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# RSCH ASTR 0297

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This 19 page Class Notes was uploaded by Olga Monahan on Friday September 4, 2015. The Class Notes belongs to ASTR 0297 at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 104 views. For similar materials see /class/177912/astr-0297-university-of-california-los-angeles in Astronomy at University of California - Los Angeles.

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Date Created: 09/04/15

Notes on Quantum Mechanics Andrew Forrester January 287 2009 Contents 1 Ideas and Questions 11 Transfer Quicknotes info into this le 12 Ideas 13 Fundamental Questions 14 More Questions 15 Answered or Partially Answered Questions 2 The Big Picture 3 Terms and Notation 31 Quantity Overview 32 Mathematical Vocabulary 33 Basics 34 Dirac Notation and Terminology 35 Continuing 36 From My Quicknotes 361 Physics Terms 362 Mathematical terms 363 Equations7 Formulas7 or Rules 364 Theorems 365 Principles 366 Approximations 367 Effects 368 Questions 4 Theoretical Summary 5 Early Quantum Theory 6 Transformation from Classical to Quantum Mechanics 61 Postulates 62 Quantization Methods 63 Interpretations 64 Other Ideas 7 Representations Expansions and Approximations 71 Schrodinger7 Heisenberg7 and Interaction Dirac Pictures 72 Special Functions7 Et Cetera 8 Angular Momenta 9 Spin 91 Pauli Spin Matrices 92 Spin Phenomena gt5 HgtOJOJOJOJCO x1xi tntn4gtugt H 15 16 16 10 Transition Theory See HydrogenTrans Lecture 101 Bound State Perturbation Theory 102 Scattering Theory 1021 Potential Scattering7 Born Series 103 Collision Theory subtopic of scattering 104 Decays of Excited States 1041 Radioactive Decay 11 Equations Laws and Formulas 12 Applications and Phenomena 13 Advanced Topics 131 Phenomenology 14 Problem Solving 19 19 20 1 Ideas and Questions 11 Transfer Quicknotes info into this le 12 Ideas 0 We start with a knowledge of advanced Classical Mechanics and Electrodynamics and perhaps some basic macroscopic Thermodynamics 0 An historical approach takes us through the old quantum theory and then into the newer theory nally entering second quantization QFT will have its own notes 0 Mathematical approach Present recipe for moving from Classical Mechanics to Quantum Mechanics present additional postulates and develop important applications 0 Many philosophical questions to be asked and various possible answers to be listed 13 Fundamental Questions 0 What are the most fundamental postulates Should derive P imam andor XP 27L 0 Why do we use complex numbers Where does the convenience rst show up Look at Sakurai again 0 Why do we declare momentum to be related to the argument of the wavefunction and position to be related to the magnitude rather than imaginary part versus real part c What are the relevant ideas from linear algebra in particular in nite dimension spaces various types of Hilbert spaces and other applicable spaces 0 With Chris s simulation of the doubleslit experiment in mind Could we perform the doubleslit experiment assuring that the same amount of energy is taken from the electron each time it is measured so that coherence is maintained and the diffraction pattern does not disappear o Is it possible that wavefunctions do not collapse 14 More Questions 0 Can the sea of vacuum uctuations be considered as a kind of aetherial medium 0 The Schrodinger Equation is like a conservation of energy equation E II or H I ih ll Why should interaction terms be placed in the Hamiltonian Could they be placed on one side of the equation representing a system with interactions where energy is not conserved Is there already a convention from HamiltonianLagrangian formalism which provides for interaction terms in the Hamil tonian When a Hamiltonian has offdiagonal interaction terms that merely means that the Hamiltonian is not being expressed in terms of eigenenergy states Diagonalize it and there will be no interaction terms 0 Hey what s Hund s Rule Hund s Rules http hyperphysics phyastr gsu eduhbaseatomichundhtmlcl 0 Term Symbols http hyperphysics phyastr gsu eduhbaseatomicterm htmlcl o How are the Bohr Sommerfeld quantizations conditions equivalent to Heisenberg s quantum mechanics to rst order 0 When should the minimal coupling rule p a p eAr be used and why is it called that Minimal coupling rule for canonical generalized momentum in relativistic notation 7m mi AMm7 7139 15 Answered or Partially Answered Questions 0 Why is the Hamiltonian formalism used but not the Lagrangian formalism Lagrangian Formalism a Path lntegral Formalism 2 The Big Picture Topics Sulotopics7 Supertopics7 Sulo elds7 Super elds7 Context of Physics as a whole 0 Quantum Statics Not really static due to uncertainty principle7 which is due to 0 Quantum Dynamics Domains 0 Classical versus Quantum o Relativistic versus Non relativistic 3 Terms and Notation 31 Quantity Overview Symbol Name Units Notes 3 angular momentum operator kg mZs 715 or n E orbital angular momentum operator kgm2s g spin angular momentum operator kg mZs magnetic dipole moment operator Am2 39yp gyromagnetic ratio for particle p Ckg pJ Am2m kgms Ckg gp g factor dimensionless X helicity kg mZs angular momentum The hat notation V denotes an operator rather than7 say7 a unit vector Some people de ne the quantum operators j 1 and g to be dimensionless Dimensionless Quantum Numbers qn s Symbol Name Notes 71 principal qn for hydrogen n 6 2 1 2 3 N radial qn for hydrogen j total angular momentum qn Z orbital angular momentum qn Z 6 0 1 2 n 7 1 for hydrogen aka orbital azimuthal or angular qn 3 spin angular momentum qn s E 0 1 g 2 g s for electron in hydrogen atom m mj secondary total angular momentum qn mj mg m5 total magnetic qn mmg secondary orbital angular momentum qn mg 6 76 76 1 76 2 6 7 2 6 716 aka magnetic qn orbital magnetic qn m mS secondary spin angular momentum qn m5 6 75 7s 1 7s 2 s 7 2 s 715 spin magnetic qn The hat notation V denotes an operator rather than say a unit vector 32 Mathematical Vocabulary 33 0 Basics State Hamiltonian The Hamiltonian doesn t always represent the energy operator it may include interaction terms that cannot strictly be called potentials in the Newtonian sense where they are strictly position dependent they may be velocity dependent in minimal coupling or timedependent antihydrogen in a laser eld or Sometimes the Hamiltonian is allowed to be non Hermitian to represent dissapative systems When it corresponds to the energy operator of a system the Hamiltonian is always Hermitian from http minty caltech eduPthSQDynamicsZ pdf HyperPhysics distinguishes between a time independent Hamiltonian H f 15 and a time dependent Hamiltonian E mat httphyperphysicsphyastrgsueduhbasequantum qmoper htmlc 1 Operators Good development at http hyperphysics phy astr gsu eduhbasequantumschrZ htmlcl Schrodinger Equation Though the Schrodinger equation cannot be derived it can be shown to be consistent with experiment httphyperphysicsphyastrgsueduhbasequantum schr2 htmlcl Hilbert Space a complete inner product space countable basis implies separable otherwise in separable 3 a 35 Physical Hilbert Space V the space of functions uncountably in nite dimensional vectors that can be normalized to either unity proper vectors or the Dirac delta function improper vectors over the eld of complex numbers C Quantum Mechanical Linear Operators T V a V T E HXKamplamp Dirac Notation and Terminology Bra Ket Properties Associativity legal expressions illegal expressions semi illegal expressions C Number classical or commuting number Q Number quantum or queer number a mathematical object that does not commute in general such as a matrix or an operator Continuing Hermitian Conjugate aka Hermitian Adjoint Ql QTY for a matrix operator Just as CV Vlci ow vmf Let s say a is an eigenvector of the operator A with eigenvalue a so A a a 04 Then WNWMMWWWWKWWW wwwrwwm Hermitian Operator Q QT Orthogonal Operator QT Q l Unitary Operator QT Q l Rotation Operator 7 We assume active rotations rather than passive rotations so that the physical system is actually rotated instantaneously with respect to its original inertial reference frame I m not sure what implications this leads to about how there must be some torque on the system to accelerate it and then decelerate it that this should take some time etc RDm o The symbol D stems from the German word Drehung meaning rotation Sakurai pg 156 Symmetry Transformation a transformation on a quantum mechanical vector space that leaves the Hamiltonian unchanged such a transformation implies a conservation law From Abers pg 159 36 From My Quicknotes 361 Physics Terms First quantized quantum mechanics nonrelativistic relativistic Second quantized theory where the EM eld is quantized at every point in space later spacetime becomes a parameter as time in the rst quantized theory 7 Canonical quantization symplectic structure see Wikipedia symplectic contact geometry F robe nius s theorem foliation Particle Quasiparticle Quark Hadron Meson Nucleon Proton Neutron Lepton Electron Photon7 Positron etc Fermion Boson Exciton Photon7 Phonon Positronium Mass Effective mass Hamiltonian interaction terms in a Hamiltonian Quantum numbers State state vector Stationary state Quasistationary state or resonance Ziman pg 131 Coherent state Asymptotic state Metastable states do decay by collisions or by what are misleadingly called forbidden transitions lasers use metastable states Pure State one energy eigenstate7 Mixed State combination of two or more energy eigenstates pure mixed ensemble Ground state Lowest energy state Excited state Zero state Reference state Vacuum state Null state Empty state ket see httpenwikipediaorgwikiCreationoperator Bound state Quasi bound state see below Continuum state Wave packet normalizable wave packet We pretend to construct nrmlzbl wv pckts77 Abers pg 292 See below after these terms for wave packet explanation 7 Localized wave train See Feynman volume 1 48 8 Dressed state Teller pg 122 Separable 141275 111quotTt 141275 111quotTt Separable with spin Tr m t xm Tt Trmt 1711quot t2mt 7 With spin Trspin t T1rspin t T2rspin t T7r spint 7 13m Rr Y0 gtTtRr 0 tY0 gtt t 7 0 Symmetry of states or quantities or spatial parity being even or odd under spatial inversion aka parity transformation apparently this is not how parity is de ned in other dimensions eg 2 D see Wikipedia time parity being even or odd under time reversal time parity transformation charge parity being even or odd under charge conjugation charge parity transf Tunneling Spin spin as angular momentum spin as magnetic moment classical spins77 Callen pg 440 Valence band Conduction band Chemical potential ideal electron gas Abers pg 446 Degeneracy pressure Abers pg 447 lnelastic scattering Quasi bound state Sakurai pg 419 Scattering length Sakurai pg 414 Scattering amplitude angular something density Partial wave amplitude lnteraction Coupling volume coupling ex Callen pg 53 Three volume coupled systems Electric coupling Magnetic coupling spin ang mom orbital ang mom coupling LS coupling spin orbit interaction Lande gyroscopic factor Form factor Sakurai pg 431 nuclear form factor pg 433 Geometrical phase factor Sakurai pf 469 Adiabatic potential Sakurai pg 474 Hannay s angle Sakurai pg 480 Decay rate Decay modes Channels Differential decay rate Lifetime time to reach 15 0368 of initial value Half life Spontaneous emission emission stimulated by vacuum uctuations Stimulated emission Absorption Magnetic resonance nuclear magnetic resonance nmr Rabi cycle Rabi frequency Fast coordinates Slow nuclear coordinates Transition Theory Transformation Theory7 Scattering Radiation Collision Etc Theories Transition transition amplitude Forbidden transition Even more forbidden transition T matrix Transition matrix the T matrix satis es certain algebraic relations which provide valuable formal connections bt elementary partial wave scattering theory Green functions and the general theory of the S matrix The evolution of quantum states over very long times in particular scattering and decays of long lived excited states requires methods more powerful than those of potential scattering or the semiclassical treatment Abers pg 291 H H0 H1 it s not important that H1 be small it s important that it have a small range 921 Abers section 921 introduce a Step function for early times and a Regulator for late times S matrix Scattering matrix K matrix or Reaction matrix Ziman pg 130 physically represents the total effect of the scattering potential when at the heart of a standing wave system ie with both incoming and outgoing spherical waves as described by the propagator 4133 non causal Green function Berry s phase Degenerate degeneracy also a mathematical term Fine Structure Splitting Emission spin orbit coupling plus relativistic energy correction Darwin term contact term depends on n and j but surprisingly not on 6 Hyper ne Structure Split Emit spin spin coupling of nucleus and electron Singlet Doublet etc Multiplets parahelium orthohelium etc spectral linesquantum states Stark shift Stark broadening Lamb shift can be computed in relativistic QFT Abers pg 217 Lamb Rutherford Regularized energy Abers pg 399 Reduced matrix element Wigner Eckart theorem math term Such a system of waves forms a crest wich propagates itself with quite a different velocity from that of its component waves this velocity being the so called group velocity Such a wave crest represents a material point which is thus either formed by it or connected with it and is called a wave packet De Broglie now found that the velocity of the material point was in fact the group velocity of the matter wave http nobelprize orgnobelprizesphysicslaureateslQBSpress html 362 Mathematical terms Vector space Hilbert space plus continuum normalized states normalizable continuum normalization Fock space a space with states containing arbitrary numbers of particles connected by creation and annihilation operators Abers pg 439 Hermitean conjugate Hermitean Degenerate Diagonalizable Diagonal Reduced matrix element Wigner Eckart theorem physics term Secular determinant Ziman pg 121 Resolvent operatormatrix Ziman pg 122 Green function Non causal Green function Ziman pg 130 4133 o Perturbation Theory 363 Equations Formulas or Rules Schrodinger s Equation Blackbodies Wein s Law Stefan s Law Fermi s golden rule golden rule number two transition rate in rst Born approximation golden rule number one transition rate in second Born approximation Wigner Eckart selection rules 0 Hund s rule 0 Transition rules Dirac Fine Structure Formula Thomas Reiche Kuhn sum rule Sakurai pg 338 o Lippman Schwinger equation Dyson equation Ziman 3128 4136 Faxen Holtsmark formula Ziman pg 127 364 Theorems 0 Bell s Theorem 0 No level crossing theorem timeindependent perturbation theory 0 H 2 E0 variational methods Sakurai pg 313 o Adiabatic Theorem 365 Principles 0 Heisenberg uncertainty indeterminacy variance principle generalized 0 Correspondence principle in the statisticallarge numberthermodynamic limit QM a CM 0 Consistency condition tot ang mom commutation rels must be same as that of orbital ang mom 0 Pauli exclusion principle 366 Approximations 0 Electric dipole approximation Sakurai pg 338 o The Born Approximation Born series far eld solution of assuming localized potential Spherically symmetric potential First Born approximation Second o Born Oppenheimer Approximation Sakurai pg 474 o Adiabatic approximation Sakurai pg 473 o Dyson Series timedep pert theory Sakurai pg 325 approx 0 Perturbation expansion 0 Brillouin Wigner perturbation expansion Ziman pg 55 367 Effects 0 Photoelectric effect c Paramagnetic resonance Gasiorowicz pg 246 o quadratic Stark effect electric eld electric dipole mom coupling nondegenerate time independent perturbation theory 0 linear Stark effect degenerate time independent perturbation theory Lande s interval rule Sakurai pg 306 o Anomalous Zeeman effect linear quadratic magnetic eld orbit coupling magnetic eld spin coupling 0 Fine structure Paschen Back effect limit van der Waals7 interaction Magnetic anomaly of the spin Lande g factor Matthews pg 84 Hyper ne structure splitting Thomas precession 0 Nuclear magnetic resonance spin magnetic resonance resonance condition Sakurai pg 321 Ramsauer Townsend effect Sakurai pg 413 0 Lamb shift 0 Dynamical Jahn Teller effect Sakurai pg 475 o Aharonov Bohm effect c Mossbauer effect resonant recoil free emit absorb of gamma rays by atom bound in solid form 0 Quantum Zeno effect Sakurai pg 311 Dominant Interaction Almost good No good Always good Weak B HLS J2 or L S Lz Sf L2 S2 JZ Strong B HB L1 51 J2 or L S L2 S2 Jz The exception is the stretched con guration for example 1932 with m i32 Here L1 and 51 are both good this is because magnetic quantum number JZ m ml 1 m5 can be satis ed in only one way Sakurai pg 319 Heisenberg picture Interaction picture Schrodinger picture State ket No change Evolution determined by VI Evolution determined by H Observable Evolution determined by H Evolution determined by H0 No change 368 Questions 0 What is spin 0 What is a singlet doublet etc parahelium orthohelium Sakurai pg 369 0 Why Since H is Hermitean it has a complete set of eigenstates such that H Ek and I 2k lkkl Abers pg 293 0 page 16 of notebook especially Why are eigenstates of H orthogonal orthonormal 4 Theoretical Summary 5 Early Quantum Theory 0 Planck s quantization was seen by him as a math trick and a physical analysis seemed out of reach 0 Einstein quantized electromagnetism o Bohr Sommer ed Model and the Sommerfeld Wilson Quantization Condition Elliptical quantized orbits f pdq nh up to rst order perturbation the Bohr Sommerfeld model and quantum mechanics make the same predictions for the spectral line splitting in the Stark effect At higher order perturbations however the Bohr Sommerfeld model and quantum mechanics differ and measurements of the Stark effect under high eld strengths helped con rm the correctness of quantum mechanics over the Bohr model The Bohr Sommerfeld quantization condition as rst formulated can be viewed as a rough early draft of the more sophisticated condition that the symplectic form of a classical phase space M be integral that is that it lies in the image of H2M Z a H2MR a HRMR where the rst map is the homomorphism of Cech cohomology groups induced by the inclusion of the integers in the reals and the second map is the natural isomorphism between the Cech cohomology and the de Rham cohomology groups This condition guarantees that the symplectic form arise as the curvature form of a connection of a Hermitian line bundle This line bundle is then called a prequantization in the theory of geometric quantization 7 http en wikipedia orgwikiBohrSommerfeldmodel o Heisenberg s Quantum Mechanics Matrix Mechanics o Schrodinger s Quantum Mechanics Wave Mechanics 6 Transformation from Classical to Quantum Mechanics 61 Postulates 0 Physical systems are made of particles molecules atoms elementary particles should I start with particle physics here From Shankar Chapter 4 Postulates 0f Nonrelativistic OneParticle Theories C lassical Mechanics Quantum Mechanics l The state of a particle at any given time is speci ed by the two variables mt and pt7 ie7 as a point in a 2D phase space I The state of a particle is represented by a vec tor lzJtgt in a Hilbert space H Every dynamical variable to is a function of z and p w wzp H The independent variables x and p of classical mechanics are represented by Hermitian opera tors X and P with the following matrix elements in the eigenbasis of X ltlelz gt ltPlPlp gt The operators corresponding to dependent vari ables wmp are given Hermitian operators m6m 7 iih6m 7 9X7P wm HX7PHP lll If the particle is in a state given by z and p7 the ideal classical measurement of the variable to will yield a value add p The stae will remain unaffected lll If the particle is in a state lib ideal quan tum measurement of the variable correspond ing to Q will yield one of the eigenvalues to with probability Pw cx MMWJMZ The state of the system will change from W to lw as a result of the measurement IV The state variables change with time accord ing to Hamilton s equations i i 892 7 8p 892 p IV The state vector lzbtgt obeys the Schrodinger equation1 mg Wt H W where HXP j x a X719 a P is the quantum Hamiltonian operator and 92 is the Hamiltonian for the corresponding classical problem Note state vectors contain information that yields the probability of every possible result for every possible measurem ent experiment a single ket represents the state of the particle in Hilbert space and it contains e statistical To extract this information for any observable we must determine the preiction for all observables eigenbasis of the corresponding operator and nd the projection of along all its eigenkets properimproper vectorsstates principle of superposition Only possible values of w are the eigenvalues of Q Complication 1 The recipe 0z a X719 a P is ambiguous Complication 2 The operator 9 may be degenerate o Complication 3 The eigenvalue spectrum of 9 may be continuous o Complicatoin 4 The quantum variable 9 may have no classical counterpart 62 Quantization Methods H Canonical quantization Dirac method77 a Weyl Quantization to Covariant canonical quantization 03 Path integral quantization F Geometric quantization U Schwinger s variational approach a Deformation Quantization Weyl Quantization 1 Quantum statistical mechanics approach 0 Canonical a Preserves symplectic structure 0 Poisson brackets a Commutators with His 0 Quantities a Operators o Liouville s Theorem a Ehrenfest Theorem 63 Interpretations o Copenhagen 7 Correspondence Principle 64 Other Ideas 0 Quantum logic7 Quasi set theory 7 Representations Expansions and Approximations 71 Schrodinger Heisenberg and Interaction Dirac Pictures Schroodinger equation becomes the Schwinger Tomonaga equation in the interaction picture Schrodinger Picture 1 tgts SIMh 110gtS Or 110gtS ethh 1 tgts xims ethhlt ll0ls or xi0M3 e thhlt lltls 0 Must it be lt Itls lt II0lsethh and similarly elsewhere Heisenberg Picture State vectors 11 given that H is timeindependent wlttgtgts m or m amth i 1 tgts Ws SimWW 0r lt14 e thW I Ws Coordinate and momentum states WU elmh i gtgt 0r i gtgt e thh WU lt gtlttgt lt eiH or lt gt eth lt gtlttgt 0 Or should it be 1 t Operators j and fr A I A 1 X7t ethh x eithh m frx t ethWx e thh we Thus lt ilt1gtlttgtgts WWW at lt gt ltxgtilt1gtlttgtgts lt gtlttgt Heisenberg Equation of Motion ethhaim5 1H lt t 1gtH ethh xeithE QgtH lt t 7xt ltIgtH a 250 4011 Interaction Picture 72 Special Functions Et Cetera The spherical harmonics Y are the spherical components of the unit vector 8 Angular Momenta 9 Spin 0 Ryder pg 40 An analysis of the Poincare group gives a true understanding of and some surprising insights into the nature of spin77 0 Thought If spin is not due to current then there should not be any torque when a spin magnetic moment is in a magnetic eld 0 Is spin due to the decay of particles into charged particles that spin around before they recombine Where would that angular momentum go when they recombine Spin is short for spin angular momentum and maybe spin magnetic moment Spin relates to angular momentum Jspm S and magnetic dipole moment pspm 79pm S The relationship between spin angular momentum and its associated magnetic dipole moment is in some ways classical their relative directions can be explained by the classical idea of rotation and in other ways not classical their relative magnitudes cannot be explained with this analogy and are explained with the Dirac equation and QFT not Thomas precession WHY DO WE FORCE THE MAGNETIC DIPOLE MOMENT VECTOR OPERATOR TO BE PARALLEL TO THE SPIN VECTOR OPERATOR The magnetic dipole moment of a particle p with spin S is It uorbuspin iWLigPLPS where gp is the g factor for particle p and MP may be the the Bohr magneton MB or the nuclear magneton MN3 i eh MB 7 2mg 7 eh MN 7 m where M is the mass of the nucleon in question The gyi ratio or t or magnetogyric ratio expresses the ratio of an object s magnetic dipole moment to its angular momentum particle gfactor e electron 95 z 2 x 1 x 20023 p proton gp z 2 x 28 56 n neutron gn z 72 x 191 7382 N nucleus gN many different values 2 3 95 04 04 Oz 7 1 7 7 0328478445 7 118311 7 2 27139 7139 7139 95 2002 319 304 376 886 91 Pauli Spin Matrices 01 07 10 17 7 27 7 3 7 a4rlt10gt aalt O harlt071gt 711f aifl Ti aiaj 611739 ieijkak 0392 I ai239 I 0207 2i61177110k a039b039 ab 1 ia x b 039 det a1 71 azam 26 tr all 0 o ls this only in a particular situation 92 Spin Phenomena o SternGerlach experiment canonical quantum mechanics experiment haVing to do with spin 0 Anomolous Zeeman Effect and its inverse o Larmor precession the precession of the magnetic moments of electrons atomic nuclei and atoms around the direction of an external magnetic eld 7pr BBoi7wovBo7v9B Larmor precession used in 7 FMR ferromagnetic resonance a spectroscopic technique to probe the magnetization of ferro magnetic materials lt is a standard tool for probing spin waves and spin dynamics FMR is very similar to nuclear magnetic resonance except FMR probes the magnetic moment of electrons and NMR probes the magnetic moment of the proton 7 NMR nuclear magnetic resonance 0 Rabi oscillation the cyclic behaviour of a two state quantum system in the presence of an oscillatory driving eld Two state wrt energy ie non energy degenerate 7 When an atom or some other two level system is illuminated by a coherent beam of photons it will cyclically absorb photons and reemit them by stimulated emission One such cycle is called a Rabi cycle and the inverse of its duration the Rabi frequency of the photon beam This mechanism is fundamental to quantum optics It can be modelled using the Jaynes Cummings model and the Bloch vector formalism 7 The effect is important in quantum optics magnetic resonance imaging and quantum computing 7 Norman F Ramsey modi ed the Rabi apparatus to increase the interaction time with the eld The extreme sensitivity due to frequency of the radiation makes this very useful for keeping accurate time and is still used today in atomic clocks On Wikipedia s Rabi Cycle article see also Larmor frequency Laser pumping Optical pumping Rabi problem Bloch sphere Atomic coherence o SpinStatistics Connection Theorem 10 Transition Theory See HydrogenTrans Lecture Non Degenerate Perturbation Theory ME 7 wiilHllwii 2 pummel W 2 Wigner Eckart theorem 1 a7j7m Tg ajm W gt 5fm 37 0j HTHH047jgt Wigner Eckart selection rules lta 7j m lT la7jmgt 0 unless 139 m 7771 and 2 37klgj jk ajm T lozjm 0 unless 139 A77 7 ql and 4 2 M962 Am E m 7 m Aj E j 7 j Zj j j Condition 2 can be stated loosely as j j and k can form a triangle Now interpret these rules physically Using timedependent perturbation theory we have H H0 H1 where H1 is localized but not necessarily small and eigenstates W1 and 117 of the unperturbed Hamiltonian H0 The system starts at time to in state Wu and we d like to know the probability that the system is in state 117 at some time 25 long after the perturbing interaction has taken place 17 The transition amplitude abat that the system initially in the state Wu be in the state W7 at time tis amt lt gtblwatgt lt gtblUt7 tol gtagt lt gtb yaw a z iooeithbawdw 00 I I 1 l glmfw wmltmli ig lw i 00 limi eilwtTtOGwbadw 1 540 27139 x more stuff Transition matrix Tm H GwG0w 1 H H GwH H H G0wTw which expressions are useful Born Series Tw H First approximation uses rst term second approximation uses rst two terms and so on The Green operator Cw COW G0wTwG0w Lippmann Schwinger equation l agt l agt GOWUH WW The transition rate in terms of the transition matrix P mm kamp j 2 dt w 7 w t7 0 27139 Tbawal 6wb 7 wa 04 to ioo Fermi s Golden Rule golden rule number two F 27139 6wb 7 4 2 Thu 27r 6wb iwa transition rate in rst Born approximation Abers 1190 formula for single photon emission rate using spinless nonrelatiVistic description of the electron and rst Born approximation Fermi s golden rule 2 F ZWf 52w 27rm2 e ik39rp Zk d0 2 P5711 r 1 awlw apparently the p s and r s from the exponent can commute since7 as explained on Abers pg 3707 A and p commute due to the gauge condition k w Furthermore7 using dipole approximation7 with r7 H0 ipm7 2 7 40qu3 7 3 rfi z e 87139 P 27139 r What does this dipole approximation mean physically 101 Bound State Perturbation Theory 102 Scattering Theory Scattering matrix 1021 Potential Scattering Born Series 103 Collision Theory subtopic of scattering 104 Decays of Excited States 1041 Radioactive Decay 11 Equations Laws and Formulas dtltAgtltmvmgtltatAgt where A is any physical quantum operator and H is the Hamiltonian This is easily derivable starting with full expression for A and using the Schrodinger equation Use the generic Hamiltonian H p22m Vzt and7 after a little work7 get a form of Newton s second law 0 Ehrenfest Theorem dt ltpgt VWW ltFgt illustrating the correspondence principle lltgt 2ltgt6l where uo is the well depth and To is the o LennardJones Potential 12 Applications and Phenomena o Lasers ie7 LASERs 13 Advanced Topics 131 Phenomenology 0 Master Eqns in physics in general a phenomenological rst order differential equation describing the timeevolution of the probability of a system to occupy each one of a discrete set of states 19

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