Fundamental Quantum Mechanics for EngineersLeon van Dommele

Fundamental Quantum Mechanics for Engineers Leon van Dommelen 12/20/07 Version beta 3.4 Copyright Copyright 2004 and on, Leon van Dommelen You are allowed to copy or print out this work in unmodified work for your personal use You are allowed to attach additional notes, corrections, and additions, as long as they are clearly identified as not being part of the original document nor written by its author Distribution of this document for pay or for any other economic gain to a general audience without permission is strictly prohibited As an exception, its unmodified web pages may be linked to freely, and may be displayed within your own frames, even for gain Conversions to html of the pdf version of this document are stupid, since there is a much better native html version already available, so try not to it iii Dedication To my parents v Preface Why Another Book on Quantum Mechanics? With the current emphasis on nanotechnology, quantum mechanics is becoming increasingly essential to engineering students Yet, the typical quantum mechanics texts for physics students are not written in a style that most engineering students would likely feel comfortable with Furthermore, an engineering education provides very little real exposure to modern physics, and introductory quantum mechanics books little to fill in the gaps The emphasis tends to be on the computation of specific examples, rather than on discussion of the broad picture Undergraduate physics students may have the luxury of years of further courses to pick up a wide physics background, engineering graduate students not really In addition, the coverage of typical introductory quantum mechanics books does not emphasize understanding of the larger-scale quantum system that a density functional computation, say, would be used for Hence this book, written by an engineer for engineers As an engineering professor with an engineering background, this is the book I wish I would have had when I started learning real quantum mechanics a few years ago The reason I like this book is not because I wrote it; the reason I wrote this book is because I like it This book is not a popular exposition: quantum mechanics can only be described properly in the terms of mathematics; suggesting anything else is crazy But the assumed background in this book is just basic undergraduate calculus and physics as taken by all engineering undergraduates There is no intention to teach students proficiency in the clever manipulation of the mathematical machinery of quantum mechanics For those engineering graduate students who may have forgotten some of their undergraduate calculus by now, there are some quick and dirty reminders in the notations For those students who may have forgotten some of the details of their undergraduate physics, frankly, I am not sure whether it makes much of a difference The ideas of quantum mechanics are that different from conventional physics But the general ideas of classical physics are assumed to be known I see no reason why a bright undergraduate student, having finished calculus and physics, should not be able to understand this book A certain maturity might help, though There are a lot of ideas to absorb vii My initial goal was to write something that would “read like a mystery novel.” Something a reader would not be able to put down until she had finished it Obviously, this goal was unrealistic I am far from a professional writer, and this is quantum mechanics, after all, not a murder mystery But I have been told that this book is very well written, so maybe there is something to be said for aiming high To prevent the reader from getting bogged down in mathematical details, I mostly avoid nontrivial derivations in the text Instead I have put the outlines of these derivations in notes at the end of this document: personally, I enjoy checking the correctness of the mathematical exposition, and I would not want to rob my students of the opportunity to so too While typical physics texts jump back and forward from issue to issue, I thought that would just be distracting for my audience Instead, I try to follow a consistent approach, with as central theme the method of separation-of-variables, a method that most mechanical graduate students have seen before already It is explained in detail anyway To cut down on the issues to be mentally absorbed at any given time, I purposely avoid bringing up new issues until I really need them Such a just-in-time learning approach also immediately answers the question why the new issue is relevant, and how it fits into the grand scheme of things The desire to keep it straightforward is the main reason that topics such as Clebsch-Gordan coefficients (except for the unavoidable introduction of singlet and triplet states) and Pauli spin matrices have been shoved out of the way to a final chapter My feeling is, if I can give my students a solid understanding of the basics of quantum mechanics, they should be in a good position to learn more about individual issues by themselves when they need them On the other hand, if they feel completely lost in all the different details, they are not likely to learn the basics either I also try to go slow on the more abstract vector notation permeating quantum mechanics, usually phrasing such issues in terms of a specific basis Abstract notation may seem to be completely general and beautiful to a mathematician, but I not think it is going to be intuitive to a typical engineer Knowledgeable readers may also note that I try to stay clear of abstract mathematical models if I can For example, the discussion of solids avoids the usual Kronig-Penney or Dirac combs sort of models in favor of a physical discussion of realistic one-dimensional crystals I try to be as consistent as possible Electrons are grey tones at the initial introduction of particles, and so they stay through the rest of the book Nuclei are red dots Occupied quantum states are red, empty ones grey That of course required all figures to be custom made They are not intended to be fancy but consistent and clear When I derive the first quantum eigenfunctions, for a pipe and for the harmonic oscillator, I make sure to emphasize that they are not supposed to look like anything that we told them before It is only natural for students to want to relate what we told them before about the motion to the completely different story we are telling them now So it should be clarified viii that (1) no, they are not going crazy, and (2) yes, we will eventually explain how what they learned before fits into the grand scheme of things Another difference of approach in this book is the way it treats classical physics concepts that the students are likely unaware about, such as canonical momentum, magnetic dipole moments, Larmor precession, and Maxwell’s equations They are largely “derived“ in quantum terms, with no appeal to classical physics I see no need to rub in the student’s lack of knowledge of specialized areas of classical physics if a satisfactory quantum derivation is readily given This book is not intended to be an exercise in mathematical skills Review questions are targeted towards understanding the ideas, with the mathematics as simple as possible I also try to keep the mathematics in successive questions uniform, to reduce the algebraic effort required I know Finally, this document faces the very real conceptual problems of quantum mechanics headon, including the collapse of the wave function, the indeterminacy, the nonlocality, and the symmetrization requirements The usual approach, and the way I was taught quantum mechanics, is to shove all these problems under the table in favor of a good sounding, but upon examination self-contradictory and superficial story Such superficiality put me off solidly when they taught me quantum mechanics, culminating in the unforgettable moment when the professor told us, seriously, that the wave function had to be symmetric with respect to exchange of bosons because they are all truly the same, and then, when I was popping my eyes back in, continued to tell us that the wave function is not symmetric when fermions are exchanged, which are all truly the same I would not the same to my own students And I really not see this professor as an exception Other introductions to the ideas of quantum mechanics that I have seen left me similarly unhappy on this point One thing that really bugs me, none had a solid discussion of the many worlds interpretation This is obviously not because the results would be incorrect, (they have not been contradicted for half a century,) but simply because the teachers just not like these results I not like the results myself, but basing teaching on what the teacher would like to be true rather on what the evidence indicates is true remains absolutely unacceptable in my book And I hope this book will manage to convince you that quantum mechanics is a very fascinating subject, whatever you think of it Our ancestors have ferreted out impressive details about how nature works, mainly out of plain curiosity Its benefits gave us the technological world we live in as well as the leisure time to appreciate what they did Going the next step is what being an engineer is all about ix Acknowledgments This book is for a large part based on my reading of the excellent book by Griffiths, [3] It includes a concise summary of the material of Griffiths’ chapters 1-5 (about 250 pages), written by an engineer who was learning the material himself at the time Somewhat to my surprise, I find that my coverage actually tends to be closer to Yariv’s book, [10] I still think Griffiths is more readable for an engineer, though Yariv has some items Griffiths does not The discussions on two-state systems are mainly based on Feynman’s notes, [2, chapters 811] Since it is hard to determine the precise statements being made, much of that has been augmented by data from web sources, mainly those referenced The nanomaterials lectures of colleague Anter El-Azab that I audited inspired me to add a bit on simple quantum confinement to the first system studied, the particle in the box That does add a bit to a section that I wanted to keep as simple as possible, but then I figure it also adds a sense that this is really relevant stuff for future engineers I also added a discussion of the effects of confinement on the density of states to the section on the free electron gas I thank Swapnil Jain for pointing out that the initial subsection on quantum confinement in the pipe was definitely unclear and is hopefully better now I thank Johann Joss for pointing out a mistake in the formula for the averaged energy of two-state systems The section on solids is mainly be based on Sproull, [8], a good source for practical knowledge about application of the concepts It is surprisingly up to date, considering it was written half a century ago Various items, however, come from Kittel [4] The discussion of ionic solids really comes straight from hyperphysics [3] I prefer hyperphysics’ example of NaCl, instead of Sproull’s equivalent discussion of KCl The section on the Born-Oppenheimer approximation comes from Wikipedia, [8}, with modifications including the inclusion of spin The section on the Hartree-Fock method is mainly based on Szabo and Ostlund [9], a wellwritten book, with some Parr and Yang [6] thrown in The many-worlds discussion is based on Everett’s exposition, [1] It is brilliant but quite impenetrable x • Generic summation index over other eigenfunctions • Integer factor in Fourier wave numbers • Probability density • Number of particles per state • A generic index • A natural number and maybe some other stuff natural Natural numbers are the numbers: 1, 2, 3, 4, P May indicate: • The linear momentum eigenfunction • A power series solution p May indicate: • Linear momentum • Linear momentum in the x-direction • Integration variable with units of linear momentum p Energy state with orbital azimuthal quantum number l = perpendicular bisector For two given points P and Q, the perpendicular bisector consists of all points R that are equally far from P as they are from Q In two dimensions, the perpendicular bisector is the line that passes through the point exactly half way in between P and Q, and that is orthogonal to the line connecting P and Q In three dimensions, the perpendicular bisector is the plane that passes through the point exactly half way in between P and Q, and that is orthogonal to the line connecting P and Q In vector notation, the perpendicular bisector of points P and Q is all points R whose radius vector r satisfies the equation: (r − rP ) · (rQ − rP ) = 21 (rQ − rP ) · (rQ − rP ) (Note that the halfway point r − rP = 21 (rQ − rP ) is included in this formula, as is the half way point plus any vector that is normal to (rQ − rP ).) photon Unit of electromagnetic radiation (which includes light, x-rays, microwaves, etcetera) A photon has a energy h ¯ ω, where ω is its natural frequency, and a wave length 2πc/ω where c is the speed of light 415 px Linear momentum in the x-direction (In the one-dimensional cases at the end of the unsteady evolution chapter, the x subscript is omitted.) Components in the y- and z-directions are py and pz Classical Newtonian physics has px = mu where m is the mass and u the velocity in the x-direction In quantum mechanics, the possible values of px are the eigenvalues of the operator px which equals h ¯ ∂/i∂x (But which becomes canonical momentum in a magnetic field.) q Charge R May indicate: • Some function of r to be determined • Some function of (x, y, z) to be determined • Rnl is a hydrogen radial wave function • Ru = 8.314472 kJ/kmol K is the universal gas constant, the equivalent of Boltzman’s constant for a kmole instead of a single atom or molecule • Rotation operator r May be radial distance from the chosen origin of the coordinate system ˆ In spherical coordir The position vector In Cartesian coordinates (x, y, z) or xˆı + yˆ  + z k nates rˆır S May indicate: • Number of states per unit volume • Number of states at a given energy level • Spin angular momentum (as an alternative to using L for generic angular momentum.) s Energy state with orbital azimuthal quantum number l = Spherically symmetric s Spin value of a particle Equals 1/2 for electrons, protons, and neutrons, is also half an odd natural number for other fermions, and is a nonnegative integer for bosons It is the azimuthal quantum number l due to spin scalar A quantity that is not a vector, a quantity that is just a single number sin The sine function, a periodic function oscillating between and -1 as shown in [7, pp 40-] Good to remember: cos2 α + sin2 α = Stokes’ Theorem This theorem, first derived by Kelvin and first published by someone else I cannot recall, says that for any reasonably smoothly varying vector v, S (∇ × v) dS = 416 v · dr where the first integral is over any smooth surface S and the second integral is over the edge of that surface How did Stokes get his name on it? He tortured his students with it, that’s why! symmetry Symmetries are operations under which an object does not change For example, a human face is almost, but not completely, mirror symmetric: it looks almost the same in a mirror as when seen directly The electrical field of a single point charge is spherically symmetric; it looks the same from whatever angle you look at it, just like a sphere does A simple smooth glass (like a glass of water) is cylindrically symmetric; it looks the same whatever way you rotate it around its vertical axis T May indicate: • Kinetic energy A hat indicates the associated operator The operator is given by the Laplacian times −¯ h2 /2m • Absolute temperature The absolute temperature in degrees K equals the temperature in centigrade plus 273.15 When the absolute temperature is zero, (i.e at −273.15◦ C), nature is in the state of lowest possible energy t The time temperature A measure of the heat motion of the particles making up macroscopic objects At absolute zero temperature, the particles are in the “ground state” of lowest possible energy u May indicate: • The first velocity component in a Cartesian coordinate system • A complex coordinate in the derivation of spherical harmonics V The potential energy V is used interchangeably for the numerical values of the potential energy and for the operator that corresponds to multiplying by V In other words, V is simply written as V v May indicate: • The second velocity component in a Cartesian coordinate system • A complex coordinate in the derivation of spherical harmonics • As v ee , a single electron pair potential v May indicate: • Velocity vector • Generic vector • Summation index of a lattice potential 417 vector A list of numbers A vector v in index notation is a set of numbers {vi } indexed by an index i In normal three-dimensional Cartesian space, i takes the values 1, 2, and 3, making the vector a list of three numbers, v1 , v2 , and v3 These numbers are called the three components of v The list of numbers can be visualized as a column, and is then called a ket vector, or as a row, in which case it is called a bra vector This convention indicates how multiplication should be conducted with them A bra times a ket produces a single number, the dot product or inner product of the vectors:     (1, 3, 5)  11  = + 11 + 13 = 105 13 To turn a ket into a bra for purposes of taking inner products, write the complex conjugates of its components as a row vectorial product An vectorial product, or cross product is a product of vectors that produces another vector If c = a × b, it means in index notation that the i-th component of vector c is c i = aı b ı − aı b ı where ı is the index following i in the sequence 123123 , and ı the one preceding it For example, c1 will equal a2 b3 − a3 b2 w May indicate the third velocity component in a Cartesian coordinate system w Generic vector X Used in this document to indicate a function of x to be determined x May indicate: • First coordinate in a Cartesian coordinate system • A generic argument of a function • An unknown value Y Used in this document to indicate a function of y to be determined Ylm Spherical harmonic Eigenfunction of both angular momentum in the z-direction and of total square angular momentum y May indicate: • Second coordinate in a Cartesian coordinate system • A generic argument of a function 418 Z May indicate: • Number of particles • Atomic number (number of protons in the nucleus) • Used in this document to indicate a function of z to be determined z May indicate: • Third coordinate in a Cartesian coordinate system • A generic argument of a function 419 Index F , 409 N , 414 T , 417 ·, 399 ×, 399 !, 399 |, 399 ↑, 399 ↓, 399 Σ, , 400 , 400 →, 400 , 400 , 400 , 400 ∇, 400 ∗ , 401 , 401 [ .], 401 ≡, 401 ∼, 401 α, 401 β, 402 γ, 402 ∆, 402 δ, 402 ∂, 402 , 403 , 403 ε, 403 η, 403 Θ, 403 θ, 403 ϑ, 403 κ, 403 λ, 403 µ, 403 ν, 404 ξ, 404 π, 404 ρ, 404 σ, 404 τ , 404 Φ, 404 φ, 404 ϕ, 405 χ, 405 Ψ, 405 ψ, 405 ω, 405 A, 405 ˚ A, 405 a, 405 a0 , 406 absolute value, absolute zero nonzero energy, 47 acceleration in quantum mechanics, 227 adiabatic surfaces, 304 Aharonov-Bohm effect, 278 angular momentum, 57 definition, 57 eigenstate normalization factors, 262 ladder operators, 257 ladders, 257 possible values, 260 uncertainty, 63 angular momentum commutation relations, 256 angular momentum components, 58 421 antisymmetrization for fermions, 121 atomic number, 128 atoms eigenfunctions, 128 eigenvalues, 128 ground state, 129 Hamiltonian, 128 Avogadro’s number, 414 azimuthal quantum number, 62 Brillouin fragment boundaries, 169 energy singularities, 200 one dimension (points), 162 Bragg’s law, 213 Brillouin zone one-dimensional, 161 three-dimensional, 169 C, 406 c, 406 canonical commutation relation, 87 canonical Hartree-Fock equations, 314 cat, Schr¨odinger’s, 24 cauchy-schwartz inequality, 406 chemical bonds, 136 covalent pi bonds, 137 covalent sigma bonds, 137 hybridization, 139 ionic bonds, 141 polar covalent bonds, 138 promotion, 139 spn hybridization, 139 chemical potential, 203 classical, 407 Clebsch-Gordan coefficients, 265 coefficients of eigenfunctions evaluating, 53 give probabilities, 23 collapse of the wave function, 21 commutation relation canonical, 87 commutation relations fundamental, 256 commutator, 84 definition, 86 commutator eigenvalue problems, 258 commuting operators, 84 common eigenfunctions, 84 complete set, 12 complex conjugate, complex numbers, component waves, 242 components of a vector, conduction band, 153 conduction of electricity B, 406 b, 406 Balmer transitions, 70 band gap, 153 band theory intro, 152 basis lithium, 154 NaCl, 151 bcc lattice, 154 Bell, 112 binding energy definition, 96 black body radiation, 205 Bloch function free electron gas, 182 one-dimensional, 160 three-dimensional, 168 Bloch’s theorem, 160 Bohm, 112 Bohr radius, 72 bond length definition, 96 Born’s statistical interpretation, 16 Born-Oppenheimer approximation, 90, 298 Born-Oppenheimer diagonal correction, 375 Bose-Einstein distribution, 204 bosons, 111 statistics, 204 bra, Bragg diffraction electrons, 218 Bragg plane X-ray diffraction, 218 Bragg planes 422 intro, 152 confined electrons, 172 confinement, 39 density of states, 178 conservation laws, 233 conventional cell, 166 Copenhagen Interpretation, 21 correlation energy, 320 cos, 407 Coulomb integrals, 313 Coulomb potential, 64 covalent bond hydrogen molecular ion, 90 covalent solids, 169 cross product, 418 crystal basis for lithium, 154 basis for NaCl, 151 bcc lattice, 154 fcc lattice, 151 one-dimensional primitive translation vector, 159 three-dimensional primitive translation vectors, 166 curl, 401 divergence theorem, 410 dot product, doublet states, 121 E, 409 e, 409 effective mass hydrogen atom electron, 65 Ehrenfest’s theorem, 227 eiax , 409 eigenfunction, 10 eigenfunctions angular momentum components, 58 atoms, 128 free electron gas, 175 harmonic oscillator, 48 hydrogen atom, 72 linear momentum, 239 position, 237 square angular momentum, 60 eigenvalue, 10 eigenvalue problems commutator type, 258 ladder operators, 258 eigenvalues angular momentum components, 58 atoms, 128 free electron gas, 175 harmonic oscillator, 46 hydrogen atom, 69 linear momentum, 239 position, 237 square angular momentum, 60 eigenvector, 10 Einstein dice, 23 Einstein Podolski Rosen, 113 electric charge electron and proton, 64 electric dipole approximation, 231 electricity conduction intro, 152 electromagnetic field Hamiltonian, 276 d, 407 D, 407 d, 407 D, 407 d, 407 Debye temperature, 206 degeneracy, 51 degeneracy pressure, 172 delta function, 238 density of states, 178 derivative, 408 determinant, 408 dipole strength molecules, 144 Dirac delta function, 238 Dirac equation, 271 Dirac notation, 14 div, 401 divergence, 401 423 Floquet theory, 160 Fock operator, 314 forbidden transitions, 231 force in quantum mechanics, 227 Fourier analysis, 161 free electron gas, 172 Bloch function, 182 eigenfunctions, 175 eigenvalues, 175 energy spectrum, 176 ground state, 176 Hamiltonian, 173 function, 4, 5, 410 functional, 410 fundamental commutation relations, 256 Maxwell’s equations, 278 electron in magnetic field, 285 electron affinity, 149 Hartree-Fock, 317 electronegativity, 134, 149 atoms, 132 energy conservation, 222 energy spectrum banded, 152 free electron gas, 176 harmonic oscillator, 46 hydrogen atom, 69 solids, 152 energy-time uncertainty principle, 225 EPR, 113 equipartition theorem, 207 Euler identity, eV, 409 Everett, III, 328 every possible combination, 99 exchange integrals, 313 exchange terms twilight terms, 110 excited determinants, 320 exclusion-principle repulsion, 135 expectation value, 76 definition, 79 simplified expression, 80 exponential function, 409 extended zone scheme, 191 g, 410 g-factor, 285 Gauss’ theorem, 410 generalized uncertainty relationship, 86 grad, 400 gradient, 400 grain, 155 grain boundaries, 155 ground state atoms, 129 free electron gas, 176 harmonic oscillator, 48 hydrogen atom, 70, 72 hydrogen molecular ion, 97 hydrogen molecule, 107, 119, 122 nonzero energy, 47 group velocity, 245 gyromagnetic ratio, 285 f , 410 fcc lattice, 152 Fermi surface confined boundary conditions, 176 periodic boundary conditions, 189 periodic zone scheme, 192 reduced zone scheme, 191 Fermi-Dirac distribution, 203 fermions, 111 statistics, 203 Fine structure, 322 fine structure constant, 322 flopping frequency, 292 H, 410 h, 411 Hamiltonian, 20 and physical symmetry, 234 atoms, 128 electromagnetic field, 276 free electron gas, 173 gives time variation, 221 harmonic oscillator, 42 424 partial, 43 hydrogen atom, 64 hydrogen molecular ion, 90 hydrogen molecule, 101 in matrix form, 127 numbering of eigenfunctions, 20 one-dimensional free space, 241 solids, 188 harmonic oscillator, 41 classical frequency, 42 eigenfunctions, 48 eigenvalues, 46 energy spectrum, 46 ground state, 48 Hamiltonian, 42 partial Hamiltonian, 43 particle motion, 250 Hartree product, 306 Hartree-Fock, 305 Coulomb integrals, 313 exchange integrals, 313 restricted closed shell, 308 open shell, 308 unrestricted, 308 h ¯ , 411 Heisenberg uncertainty principle, 17 Heisenberg uncertainty relationship, 87 Hermitian adjoint, 414 Hermitian operators, 11 hidden variables, 23, 113 hidden versus nonexisting, 63 hybridization, 139 hydrogen atom, 64 eigenfunctions, 72 eigenvalues, 69 energy spectrum, 69 ground state, 70, 72 Hamiltonian, 64 hydrogen bonds, 138, 145 hydrogen molecular ion, 90 bond length, 97 experimental binding energy, 97 ground state, 97 Hamiltonian, 90 shared states, 93 hydrogen molecule, 100 binding energy, 107 bond length, 107 ground state, 107, 119, 122 Hamiltonian, 101 I, 411 i, 1, 411 inverse, i index, i, 411 identical particles, 121 iff, 8, 411 imaginary part, index notation, 411 inner product multiple variables, 14 inner product of functions, inner product of vectors, integer, 411 interpretation interpretations, 22 many worlds, 327 orthodox, 21 relative state, 327 statistical, 21 ionic bonds, 141 ionic molecules, 148 ionic solids, 148 ionization, 70 ionization energy, 149 atoms, 132 Hartree-Fock, 316 hydrogen atom, 70 J, 411 j, 411 K, 412 K, 412 k, 412 kB , 412 ket, ket notation spherical harmonics, 61 425 spin states, 111 kinetic energy operator, 19 kinetic energy operator in spherical coordinates, 64 kmol, 412 Koopman’s theorem, 316 M , 413 m, 413 me , 413 mp , 413 Madelung constant, 150 magnetic dipole moment, 285 magnetic quantum number, 59 magnitude, matrix, 9, 413 Maxwell’s equations, 278 Maxwell-Boltzmann distribution, 205 measurable values, 21 measurement, 22 metals, 153 molecular solids, 143 molecules ionic, 148 momentum space wave function, 240 L, 412 L, 412 l, 412 , 412 ladder operators angular momentum, 257 Lagrangian multipliers, 297 Laplacian, 401 Larmor frequency definition, 289 Larmor precession, 291 laser, 230 lattice body centered cubic, 154 face centered cubic, 152 one-dimensional primitive translation vector, 159 reciprocal primitive translation vectors, 169 three-dimensional primitive translation vectors, 166 law of Dulong and Petit, 206 length of a vector, Lennard-Jones potential, 143 light waves classical, 284 lim, 413 linear momentum classical, 17 eigenfunctions, 239 eigenvalues, 239 operator, 19 localization absence of, 243 London forces, 143 Lyman transitions, 70 N, 414 n, 414 nabla, 400 natural, 415 nearly free electron model, 184 Newton’s second law in quantum mechanics, 227 Newtonian analogy, 19 Newtonian mechanics, 15 in quantum mechanics, 226 noble gas, 131 noncanonical Hartree-Fock equations, 386 nonexisting versus hidden, 63 norm of a function, normalized, normalized wave functions, 16 nuclear magnetic resonance, 287 observable values, 21 one-dimensional free space Hamiltonian, 241 operators, angular momentum components, 58 Hamiltonian, 20 kinetic energy, 19 in spherical coordinates, 64 M, 413 426 linear momentum, 19 position, 19 potential energy, 20 quantum mechanics, 19 square angular momentum, 60 total energy, 20 orbitals, 305 orthodox interpretation, 21 orthogonal, orthonormal, primitive translation vector one-dimensional, 159 reciprocal lattice, 169 three-dimensional, 166 principal quantum number, 66 probabilities evaluating, 53 from coefficients, 23 probability density, 102 probability to find the particle, 16 promotion, 139 px , 415 P , 415 p states, 73 p, 415 p-state, 415 Paschen transitions, 70 Pauli exclusion principle, 125 atoms, 131 common phrasing, 132 free electron gas, 172 Pauli repulsion, 135 Pauli spin matrices, 269 periodic zone scheme, 192 permittivity of space, 64 perpendicular bisector, 415 phonons, 206 photon, 415 physical symmetry commutes with Hamiltonian, 234 pi bonds, 137 Planck formula, 70 Planck’s constant, 19 pointer states, 74 polar bonds, 138 poly-crystalline, 155 population inversion, 230 position eigenfunctions, 237 eigenvalues, 237 operator, 19 possible values, 21 potential energy operator, 20 potential energy surfaces, 304 primitive cell, 166 q, 416 quantum dot, 40 quantum mechanics acceleration, 227 force, 227 Newton’s second law, 227 Newtonian mechanics, 226 velocity, 226 wave packet velocity, 244 quantum well, 40 quantum wire, 40 R, 416 r, 416 r, 416 Rabi flopping frequency, 292 random number generator, 23 real part, reciprocal lattice one-dimensional, 161 three-dimensional, 169 reduced zone scheme, 191 relative state formulation, 330 relative state interpretation, 327 Relativistic effects Dirac equation, 271 resonance factor, 292 restricted Hartree-Fock, 308 RHF, 308 rot, 401 S, 416 s state, 416 427 s states, 73 scalar, 416 scattering, 251 Schr¨odinger equation, 221 failure?, 325 Schr¨odinger’s cat, 24 self-consistent field method, 315 separation of variables, 42 for atoms, 128 for free electron gas, 173 linear momentum, 239 position, 237 shielding approximation, 128 sigma bonds, 137 simple cubic lattice, 187 sin, 416 singlet state, 120 derivation, 263 Slater determinants, 125 small perturbation theory, 184, 188 solids, 143 band theory intro, 152 covalent, 169 energy spectrum, 152 Hamiltonian, 188 ionic, 148 molecular, 143 n sp hybridization, 139 specific heat, 206 spectral line broadening, 233 spectrum hydrogen, 71 spherical coordinates, 58 spherical harmonics derivation, 362 spin, 110 value, 111 x- and y-eigenstates, 270 spin down, 111 spin orbitals, 305 spin states ambiguity in sign, 366 axis rotation, 365 spin up, 111 spinor, 117 square angular momentum, 60 eigenfunctions, 60 eigenvalues, 60 standard deviation, 76 definition, 78 simplified expression, 80 stationary states, 223 statistical interpretation, 21 statistical mechanics, 203 statistics bosons, 204 fermions, 203 Stokes’ theorem, 416 superluminal interactions, 111 symmetrization requirement identical bosons, 121 identical fermions, 121 symmetry, 417 t, 417 temperature, 417 temperatures above absolute zero, 203 throw the dice, 23 time variation Hamiltonian, 221 total energy operator, 20 transitions hydrogen atom, 70 transpose of a matrix, 409 triplet states, 120 derivation, 263 tunneling, 252 twilight terms, 109 exchange terms, 110 Hartree-Fock, 313 particle exchange, 224 two state systems ground state energy, 107 time variation, 224 unsteady perturbations, 227 u, 417 UHF, 308 428 free space, 241, 248 harmonic oscillator, 250 partial reflection, 251 physical interpretation, 243 reflection, 249 Wigner-Seitz cell, 167 uncertainty principle angular momentum, 63 energy, 50, 223 Heisenberg, 17 position and linear momentum, 17 uncertainty relationship generalized, 86 Heisenberg, 87 unit cell bcc, 154 fcc, 152 unit matrix, 414 universal gas constant, 206 unrestricted Hartree-Fock, 308 X, 418 x, 418 X-ray diffraction, 213 Y , 418 y, 418 Ylm , 418 Z, 418 z, 419 zero matrix, 414 zero point energy, 302 V , 417 v, 417 v, 417 vacuum energy, 393 valence band, 153 values measurable, 21 observable, 21 possible, 21 Van der Waals forces, 143 variational method, 95 vector, 4, 417 vectorial product, 418 velocity in quantum mechanics, 226 wave packet, 244 vibronic coupling terms, 374 virial theorem, 225 w, 418 w, 418 wave function, 15 multiple particles, 99 multiple particles with spin, 119 with spin, 116 wave number vector Bloch function, 168 wave packet accelerated motion, 249 definition, 243 429 [...]... discussions I am an engineer Feedback can be e-mailed to me at quantum@ dommelen.net This is a living document I am still adding some things here and there, and fixing various mistakes and doubtful phrasing Even before every comma is perfect, I think the document can be of value to people looking for an easy to read introduction to quantum mechanics at a calculus level So I am treating it as software,... radial wave functions for hydrogen, from [3, p 154] 61 68 4.1 Abbreviated periodic table of the elements, showing element symbol, atomic number, ionization energy, and electronegativity 132 xxvii Chapter 1 Mathematical Prerequisites Quantum mechanics is based on a number of advanced mathematical ideas that are described in this chapter 1.1 Complex Numbers Quantum mechanics is full... PREREQUISITES Functions as Vectors The second mathematical idea that is critical for quantum mechanics is that functions can be treated in a way that is fundamentally not that much different from vectors A vector f (which might be velocity v, linear momentum p = mv, force F , or whatever) is usually shown in physics in the form of an arrow: Figure 1.1: The classical picture of a vector However, the same... So, any complex number can be written in “polar form” as c = |c|eiα (1.6) where both the magnitude |c| and the angle α are real numbers Any complex number of magnitude one can therefor be written as eiα Note that the only two real numbers of magnitude one, 1 and −1, are included for α = 0, respectively α = π The number i is obtained for α = π/2 and −i for α = −π/2 (See note {A.1.1} if you want to know... 3 beta 3.2 adds a section on Hartree-Fock It took forever My main regret is that most of them who wasted my time in this major way are probably no longer around to be properly blasted Writing a book on quantum mechanics by an engineer for engineers is a minefield of having to see through countless poor definitions and dubious explanations It takes forever In view of the fact that many of those physicist... | or simply by f , is normally computed as |f | = f ·f = fi2 all i However, this does not work correctly for complex vectors The difficulty is that terms of the form fi2 are no longer necessarily positive numbers For example, i2 = −1 Therefore, it is necessary to use a generalized “inner product” for complex vectors, which puts a complex conjugate on the first vector: f |g ≡ fi∗ gi all i (1.7) 1.3 THE... Operators 1.7 Additional Points 1.7.1 Dirac notation 1.7.2 Additional independent variables 2 Basic Ideas of Quantum Mechanics 2.1 The Revised Picture of Nature 2.2 The Heisenberg Uncertainty Principle 2.3 The Operators of Quantum Mechanics 2.4 The Orthodox Statistical Interpretation 2.4.1 Only eigenvalues 2.4.2 Statistical selection 2.5 Schr¨odinger’s... confinement [Advanced] 5.6.6 Relation to Bloch functions Nearly-Free Electrons [Advanced] 5.7.1 The lattice structure 5.7.2 The small perturbation approach 5.7.3 Zeroth order solution 5.7.4 First order solution 5.7.5 Second order solution 5.7.6 Discussion of the energy changes Quantum Statistical Mechanics. .. Method [Advanced] 7.5.1 Basic variational statement 7.5.2 Differential form of the statement 7.5.3 Example application using Lagrangian multipliers 7.6 The Born-Oppenheimer Approximation [Advanced] 7.6.1 The Hamiltonian 7.6.2 The basic Born-Oppenheimer approximation 7.6.3 Going one better 7.7 The Hartree-Fock Approximation [Advanced]... coefficients for lb equal to one half Clebsch-Gordan coefficients for lb equal to one Relationship of Maxwell’s first equation to Coulomb’s law Maxwell’s first equation for a more arbitrary region The figure to the right includes the field lines through the selected points The net number of field lines leaving a region is a measure for the net ...Copyright Copyright 2004 and on, Leon van Dommelen You are allowed to copy or print out this work in unmodified work for your personal use... 5.6.5 The density of states and confinement [Advanced] 5.6.6 Relation to Bloch functions Nearly-Free Electrons [Advanced] 5.7.1 The lattice structure ... Relativistic Dirac Equation [Advanced] 7.2.1 The Dirac idea 7.2.2 Emergence of spin from relativity 7.3 The Electromagnetic Field [Advanced] 7.3.1 The