 + cos 2 = 1 ,2F22= m ( 2 2 ) c + k + 2 ( c 2m )2AFA=cfw_m ( ) c + k + ( c 2m ) 2222122and x ( t ) = Ae t cos ( t ) + the transient term.3.19l A2 T 21 g8(a)for A =(b)412l1.041g,
UNF - PHYSICS - 4360C
1 Tf ( t ) = c + 2T f ( t ) cos ( n t ) dt cos n tn T2T1+ 2T f ( t ) sin ( n t ) dt sin n t ,n T21Tn = 1, 2, . . ., and c = 2T f ( t ) dtT 2Now, due to the equality of terms in n :2 Tf ( t ) = c + 2T f ( t ) cos ( n t ) dt cos n tn T2T2
UNF - PHYSICS - 4360C
3.22In steady state, x ( t ) = An e (i n t n )nAn =Fnm1( 2 n 2 2 )2 + 4 2 n 2 2 24F,n = 1, 3, 5, . . .andNow Fn =n9 2Q = 100 so 2 40, 00024F1A1 =1m22229 22( 9 ) + 4 40000 FA1 2m 24F1A3 =13m2 2 22 9 ( 9 2 9 2 ) +