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### Fund Quantum Mechanics Lect & HW Solutions 59

Course: PHY 3604, Fall 2011
School: UNF
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Word Count: 314

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ANGULAR 4.1. MOMENTUM 4.1.3.1 41 Solution anguc-a Question: The general wave function of a state with azimuthal quantum number l and magnetic quantum number m is = R(r)Ylm (, ), where R(r) is some further arbitrary function of r. Show that the condition for this wave function to be normalized, so that the total probability of nding the particle integrated over all possible positions is one, is that r =0 R(r)...

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ANGULAR 4.1. MOMENTUM 4.1.3.1 41 Solution anguc-a Question: The general wave function of a state with azimuthal quantum number l and magnetic quantum number m is = R(r)Ylm (, ), where R(r) is some further arbitrary function of r. Show that the condition for this wave function to be normalized, so that the total probability of nding the particle integrated over all possible positions is one, is that r =0 R(r) R(r)r2 dr = 1. Answer: You need to have | = d3 r = 1 for the wave function to be normalized. Now the volume element d3 r is in spherical coordinates given by r2 sin drdd, so you must have 2 R(r) Ylm (, ) R(r)Ylm (, )r2 sin drdd = 1. r =0 =0 =0 Taking this apart into two separate integrals: r =0 R(r) R(r)r2 dr 2 =0 =0 Ylm (, ) Ylm (, ) sin dd = 1. The second integral is one on account of the normalization of the spherical harmonics, so you must have R(r) R(r)r2 dr = 1. r anguc-b Question: =0 4.1.3.2 Solution Can you invert the statement about zero angular momentum and say: if a particle can be found at all angular positions compared to the origin with equal probability, it will have zero angular momentum? Answer: No. To be at zero angular momentum, not just the probability ||2 , but itself must be independent of the spherical coordinate angles and . As an arbitrary example, = R(r)ei sin would have a probability of nding the particle independent of and , but not zero angular momentum. 4.1.3.3 Solution anguc-c Question: What is the minimum amount that the total square angular momentum is larger than just the square angular momentum in the z -direction for a given value of l? Answer: The total square angular momentum is l(l + 1) 2 and the square angular z h 22 momentum is m h . Since for a given value of l, the largest that |m| can be is l, the dierence
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UNF - PHY - 3604
42CHAPTER 4. SINGLE-PARTICLE SYSTEMSis at leastl(l + 1) 2 l2 h2 = lh2 .h4.1.44.2Angular momentum uncertaintyThe Hydrogen Atom4.2.1The Hamiltonian4.2.2Solution using separation of variables4.2.2.1Solution hydb-aQuestion: Use the tables for t
UNF - PHY - 3604
4.2. THE HYDROGEN ATOM43The total probability of nding the particle integrated over all possible positions is, using thetechniques of volume integration in spherical coordinates:2r =0|100 |2 d3 r ==0=0or rearranging12r/a0r/a0 =0egivingr2 r
UNF - PHY - 3604
44CHAPTER 4. SINGLE-PARTICLE SYSTEMSwhere a0 = 40 h2 /me e2 , to nd the ground state energy. Express in eV, where 1 eV equals1.602,2 1019 J. Values for the physical constants can be found at the start of this section andin the notations section.Answe
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4.2. THE HYDROGEN ATOM4.2.3.345Solution hydc-cQuestion: Based on the results of the previous question, what is the color of the lightemitted in a Balmer transition from energy E3 to E2 ? The Planck-Einstein relation says thatthe angular frequency of
UNF - PHY - 3604
46CHAPTER 4. SINGLE-PARTICLE SYSTEMSAnswer: The square wave function is|100 (r)|2 =1 2r/a0ea30and the value at the nucleus r = 0 is then|100 (0)|2 =1a30For the value at r above to be one percent of this, you must havee2r/a0 = 0.01or taking
UNF - PHY - 3604
4.3. EXPECTATION VALUE AND STANDARD DEVIATION47or multiplying out2px |2px =1( 211 |211 + 211 |211 + 211 | 211 + 211 |211 )2or using the orthonormality of 211 and 211 .2px |2px =1(1 + 0 + 0 + 1) = 1.2For the state 2py , remember that i comes ou
UNF - PHY - 3604
484.3.1.3CHAPTER 4. SINGLE-PARTICLE SYSTEMSSolution esda-cQuestion: Continuing this example, what will be the standard deviation?Answer: The average square deviation from 1.5 is:111(1 1.5)2 + (2 1.5)2 =224Taking a square root, the standard de
UNF - PHY - 3604
4.3. EXPECTATION VALUE AND STANDARD DEVIATION49What are the expectation values of energy, square angular momentum, and z -angular momentum for this state?Answer: Note that the square coecients of the eigenfunctions 211 and 211 are each 1 ,21so each
UNF - PHY - 3604
50CHAPTER 4. SINGLE-PARTICLE SYSTEMSFor the z -angular momentum, the expectation value is zero but the two states have eigenvaluesh and h, so11( 0)2 + (h 0)2 = h.hLz =22Whether h or h is measured, the deviation from zero has magnitude h.4.3.3
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4.3. EXPECTATION VALUE AND STANDARD DEVIATION51or multiplying outL2 =1211 + 211 | 2 2 211 + 2 2 211 .hh2multiplying out further to L2 = 2 2 .hFor the z -angular momentum,Lz =or multiplying outLz =multiplying out further to Lz4.3.3.21211
UNF - PHY - 3604
524.4CHAPTER 4. SINGLE-PARTICLE SYSTEMSThe Commutator4.4.1Commuting operators4.4.1.1Solution commutea-aQuestion: The pointer state12px = (211 + 211 ) .2is one of the eigenstates that H , L2 , and Lx have in common. Check that it is not an eige
UNF - PHY - 3604
4.5. THE HYDROGEN MOLECULAR ION53Answer: On second thought, maybe I can relax.According to the uncertainty relationship, the uncertainties could be as small as, for example,0.5 1010 m in position and 1024 kg m/s for linear momentum. I am not going to
UNF - PHY - 3604
54CHAPTER 4. SINGLE-PARTICLE SYSTEMS4.5.4States that share the electron4.5.5Comparative energies of the states4.5.6Variational approximation of the ground state4.5.6.1Solution hione-aQuestion: The solution for the hydrogen molecular ion requires
UNF - PHY - 3604
4.5. THE HYDROGEN MOLECULAR ION55Now evaluate the expectation energy:E = ax( x)|H |ax( x) = |a|2 x( x) h2 2x( x)2m x2You can substitute in the value of |a|2 from the normalization requirement above and applythe Hamiltonian on the function to its r
UNF - PHY - 3604
Chapter 5Multiple-Particle Systems5.15.1.1Wave Function for Multiple ParticlesSolution complex-aQuestion: A simple form that a six-dimensional wave function can take is a product of twothree-dimensional ones, as in (r1 , r2 ) = 1 (r1 )2 (r2 ). Show
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
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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
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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
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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
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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:
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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
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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
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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=
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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