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### Fund Quantum Mechanics Lect & HW Solutions 75

Course: PHY 3604, Fall 2011
School: UNF
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Word Count: 172

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5 Multiple-Particle Chapter Systems 5.1 5.1.1 Wave Function for Multiple Particles Solution complex-a Question: A simple form that a six-dimensional wave function can take is a product of two three-dimensional ones, as in (r1 , r2 ) = 1 (r1 )2 (r2 ). Show that if 1 and 2 are normalized, then so is . Answer: This is a direct consequence of the fact that integrals can be factored if their integrands can be and...

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5 Multiple-Particle Chapter Systems 5.1 5.1.1 Wave Function for Multiple Particles Solution complex-a Question: A simple form that a six-dimensional wave function can take is a product of two three-dimensional ones, as in (r1 , r2 ) = 1 (r1 )2 (r2 ). Show that if 1 and 2 are normalized, then so is . Answer: This is a direct consequence of the fact that integrals can be factored if their integrands can be and the limits of integration are independent of the other variable: 2 all r1 5.1.2 all r 2 1 (r1 )2 (r2 ) d3 r1 d3 r2 = 2 all 1 1 r (r1 ) d3 r1 2 all r 2 2 (r2 ) d3 r2 = 1 Solution complex-b Question: Show that for a simple product wave function as in the previous question, the relative probabilities of nding particle 1 near a position ra versus nding it near another position rb is the same regardless where particle 2 is. (Or rather, where particle 2 is likely to be found.) Note: This is the reason that a simple product wave function is called uncorrelated. For particles that interact with each other, an uncorrelated wave function is often not a good 57
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UNF - PHY - 3604
58CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSapproximation. For example, two electrons repel each other. All else being the same, theelectrons would rather be at positions where the other electron is nowhere close. As a result,it really makes a dierence for
UNF - PHY - 3604
5.2. THE HYDROGEN MOLECULE59general, it is just not worth the trouble for the electrons to stay away from the same position:that would reduce their uncertainty in position, increasing their uncertainty-demanded kineticenergy.Answer: The repulsive pot
UNF - PHY - 3604
605.2.2.2CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSSolution hmolb-aQuestion: When the protons are close to each other, the electrons do aect each other, andthe wave function above is no longer valid. But suppose you were given the true wave function,and y
UNF - PHY - 3604
5.2. THE HYDROGEN MOLECULE5.2.4.161Solution hmold-aQuestion: Obviously, the visual dierence between the various states is minor. It may evenseem counter-intuitive that there is any dierence at all: the states l r and r l are exactlythe same physical
UNF - PHY - 3604
62CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSvice-versa. But there is still almost no probability of nding both protons in the rst quadrant,both near the right proton. Nor are you likely to nd them in the third quadrant, both nearthe left proton.If you aver
UNF - PHY - 3604
5.2. THE HYDROGEN MOLECULEz263z2z1n1z2z1n1z2z1n1z1n1n2zn2zn2zn2znznznznznznznznzzzzzFigure 5.2: Probability density functions on the z -axis through the nuclei. From left toright: l r , r l , the symmetric combin
UNF - PHY - 3604
64CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSof the symmetric and antisymmetric states are quite dierent, though they look qualitativelythe same.5.2.5Variational approximation of the ground state5.2.6Comparison with the exact ground state5.35.3.1Two-St
UNF - PHY - 3604
5.3. TWO-STATE SYSTEMS65and this can be multiplied out, dropping the common factor 2 and noting that 1 |1 and2 |2 are one, as1 |2 2 2 + 1 |2 = 0for which the smallest root can be written as=1+1 |21 1 |22.That is less than 1 |2 , hence is small
UNF - PHY - 3604
66CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSIf this is plugged into the expression for the solution, c1 1 + c2 2 , it takes the formc1 1 + c2 2wherec1 =c1 + c2(1 2 )c2 =c2 + c1.(1 2 )So, while the constants c1 and c2 are dierent from c1 and c2 , the
UNF - PHY - 3604
5.5. MULTIPLE-PARTICLE SYSTEMS INCLUDING SPIN67Answer:s(s + 1) 2 =h5.515 2h.4Multiple-Particle Systems Including Spin5.5.1Wave function for a single particle with spin5.5.1.1Solution complexsa-aQuestion: What is the normalization requirement
UNF - PHY - 3604
68CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSof nding it somewhere with spin down is | |2 d3 r. The sum of the two integrals must beone to express the fact that the probability of nding the particle somewhere, either with spinup or spin down, must be one, ce
UNF - PHY - 3604
5.5. MULTIPLE-PARTICLE SYSTEMS INCLUDING SPIN695.5.4Wave function for multiple particles with spin5.5.4.1Solution complexsb-aQuestion: As an example of the orthonormality of the two-particle spin states, verify that| is zero, so that and are indeed
UNF - PHY - 3604
70CHAPTER 5. MULTIPLE-PARTICLE SYSTEMSThis is similar to the observation in calculus that integrals of products can be factored intoseparate integrals:all r 1all r 2all r 1f (r1 )g (r2 ) d3 r1 d3 r2 =f (r1 ) d3 r1all r 2g ( r 2 ) d3 r 2Answer:
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
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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
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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
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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
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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
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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
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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
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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
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96CHAPTER 7. TIME EVOLUTION7.12Scattering7.12.1Partial reection7.12.2Tunneling7.13Reection and Transmission Coecients
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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
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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
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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
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102CHAPTER 9. NUMERICAL PROCEDURES
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Chapter 10Solids10.1Molecular Solids10.2Ionic Solids10.3Metals10.3.1Lithium10.3.2One-dimensional crystals103
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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
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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
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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
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110CHAPTER 11. BASIC AND QUANTUM THERMODYNAMICS11.14.5Blackbody radiation11.14.6The Debye model11.15Specic Heats
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Chapter 12Angular momentum12.1Introduction12.2The fundamental commutation relations12.3Ladders12.4Possible values of angular momentum111
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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
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12.12. THE RELATIVISTIC DIRAC EQUATION12.12The Relativistic Dirac Equation113
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114CHAPTER 12. ANGULAR MOMENTUM
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Chapter 13Electromagnetism13.1The Electromagnetic Hamiltonian13.2Maxwells Equations13.3Example Static Electromagnetic Fields13.3.1Point charge at the origin13.3.2Dipoles115
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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
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13.6. NUCLEAR MAGNETIC RESONANCE13.6.1Description of the method13.6.2The Hamiltonian13.6.3The unperturbed system13.6.4Eect of the perturbation117
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118CHAPTER 13. ELECTROMAGNETISM
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Chapter 14Nuclei [Unnished Draft]14.1Fundamental Concepts14.2The Simplest Nuclei14.2.1The proton14.2.2The neutron14.2.3The deuteron119