# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

1 Page

### Fund Quantum Mechanics Lect & HW Solutions 88

Course: PHY 3604, Fall 2011
School: UNF
Rating:

Word Count: 156

#### Document Preview

5. 70 CHAPTER MULTIPLE-PARTICLE SYSTEMS This is similar to the observation in calculus that integrals of products can be factored into separate integrals: all r 1 all r 2 all r 1 f (r1 )g (r2 ) d3 r1 d3 r2 = f (r1 ) d3 r1 all r 2 g ( r 2 ) d3 r 2 Answer: | | = 1 h Sz1 = 2 (Sz1 )(Sz1 ) 1 Sz2 = 2 h (Sz2 )(Sz2 ) and written out 1 1 | | = (+ 1 h)(+ 2 h) + ( 1 h)( 1 h) (+ 1 h)(+ 1 h) + ( 2...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Florida >> UNF >> PHY 3604

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
5. 70 CHAPTER MULTIPLE-PARTICLE SYSTEMS This is similar to the observation in calculus that integrals of products can be factored into separate integrals: all r 1 all r 2 all r 1 f (r1 )g (r2 ) d3 r1 d3 r2 = f (r1 ) d3 r1 all r 2 g ( r 2 ) d3 r 2 Answer: | | = 1 h Sz1 = 2 (Sz1 )(Sz1 ) 1 Sz2 = 2 h (Sz2 )(Sz2 ) and written out 1 1 | | = (+ 1 h)(+ 2 h) + ( 1 h)( 1 h) (+ 1 h)(+ 1 h) + ( 2 h)( 1 h) 2 2 2 2 2 2 and multiplying out, and reordering the second and third factor in each term, you see it is the same the as expression obtained in the answer to the previous question, | 1 1 1 = (+ 2 h)(+ 2 h) (+ 1 h)(+ 1 h) + (+ 1 h)( 2 h) (+ 1 h)( 1 h) + 2 2 2 2 2 1 1 ( 1 h)(+ 1 h) ( 2 h)(+ 2 h) + ( 1 h)( 1 h) ( 1 h)( 1 h). 2 2 2 2 2 2 5.5.5 Example: the hydrogen molecule 5.5.5.1 Solution complexsc-a Question: Show that the normalization requirement for gs means that |a++ |2 + |a+ |2 + |a+ |2 + |a |2 = 1 Answer: For brevity, write gs,0 = a [l (r1 )r (r2 ) + r (r1 )l (r2 )] so that gs = a++ gs,0 + a+ gs,0 + a+ gs,0 + a gs,0 .
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

UNF - PHY - 3604
5.6. IDENTICAL PARTICLES71For gs to be normalized, its square norm must be one:gs |gs = 1.According to the previous subsection, this inner product evaluates as the sum of the innerproducts of the matching spin components:a+ gs,0 |a+ gs,0 + a+ gs,0 |
UNF - PHY - 3604
725.6.1CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSSolution ident-aQuestion: Check that indeed any linear combination of the triplet states is unchanged underparticle exchange.Answer: In the notations of the previous section, the most general linear combina
UNF - PHY - 3604
5.7. WAYS TO SYMMETRIZE THE WAVE FUNCTION5.75.7.173Ways to Symmetrize the Wave FunctionSolution symways-aQuestion: How many single-particle states would a basic Hartree-Fock approximation useto compute the electron structure of an arsenic atom? How
UNF - PHY - 3604
74CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSall r1all r 2l (r1 )r (r2 ) r (r1 )l (r2 ) d3 r1 d3 r2can according to the rules of calculus be factored into three-dimensional integrals asSS1 |2=all r 1l (r1 ) r (r1 ) d3 r1all r 2r (r2 ) l (r2 ) d3 r2=
UNF - PHY - 3604
5.8. MATRIX FORMULATION75So there are two energy eigenvalues:E1 = J LandE2 = J + L.In the rst case, since according to the equations above(J E1 ) a1 La2 = 0=La1 La2 = 0it follows that a1 and a2 must be equal, producing the eigenfunctionSSa1 1
UNF - PHY - 3604
76CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSelectrons around the same nucleus, the remaining C = 4 Slater determinants can be writtenout explicitly to give the two-particle statesS1 =l r r l 2S2 =l r r l 2S3 =l r r l 2S4 =l r r l 2Note that
UNF - PHY - 3604
5.9. HEAVIER ATOMS77Seigenfunction 1 = 1 has, according to its denitionabove, both electrons spin-up and inthe excited antisymmetric spatial state (l r r l ) / 2.A similar guess that works is(a1 , a2 , a3 , a4 ) = (0, 0, 0, 1).SThis corresponds t
UNF - PHY - 3604
78CHAPTER 5. MULTIPLE-PARTICLE SYSTEMS5.9.4Lithium to neon5.9.5Sodium to argon5.9.6Potassium to krypton5.9.7Full periodic table5.10Pauli Repulsion5.11Chemical Bonds5.11.1Covalent sigma bonds
UNF - PHY - 3604
5.11. CHEMICAL BONDS5.11.2Covalent pi bonds5.11.3Polar covalent bonds and hydrogen bonds5.11.4Promotion and hybridization5.11.5Ionic bonds5.11.6Limitations of valence bond theory79
UNF - PHY - 3604
Chapter 6Macroscopic Systems6.1Intro to Particles in a Box6.2The Single-Particle States6.3Density of States6.4Ground State of a System of Bosons81
UNF - PHY - 3604
82CHAPTER 6. MACROSCOPIC SYSTEMS6.5About Temperature6.6Bose-Einstein Condensation6.6.1Rough explanation of the condensation6.7Bose-Einstein Distribution6.8Blackbody Radiation6.9Ground State of a System of Electrons6.10Fermi Energy of the Fr
UNF - PHY - 3604
6.11. DEGENERACY PRESSURE6.11Degeneracy Pressure6.12Connement and the DOS6.13Fermi-Dirac Distribution6.14Maxwell-Boltzmann Distribution6.15Thermionic Emission6.16Chemical Potential and Diusion6.17Intro to the Periodic Box83
UNF - PHY - 3604
84CHAPTER 6. MACROSCOPIC SYSTEMS6.18Periodic Single-Particle States6.19DOS for a Periodic Box6.20Intro to Electrical Conduction6.21Intro to Band Structure6.21.1Metals and insulators6.21.2Typical metals and insulators6.21.3Semiconductors
UNF - PHY - 3604
6.22. ELECTRONS IN CRYSTALS6.21.4Semimetals6.21.5Electronic heat conduction6.21.6Ionic conductivity6.22Electrons in Crystals6.22.1Bloch waves6.22.2Example spectra6.22.3Eective mass85
UNF - PHY - 3604
86CHAPTER 6. MACROSCOPIC SYSTEMS6.22.4Crystal momentum6.22.5Three-dimensional crystals6.23Semiconductors6.24The P-N Junction6.25The Transistor6.26Zener and Avalanche Diodes6.27Optical Applications
UNF - PHY - 3604
6.27. OPTICAL APPLICATIONS6.27.1Atomic spectra6.27.2Spectra of solids6.27.3Band gap eects6.27.4Eects of crystal imperfections6.27.5Photoconductivity6.27.6Photovoltaic cells6.27.7Light-emitting diodes87
UNF - PHY - 3604
88CHAPTER 6. MACROSCOPIC SYSTEMS6.28Thermoelectric Applications6.28.1Peltier eect6.28.2Seebeck eect6.28.3Thomson eect
UNF - PHY - 3604
Chapter 7Time Evolution7.1The Schrdinger Equationo7.1.1The equation7.1.2Solution of the equation7.1.2.1Solution schrodsol-aQuestion: The energy of a photon is h where is the classical frequency of the electrohmagnetic eld produced by the phot
UNF - PHY - 3604
90CHAPTER 7. TIME EVOLUTIONClassical physics understands the wave nature of light well, and not its particle nature. This isthe opposite of the situation for an electron, where classical physics understands the particlenature, and not the wave nature.
UNF - PHY - 3604
7.2. TIME VARIATION OF EXPECTATION VALUES7.1.4Stationary states7.1.5The adiabatic approximation7.2Time Variation of Expectation Values7.2.1Newtonian motion7.2.2Energy-time uncertainty relation7.3Conservation Laws and Symmetries7.4Conservatio
UNF - PHY - 3604
92CHAPTER 7. TIME EVOLUTION7.4.1Conservation of energy7.4.2Combining angular momenta and parities7.4.3Transition types and their photons7.4.4Selection rules7.5Symmetric Two-State Systems7.5.1A graphical example7.5.2Particle exchange and for
UNF - PHY - 3604
7.6. ASYMMETRIC TWO-STATE SYSTEMS7.5.37.67.6.17.7Spontaneous emissionAsymmetric Two-State SystemsSpontaneous emission revisitedAbsorption and Stimulated Emission7.7.1The Hamiltonian7.7.2The two-state model7.8General Interaction with Radiatio
UNF - PHY - 3604
947.9CHAPTER 7. TIME EVOLUTIONPosition and Linear Momentum7.9.1The position eigenfunction7.9.2The linear momentum eigenfunction7.10Wave Packets7.10.1Solution of the Schrdinger equation.o7.10.2Component wave solutions7.10.3Wave packets
UNF - PHY - 3604
7.11. ALMOST CLASSICAL MOTION7.10.4Group velocity7.10.5Electron motion through crystals7.11Almost Classical Motion7.11.1Motion through free space7.11.2Accelerated motion7.11.3Decelerated motion7.11.4The harmonic oscillator95
UNF - PHY - 3604
96CHAPTER 7. TIME EVOLUTION7.12Scattering7.12.1Partial reection7.12.2Tunneling7.13Reection and Transmission Coecients
UNF - PHY - 3604
Chapter 8The Meaning of Quantum Mechanics8.1Schrdingers Cato8.2Instantaneous Interactions8.3Global Symmetrization8.4A story by Wheeler97
UNF - PHY - 3604
98CHAPTER 8. THE MEANING OF QUANTUM MECHANICS8.5Failure of the Schrdinger Equation?o8.6The Many-Worlds Interpretation8.7The Arrow of Time
UNF - PHY - 3604
Chapter 9Numerical Procedures9.1The Variational Method9.1.1Basic variational statement9.1.2Dierential form of the statement9.1.3Example application using Lagrangian multipliers9.2The Born-Oppenheimer Approximation99
UNF - PHY - 3604
100CHAPTER 9. NUMERICAL PROCEDURES9.2.1The Hamiltonian9.2.2The basic Born-Oppenheimer approximation9.2.3Going one better9.3The Hartree-Fock Approximation9.3.1Wave function approximation9.3.2The Hamiltonian9.3.3The expectation value of energ
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
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]