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This is an introduction to modern quantum mechanics – albeit for those already familiar with vector calculus and modern physics – based on my personal understanding of the subject that emphasizes the concepts from first principles. Nothing of this is new or even developed first hand but the content (or maybe its clarity) is original in the fact that it displays an abridged yet concise and straightforward mathematical development that provides for a solid foundation in the tools and techniques to better understand and have a good appreciation for the physics involved in quantum theory and in an atom!

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- 1. From First Principles PART III – QUANTUM MECHANICS March 2017 – R3.0 Maurice R. TREMBLAY
- 2. Feynman’s last chalk boards – in and around the red box we can read: ‘SPIN COUPLING TO SELF KLEIN pk’ and ‘DO I WANT BBC INTERVIEW’; and above ‘What I cannot create, I do not understand’ & ‘Know how to solve every problem that has been solved’. Richard P. Feynman * 1918-1988 Circa 1966 * Notable awards: Albert Einstein Award (1954); E. O. Lawrence Award (1962); Nobel Prize in Physics (1965); Oersted Medal (1972); National Medal of Science (1979) – Influences: Paul Dirac – Known for: Feynman diagrams; Feynman point; Feynman-Kac formula; Wheeler-Feynman absorber theory; Feynman sprinkler; Feynman Long Division Puzzles; Hellmann-Feynman theorem; Feynman slash notation; Feynman parametrization; Sticky bead argument; One-electron universe; Quantum cellular automata. Prolog “I think I can safely say that nobody understands quantum mechanics.” Richard Feynman, The Character of Physical Law (1965). 2017 MRT 2
- 3. βsinoLpτ = ωωωω,Lo ΩΩΩΩp ττττ β pp p p t I lgm L lgm L lgm L Lt L t I td d I td Id td d t I ω sin sin sin sin )( lim o oo o o o oo 0 o === == ∆ ∆ = ∆ ∆ = = === = ∆ ∆ = = →∆ top toptop top top Lτ LL τ L β β β τ β α :VelocityAngularPrecession :Force)of(MomentTorque :MomentumAngular ×××× αααα ωωωωωωωω ωωωω At its center, a thin, solid disk of radius r and mass m. = z x x I I I I 00 00 00 Disk )(cos sin )cos( 2 1 2 2 22 PotentialKinetic ++ + − += i.e.β β β γγαβ mg I L I LL I L H zxx Disk Inertia (Symmetry) : Generating angular momentum using a toy top! A top is a toy that can be spun on an axis (z ), balancing on a point. The ‘action’ of a top relies on the gyroscopic effect for its operation... ωωωω,Lo L,HI,ττττ 2π π 3π • • 0 • 2π/4 or π/2 3π/2 5π/2π/4 3π/4 mg βsinoL α∆ oL∆ βsinl A 100 g top spinning at 14 rev/s makes an angle of 30° to the vertical and precesses at a rate of 1 rev per 8.0 s. If its CM is 2 cm from its tip along its symmetry axis, what is the moment of inertia of the top? CM Hamiltonian (Euler anglesα, β &γ ): 2 2 1 2 4 1 rmI rmII z yx = == Typical Problem : l Ans.: Itop = mgl/Ωpω≈ 0.01kg⋅m2. The Center of Mass (CM) is located a distance l away from the tip. 2017 MRT 3
- 4. E1 Here’s a hydrogen atom… according to Neils Bohr (circa 1907-13) The electron is held in a circular orbit by electrostatic attraction. The total energy is negative and inver- sely proportional to the absolute distance r. Simple! Right? Like the motion of planet Earth around the Sun they say… Well, unfortunately, things are a little bit more complicated than this… As we shall soon see! Proton (Nucleus) +e (Z =1) 1s Electron in orbit around the nucleus! second).Joule( constantsPlanck’iswhere ⋅× ==−≡∆ −34 23 106.626 )( h ch hEEE λ ν :RelationPlanck-Einstein hn h nL == π2 :MomentumAngular rπ2=λn :ConditionBohr r).Henry/mete( typermeabilimagnetictheis andnd)meter/seco( vacuuminlightofspeedtheisand Coulomb meterNewton :constant dielectricsCoulomb’iswhere :radiusanyatpotential) minus(kineticenergytotaltheasor only)orbit(circularforceCoulomb thetoequalisforcelcentripetaThe 2 2 7 o 9 o 2 o 2 2 2 e e 2 2 22 e 104π µ8299,792,45 10987.8 π4 µ επ4 1 2 2 1 ˆˆ : − × ⋅ ×= == −=⇒ −=−= =⇒−=− c c k k r ek E ek mVTE m ekekm E E E E EE r v r vr r r r v r me −e v kilogram31 pe 1031.9 1836 1 − ×== mm :electrontheofMass Coulomb19 1067.1 − ×=e :electrontheofCharge Voltselectron :numberquantumandradius thebydeterminedisprotons)(withatomanyFor 2 2 22 e 2222 th 6.13 2 )( 2 n Z n mekZ r Zek E nr Z n n n n −≅−=−= h EE :leveltheofenergyThe n =1 E3 E2 mp 2017 MRT 4
- 5. Foreword This is an introduction to modern quantum mechanics – albeit for those already familiar with vector calculus, quantum physics and modern physics – which is based on my personal understanding of the subject that emphasizes the concepts from first principles. Nothing of this is new or even developed first hand but the content (or maybe its clarity and the way it is organized) is original in the fact that it displays an abridged yet concise and straightforward mathematical development that provides for a solid foundation in the tools and techniques of quantum mechanics to better understand and have a good appreciation for the physics involved in quantum theory and in an atom such as that of hydrogen! To me, the chief aim was to generate a consolidated review of the fundamentals in a presentation medium such as Microsoft® PowerPoint®. Starting with an historical appetizer on how quantum theory comes about we describe it using particle-wave concepts to generate the path integral formulation by reviewing 3 approaches to the double-slit experiment. We then set forth to present a revised and original formulation of rotation transformation theory where we develop rotation generators first hand. We then show that the generators of rotation lead inevitably to the total angular momentum operator. This will close the first part of this consolidated review of quantum mechanics. In the second part will consider quantum fields and the last (third) part will deal exclusively with the hydrogen atom. We will finish this with an appendix on the harmonic oscillator, electromagnetic interactions, &c. 2017 MRT When finished with these prerequisites, we are ready to formulate angular momentum in quantum theory by using the mathematics and postulates of quantum mechanics to generate tools such as the spherical harmonics to represent them. Special cases such as the electron spin angular momentum and adding it to the angular momentum gives us the total angular momentum. We then present the total angular momentum ladder operator matrix elements and Euler angle-based rotations to represent those calculations throughout. Angular momentum coupling is then discussed with a few examples. After all this mathematical basis is provided, the presentation of the five postulates of quantum mechanics comes next where we associate the particle description to the wave packet point of view and then show the non-relativistic particle formulation with it’s generalized Feynman path integral. 5
- 6. “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that the exact appli- cation of these laws leads to equations much too complicated to be soluble.” Paul Dirac, ‘Quantum Me- chanics of Many-Electron Systems’ Proceedings of the Royal Society (London), 123 (1929) pp. 714-733. Contents 2017 MRT For the two-slit experiment, one can imagine firing classical ‘particles’ (e.g. bullets) at a panel containing two slits in Figure 1 and observing the distribution of the particles against a screen (target). As shown at the far right of Figure 1 the target will show to have two ‘classical mounds’ of particles. Next, one allows classical ‘waves’ (e.g. water waves) to propagate toward a panel with two slits as in Figure 2. The waves will ‘diffract’ at the slits and produce the classical interference patterns shown at the far right of Figure 2. Finally, one can fire ‘electrons’ from a heated tungsten filament toward the two-split panel as in Figure 3. It appears that we are firing bullet-like objects toward the panel, but when we look at the distribution at the far right of Figure 3 we see the apparance of the quantum interference patterns! Figure 3 Figure 2 Tungsten Filament Figure 1 P1 + P2 I1 + I2 +2√I1 I2 cosδ abba abbaab 222 111 viatopath viatopath :where + = 2 ),( abK= Mathematically 〈b|a〉2 Source at a Screen at b Slits Paddle Wheel Machine Gun Water Waves Electrons Bullets I1 I2 K1 2 K2 2 P1 P2 P(b,a) P12 I12 PART III – QUANTUM MECHANICS Introduction Symmetries and Probabilities Angular Momentum Quantum Behavior Postulates Quantum Angular Momentum Spherical Harmonics Spin Angular Momentum Total Angular Momentum Momentum Coupling General Propagator Free Particle Propagator Wave Packets Non-Relativistic Particle Appendix: Why Quantum? References 6 ProbabilitisticallyPhysically
- 7. PART V – THE HYDROGEN ATOM What happens at 10−−−−10 m? The Hydrogen Atom Spin-Orbit Coupling Other Interactions Magnetic & Electric Fields Hyperfine Interactions Multi-Electron Atoms and Molecules Appendix – Interactions The Harmonic Oscillator Electromagnetic Interactions Quantization of the Radiation Field Transition Probabilities Einstein’s Coefficients Planck’s Law A Note on Line Broadening The Photoelectric Effect Higher Order Electromagnetic Interactions References 2017 MRT “The development of quantum mechanics in the 1920s was the greatest advance in physical science since the work of Isaac Newton. It was not easy; the ideas of quantum mechanics present a profound departure from ordi- nary human intuition. […] I like to give a much greater emphasis than usual to principles of symmetry, including their role in motivating commutation rules.” Steven Weinberg, Preface to Lectures on Quantum Mechanics. PART IV – QUANTUM FIELDS Review of Quantum Mechanics Galilean Invariance Lorentz Invariance The Relativity Principle Poincaré Transformations The Poincaré Algebra Lorentz Transformations Lorentz Invariant Scalar Klein-Gordon & Dirac One-Particle States Wigner’s Little Group Normalization Factor Mass Positive-Definite Boosts & Rotations Mass Zero The Klein-Gordon Equation The Dirac Equation References 7
- 8. Introduction The revolutionary change in our understanding of microscopic phenomena that took place during the first 27 years of the twentieth century is unprecedented in the history of natural science. Not only did we witness severe limitations in the validity of classical physics, but we found the alternative theory that replaced the classical physics theories to be far richer in scope and far richer in its range of applicability (e.g., the transistor!) Quantum theory was born in 1900, when Max Planck announced his theoretical derivation of the distribution law for black-body radiation (c.f., Appendix). Planck (1858-1947) showed that the results of experiment on the distribution of energy with frequency of radiation in equilibrium with matter at a given temperature can be accounted for by postulating that the vibrating particles of matter do not emit or absorb light continuously but instead only in discrete quantities of magnitude hν proportional to the frequency ν of the light, with h being a constant. 1e π8 ),( 1e 12 ),( 3 3 eq 5 2 − == − = TkhTkch BB c h UT hc TI ννλ ν νρ λ λ or This constant of proportionality, h, is a new constant of nature; it is called Planck’s constant and has the magnitude 6.626 × 10−34 Joules- seconds. Its dimensions (energy × time) are those of the old (or classi- cal) dynamical quantity called action; they are such that the product of the constant h and the frequency (sec−1) has the dimensions of energy. The dimensions of h are also those of angular momentum and h/2π is a natural unit – or quantum – of angular momentum. 2017 MRT And it was not until 1905 that Albert Einstein suggested that the quantity of radiant energy hν was sent out in the process of emission of light. The name photon was applied to such a ‘particle’. Einstein (1879-1955) also discussed the photoelectric effect (e.g., when light falls on a metal plate, electrons are emitted from it.) “For his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect” Einstein was awarded the 1921 Nobel Prize. Planck’s law I(λ,T ) describes the spectral radiance of unpolarized electromagnetic radiation at all wavelengths (λ) emitted from a black body at absolute temperature T. 0 500 1000 1500 2000 λ [nm] 800 600 400 200 0 I(λ)[kJ/nm] T = 3500K T = 4000K T = 4500K T = 5000K T = 5500K ∫ − = −λ λ λλ λ 0 5 2 1e 2),( Tkch B d chTI 1e e e 0 0 − = = − ∞ = − ∞ = − ∑ ∑ Tkh n Tknh n Tknh B B B h nh E λν λν λν ν ν ννρ d L N Ed 3 = N(ν)=2×4π|n|2d|n|=8π(L/c)3ν2dν L c c c n q = = = π2 λ ν q n q π2 π2 = = λ L xq•i e 2ˆn 3ˆn 1ˆn 8
- 9. Bohr (1885-1962), in 1913, used a mixture of classical physics (mechanics and electromagnetism) and quantization of energy concept to formulate a satisfactory theory for the observed spectrum of the hydrogen atom. Bohr’s theory is based on the following three postulates (c.f., Appendix): 1. The electron in the hydrogen atom moves about the nucleus (i.e., a lone proton) in certain circular orbits (called stationary states) without radiating energy; 2. The allowed stationary states are such that: 2017 MRT 3. When the electron makes a transition from an initial state of energy Ei to a final state of energy Ef (where Ei >Ef ) electromagnetic radiation (i.e., photons) is emitted from the hydrogen atom. The frequency ν of this radiation is given by: where L is the angular momentum of the electron, r is the radius of the orbit, me is the mass of the electron, h is Planck’s constant, v is the speed of the electron, and n is the principal quantum number; ( )...,3,2,1 π2 e === n h nrvmL h EE h E fi − = ∆ =ν Bohr’s theory was generalized by Wilson and Sommerfield and applied to other atoms with limited success. By 1924, it was clear that a new theory was needed to explain the basic properties of atoms and molecules in a systematic manner. 9
- 10. In an historic paper Uber quantentheoretische Umdeutung kinematischer und mecha- nischer Beziehungen (Quantum-mechanical re-interpretation of kinematic and mechani- cal relations), Z. Phys. 33, 879-893 (1925) which led to the development of matrix mechanics, Heisenberg (1901-1975) introduced a system of mechanics in which classi- cal concepts of mechanics were drastically revised. Heisenberg assumed that atomic theory should emphasize the observable quantities rather than the shape of electronic orbitals (Bohr’s theory). This theory was rapidly developed by means of matrix algebra. 2017 MRT Parallel to the development of matrix mechanics, Schrödinger initiated (1926) a new line of study which evolved into wave mechanics. Wave mechanics was inspired by de Broglie’s (1892-1987) wave theory of matter, λ=h/p, where p is the momentum of the particle and λ is the wavelength of its associated wave (matter wave). Schrödinger (1887-1961) introduced an equation of motion, the Schrödinger wave equation, for matter waves and proved that wave mechanics was mathematically equivalent to matrix mechanics. However, the physical meaning of wave mechanics was not clear at first. Schrödinger first considered the de Broglie wave as a physical entity (the particle, e.g., an electron, is actually a wave). This interpretation soon led to difficulty since a wave may be partially reflected and partially transmitted at a boundary, but an electron cannot be split into two parts for transmission and reflection. This difficulty was removed by Born (1882-1970) who proposed a statistical interpretation of the de Broglie waves which is now generally accepted. The new theory based on the statistical interpretation was very rapidly developped into a general coherent system of mechanics (called quantum mechanics). 10
- 11. In the first paragraph of his paper Quantisierung als Eigenewertproblem,communicated to the Annalen der Physik on January 27, 1926, Erwin Schrödinger stated essentially:“In this communication I wish to show, first for the simplest case of the non-relativistic and unperturbed hydrogen atom, that the usual rules of quantization can be replaced by another postulate, in which there occurs no mention to whole numbers. Instead, the introduction of integers arises in the same natural way as, for example, in a vibrating string, for which the number of nodes is integral. The new conception can be generalized, and I believe that it penetrates deeply into the true nature of the quantum rules.” (E. Schrödinger, Ann. D. Phys. 79, 361 (1926)) 2017 MRT For our purpose, the Schrödinger equation, the auxiliary restrictions upon the wave function ΨΨΨΨ, and the interpretation of the wave function are taken as fundamental postulates (i.e., they are true!), with no derivation from other principles necessary. The Schrödinger wave equation and its auxiliary postulates enable us to determine certain functions ΨΨΨΨ of the coordinates of a system and the time. These functions are called the Schrödinger wave functions or probability amplitude functions. The square of the absolute value of a given wave function is interpreted as a probability distribution function for the coordinates of the system in the state represented by this wave function. Besides yielding the probability amplitude or wave function ΨΨΨΨ, the Schrödinger equa- tion provides a method of calculatingvaluesoftheenergyofastationarystate of a system. In this and four other papers, published during the first half of 1926, Schrödinger communicated his wave equation and applied it to a number of problems, including the hydrogen atom, the harmonic oscillator, the rigid rotator, the diatomic molecule, and the hydrogen atom in an electric field (e.g., Stark effect – See Magnetic & Electric Fields). 11
- 12. As such,the wave equationisnotderived from other physical laws nor obtained as a ne- cessary consequence of any experiment; instead, it is assumed to be correct (e.g., like the 2nd law F=ma), and results predicted by it are compared with data from a laboratory. But the possibility that material particles can– like the photon– be described in terms of waves was first suggested in 1924 by Louis de Broglie. This was based on Lorentz invari- ance: If particles are described by a wave whose phase at position r and time t is of the form 2π(k•r−νt), and if this phase is to be Lorentz invariant, then the wave vector k and the frequency ν must transform like r and t, and hence like the momentum p and the total energy E. In order for this to be possible, k and ν must have the same velocity dependence as p and E, and therefore must be proportional to them, with the same constant of proportionality. For photons, one had Einstein’s relation E=hν, so it was na- tural to assume that, for material particles: k=p/h =1/λ and ν =E/h – just as for photons. 2017 MRT Thus quantum mechanics exposes us to new quantities and entirely new effects since it requires added dimensions to describe physical states. Also, in retrospect, wave mechanics was by-passed in the next step in the history of quantum mechanics by the development of matrix mechanics by Werner Heisenberg, Max Born, Pascual Jordan and Wolfgang Pauli in the years 1925-1926. At least part of the inspiration for matrix mechanics was the insistence that the theory should involve only observables such as the energy levels, or emission and absorption rates. e.g., Heisenberg’s 1925 paper opens with the manifesto: “The present paper seeks to establish a basis for theoretical quantum mechanics founded upon relationships between quantities that in principle are observable.” (W. Heisenberg, A. Phys. 33, 879 (1925)) 12
- 13. One note on the measurements of ‘quantum’ effects. It was Heisenberg who first pointed out that the new laws of quantum mechanics imply a fundamental limitation to the accuracy of experimental measurements (viz, only at the quantum level!) In making measurements on a quantum system, it is not possible to measure the (e.g., the ‘vector’) quantities r and p as accurately as we would wish to. There is always some minimum error – or uncertainty ∆r, ∆p – in r and p associated with their measurement. 2017 MRT We see that in order to locate the particle very precisely we must use high frequency light (or – since ν =c/λ – small wavelength ‘electromagnetic waves’). But, such high fre- quency light will arrive in photons with very large energy (because of the Planck’s rela- tion E=hν ) which therefore give the quantum system a very large ‘kick’! Similarly, if we want to know the momentum very accurately, we must give the systemaverysmall ‘kick’. Due to Planck’s relation, this means using light of low frequency.Low frequency means long wavelengthand this in turn means large uncertainty in the measurement of position! Heisenberg’s uncertainty principle relates the uncertainties in position and momentum in the following way (and limiting ourselves to measurements along the x-axis,i.e., r=xî): hpx x ≈∆⋅∆ So, if you want to make the uncertainty in position, ∆x, very small, then the uncertainty in momentum, ∆px, cannot also be, by default, also as small – as you would want it to be! We again come across the realization that Planck’s constant h (with its magnitude of 6.626×10–34 J⋅s) and its ‘smallness’ is of course the reason it took so long before physicists were able to observe any of the strange quantum effects such as this one! 13
- 14. Now, let us digress a little and consider an appropriate application of coordinate differen- tials and introduce the concept of an operator into practical (and physical) applications. Loosely speaking, an operator is some instruction that, when applied to a function f(r), changes it into another function f (r) (e.g., note the result of applying the gradient ∇∇∇∇r operator considered earlier). Take for example an infinitesimal translation Tr(dr) (i.e., we will first use the ‘infinitesimal’ dr – or sometime you’ll see ε – instead of the ‘leap’ a) that generates infinitesimal translations, i.e., shifts by dr a function at r to a function at r++++dr: )()()( rrrrr dffdT ++++= One of the greatest mathematical tools available to break down functions f(x) about a point (e.g., say xo) is the Taylor (1685-1731) expansion (or series approximation): Consider a translation of the coordinates r by an amount represented by a vector a: arrr ++++=→ This is a continuous translation (to contrast with a discret translation such as r→r=−r, i.e., space inversions) and we’ve chosen to represent the new coordinate system by r. f (N)(xo) is the N-th derivative of f (x) (e.g., f ′=d/dx) evaluated at xo. For Tr(a) f(r), we have: ∑ ∞ = =+ ⋅⋅ ′′′ ⋅+ ⋅ ′′ ⋅+′⋅+= 0 o )(oo3 o o2 oooo )( ! )( 321 )( )( 21 )( )()()()()( N N N f N ff fff x xxx xx x xxxxxxx −−−− −−−−−−−−−−−− K ∑ ∞ = = == • =+••+•+= 0 )( ! )( )]([ 2 1 )()()( N N f N ffff rr r rrrrrrr r a raararar ∇∇∇∇ ∇∇∇∇∇∇∇∇∇∇∇∇++++ K 2017 MRT where the differential operator ∇∇∇∇r ≡Σiêi∂/∂xi with êi an appropriatecoordinateunitvector. Symmetries and Probabilities 14
- 15. where the vector dot product • is defined by (e.g., by using two Cartesian coordinates systems r and r) r•r=Σkrk rk=x1 x1 +x2 x2 +x3x3≡xx+yy+zz giving us: Applying this Taylor series to r +dr and keeping only the first-order term, we get: K+•+== =rrr rrrrrr )()()()( fdfdff ∇∇∇∇++++ 2017 MRT ∑∑∑ ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ • ∂ ∂ =• = = k k k i j ji ji r rd z zd y yd x xdudud uu d 3 1 3 1 r r r r∇∇∇∇ We thus obtain: rr r rrrr = ∑ ∂ ∂ += )()()()( f r rdffdT k k k ∑∑ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = j j k j kj j j k k rr r r rd r r rd and With the differential found for the element of displacement dr (assuming, e.g., ui →r j ): we get: rr r rrrr = ∑ ∑ ∂ ∂ ∂ ∂ += )()()()( f rr r rdffdT k j j k j k 15
- 16. 2017 MRT Inserting i =√(−1), the imaginary number (e.g., to ensure a unitary representation), and by rearranging everything to separate the infinitesimal part dr=drk from the rest, we get: kk rr k j j k j k f rr r rdffdT = ∑ ∑ ∂ ∂ ∂ ∂ += )( 1 )()()( rrrrr i i and again rearranging (i2 =−1): )()()()( rrrrr f rr r irdiffdT k G j j rrk j k k kk ∑ ∑ ∂ ∂ ∂ ∂ −+= = 4444 34444 21 generator)on(translati We now have the final result of an infinitesimal translation of the coordinates: )()()()( rrrrr fGrdiffdT k kk += ∑ where Gk is the infinitesimal translation generator (i.e., a Hermitian operator): ∑ ∂ ∂ ∂ ∂ −= =j j rrk j k rr r iG kk In the introduction, we said that: ‘If particles are described by a wave whose phase at position r […] is of the form 2πk•r,and if this phase is to be Lorentz invariant,then the wave vector k […] must transform like r […] and hence like the momentum p.’ 16
- 17. Now for the general‘leap’ a andsimpletranslationthree-vector,ai, of the coordinates ri: jjjj arrr +=→⇔=→ arrr ++++ where for all practical 3-dimensional purposes, r j ≡rj. We evaluate the partial derivative: k j ak j j j ak j k kk a r ra r iG δ= ∂ ∂ ⇒ ∂ ∂ ∂ ∂ −= == ∑ 00 where the Kronecker (1823-1891) delta δ j k (=δij) has the definition δij=1 when i=j and δij=0 when i≠ j. Inserting this in the expression for the translation operator, we obtain: where ∂k ≡∂/∂rk and it can be shown just by expansion that Σjδ j k∂j =∂k. Thus we obtain: )()()()( r1r1rar fakif r iaifT k kk k kk += ∂ ∂ −+= ∑∑ Restoring the vector notation,weget the representationof the translation operator: ap1ak1ar •+=•+= h i iT π2 )( The outcome is that the generator of translations is k≡−i∇∇∇∇r and it is proportional to themomentum operator p≡−ih∇∇∇∇r /2π since p=(h/2π)k,whereh is Planck’s constant. 2017 MRT )()()()()()()( rrrrrar fiaiff r iaiffT k kk k j j k j k ∑∑ ∑ ∂−+= ∂ ∂ −+= δ 17
- 18. The development is the same for a Taylor series expansion of a time displacement,dt, to t +dt. So, pray tell, let us now consider a ‘jump’ forward, τ, of the time coordinate t: τ+=→ ttt Applying things like we did earlier (i.e., for f (t)= f (t+τ )= f (t)+(τ ∂/∂t) f (t)|t=t), we get: 1= ∂ ∂ ⇒ ∂ ∂ ∂ ∂ −= == tttt t t t tt t iG we obtain, like, I mean: where ∂t ≡∂/∂t and we finallyget the representation of the time translation operator: ττντ E h i iTt π2 )( −=−= 11 2017 MRT )()]([)(1)()()( tfiitf t iitftfT tt ∂−= ∂ ∂ ⋅⋅−+= τττ 1 The time translation generator is ν =i∂t and proportional to E≡ih∂t /2π, i.e., the energy operator. Again, from the introduction: ‘If particles are described by a wave whose phase at […] time t is of the form −2πνt, and if this phase is to be Lorentz invariant, then the frequency ν must transform like t,and hence like the total energy E (=hν)…’ )()()()( tf tt t iitftfT tt t ∂ ∂ ∂ ∂ −+= = ττ Then evaluating the partial derivative, you know, like: 18
- 19. Galilean boosts are transformations which look at the system from the point of view of an observer moving withaconstant velocity v (while the time coordinate t is unchanged): tvrrrt jjjj +=→⇔=→ vrrr ++++ We evaluate the partial derivative (i.e., for f (r)= f (r++++vt)= f (r)+(vt •∇∇∇∇r) f (r)|r=r): k j rrk j k j j rrk j k kkkk r r f rr r itviffB δ= ∂ ∂ ⇒ ∂ ∂ ∂ ∂ −+= == ∑ ∑ )()()()( rrrvr Inserting this in the expression for the boost operator, we obtain: and thus we get: )()()()( r1r1rvr fKvif r tivifB k kk k kk −= ∂ ∂ −= ∑∑ Restoring the vector notation,wefinallyget the representationof the boost operator: Kv1vr •−= iB )( The outcome is that the generator of Galilean boosts is K≡it∇∇∇∇r for infinitesimal v. 2017 MRT )()()()()()()( rrrrrvr ftiviff r itviffB k kk k j j k j k ∑∑ ∑ ∂−= ∂ ∂ −+= δ )()()( tffB vrrvr ++++= So, for the general Galilean‘boost’ vt: 19
- 20. 20 2017 MRT )()( 2 vKv1vr OiB +•−= )()]([)()1( rrr v1v1Kv1v ∇∇∇∇∇∇∇∇ •+=•−=•−= iiiB The Galilean boost was just given by: To first order for infinitesimal velocity v and using K=it∇∇∇∇r : To second order, we use f (r)= f (r++++vt)= f (r)+(vt•∇∇∇∇r) f (r)|r=r +½[vt•(vt•∇∇∇∇r∇∇∇∇r)] f (r)|r=r: )( 1 2 )( 1 )()()()2( r rrrvr f rrr r r r i vv i f rr r i viffB m l k j jk rr rrl j m k lm k j j rrk j k ll kk kk ∑ ∑ ∑∑ ∑ ∑ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ += = = = t t t t t Now, also notice that Br (1)(v) above can be rewritten as: )( π2 π2 π2 π2 π2 )()( 2 )1( pv1v1v1v1v rrrr •+= −•+= •−=•−= hi ihi hihi hi B ∇∇∇∇∇∇∇∇∇∇∇∇ But what is this scalar product of the velocity and quantum momentum operator? t t t t )( π2 )( pv1vp •−= h i B t if we use p≡−ih∇∇∇∇r /2π. So, we finally get this weird expression for the first order boost: −= ττ E h i Tt π2 )( 1viz
- 21. 21 2017 MRT we get: l j m k rr rrl j m k k j rrk j mm ll kk r r r r r r δδδ = ∂ ∂ ∂ ∂ = ∂ ∂ = = = and )( 2 )()()()()2( rrrrvr f rr ivv i f r iviffB m l jk jk l j m k lm k j j k j k ∑ ∑ ∑∑ ∑ ∂ ∂ ∂ ∂ −+ ∂ ∂ −+= δδδ Since: and then: )()( 2 )()()()()()2( rrrrvr fivv i fiviffB lm lmlm k kk ∑∑ ∂∂−∂−= Then: ∑∑ ∂∂ ∂ − ∂ ∂ −= lm lmlm k kk rr ivv i r iviB 2 )2( 2 )( 1vr and finally using the generators Kk =it∂/∂r k and H ≡Hml = it2∂2/∂r m∂r l (a pseudo-tensor): ∑∑ −−= lm lmlm k kk Hvv i KviB 2 )()2( 1vr whereΣk vk Kk ≡v•K,KbeingtheGalileanboost.Remembers=so ++++vot−−−−½gt2 vs i=√(−1)? t2t t2t t t t
- 22. Since, to first order, we already obtained: 22 2017 MRT rr v1Kv1v ∇∇∇∇•+→•−= tiB )()1( ∑−•+= lm lmlm Hvv i ttB 2 )()2( rr v1v ∇∇∇∇ using K=it∇∇∇∇r . we now get to second order: that is, a general Galilean‘boost’ vt: tvrrr ++++=→ ))(()()( 2)2( vvrrvr OtffB ++++++++= where t is the time, giving us: 2 )( 2 1 )( rrr vv1v ∇•+•+= tttB ∇∇∇∇ We will now see how a ‘nudge’ affects the Galilean boost to second order: Using K=it∇∇∇∇r (instead of H= it2∇2 r where ∇2 r ≡ ∇∇∇∇r •∇∇∇∇r is the Laplace operator and ∇∇∇∇r the gradient operator at position r), we obtain: 2)2( )( 2 )( KvKv1vr •−•−= i iB where the Galilean boost provides only a classical effect measured by the scalar product of the distance travelled during an infinitesimal amount of time t and the gradient, vt •∇r (i.e., the projection of the distance travelled onto the gradient vector).
- 23. We just saw that after applying an infinitesimal translation to the space coordinates,dr, a differentiable function f (r) of position r changes to: rrr rrrrrr = •+== )()()()( fdfdff ∇∇∇∇++++ )()()()()2( 2 rr1rrr1rr rr fddfddf ∇∇∇∇++++∇∇∇∇++++ •+=•+= )(e)(lim)()()( rr a 1arra ra rr ff N ffT N N ∇∇∇∇ ∇∇∇∇++++ • ∞→ = •+== with k≡−i∇∇∇∇r ,thegeneratorofspacetranslations.Applying the change k=(2π/h)p we get: ka r a • = i T e)( After another infinitesimal translation dr, we have: with f(r+3dr) left as an Exercise to the reader, and so on and so forth. since the series, mathematically in its power development, is represented by an exponential function expξ =limN→∞(1+ξ/N)N (e.g., e=limN→∞(1+1/N)N!) We have then: pa r a • = h i T π2 e)( 2017 MRT with p≡−ih∇∇∇∇r /2π the momentum operator. Finding out the meaning of this is left as an Exercise to the reader(…if you just can’t figure this one out, start over with PART II!) Now, since a finite translation a can be constructed from an infinite number of successive infinitesimal space translations, we have: 23
- 24. Now, doing the same thing by (this time around) applying an infinitesimal time displacement,dt, a continuously differentiable function f (t) of time t changes to: tt tf t tdtftdtftf = ∂ ∂ +=+= )()()()( )()()2( 2 tf t tdtdtf t tdtdtf ∂ ∂ +=+ ∂ ∂ +=+ 11 )(e)(lim)()()( tftf tN tftfT t N N t ∂ ∞→ = ∂ ∂ +=+= ττ ττ 1 with ν ≡i∂t , the generator of time translations. Applying the change E=hν we get: ντ τ i tT − = e)( After another infinitesimal time displacement dt, we have: Since a finite time displacement τ can be constructed from an infinite number of successive infinitesimal time displacements, we have: where: 2017 MRT with E ≡ih∂t /2π the total energy operator. E h i tT τ τ π2 e)( − = 24
- 25. Finally, for the Galilean boost,vt, a differentiable function f (r) of position r changes to: rrr rvrvrr = •+== )()()()( ftftff ∇∇∇∇++++ )()()()()2( 2 rv1rrv1vr rr ftdfttf ∇∇∇∇++++∇∇∇∇++++ •+=•+= )(e)(lim)()()( )( rr v 1vrrv rv rr ff N t tffB t N N ∇∇∇∇ ∇∇∇∇++++ • ∞→ = •+== with K≡it∇∇∇∇r, the generator of Galilean boosts, yielding the final result: After another boost vt, we have: In the same way as a finite translation or a finite time displacement, boosts Br(v) can be constructed from an infinite number of successive boosts. We eventually get: Kv r v •− = i B e)( 2017 MRT In the final analysis, ‘Any kind of wave of frequency ν and wave number k has a spacetime dependence exp(−iωt+ik•r), where ω=2πv. Lorentz invariance requires that [ω,k] transform as a four-vector, just likethemomentumfour-vector [E,p].’ Within the realm of special relativity, it is required of us to consider boosting things (e.g., particles such as the W+-boson in an experiment at CERN.) Unfortunately, althought trivial in 3- dimensional considerations, Galilean boosts as described above will not work in 4-dim- ensional space (e.g., quantum field theory is needed to calculate W+-boson scat- tering with whatever particles combinations they so choose to smash to produce it.) 25
- 26. ∫∫ ∞ ∞− −•− ∞ ∞− •− == rrrrk rkrk ddtt tii )ω( 3/23/2 e)( π)2( 1 e),( π)2( 1 ),( ψΨΨΨΨΦΦΦΦ A note on the notation of the Fourier Transforms forms (i.e., from coordinate space r represented by the wave function ΨΨΨΨ(r,t) to momentum space p represented by the wave function ΦΦΦΦ (p,t) and vice versa) that are the most often used in quantum mechanics are given in the following useful expressions and key to understanding physical phenomena. 2017 MRT wherep =hk/2π is themomentum (alongthe directionof a standard momentumk) and E=hω/2π isthetotalenergy(withangularfrequencyω)andfinally,h isPlanck’sconstant. ∫∫ ∞ ∞− −•− ∞ ∞− •− == rrrrp rprp d h dt h t htEihi )(π2 2/3 π2 2/3 e)( 1 e),( 1 ),( ψΨΨΨΨΦΦΦΦ or better yet: and inversly as a result of the Fourier transform: ∫∫ ∞ ∞− −• ∞ ∞− • == kkkkr rkrk ddtt tii )ω( 3/23/2 e)( π)2( 1 e),( π)2( 1 ),( ϕΦΦΦΦΨΨΨΨ ∫∫ ∞ ∞− −• ∞ ∞− • == ppppr rprp d h dt h t htEihi )(π2 2/3 π2 2/3 e)( 1 e),( 1 ),( ϕΦΦΦΦΨΨΨΨ thus: Here the corresponding transform pairs are (considering a standard momentum ki =k): 26
- 27. In developing a wave equation for particles, Schrödinger knew that: 1. Hamilton (1805-1865) had established an analogy between the Newtonian mechanics of a particle and geometry (ray) optics (i.e., Hamiltonian mechanics), and; 2. Equations of wave optics reduced to those of geometrical optics if the wavelength in the former is equal to zero. Schrödinger postulated that classical Newtonian mechanics was the limiting case of a more general wave mechanics and proceeded to obtain a wave equation for particles. 2017 MRT The conditions for the existence of a wave equation for particles will now be established using some of the items used in the calculus of variations. 0=∫ B A sdpδ where pds is called an element of action (MKS units of Joules⋅second). δ ∫pds is referred to as the principle of least action and can be derived from Hamilton’s more general principle of stationary action which is given by: 0=∫ B A tdLδ where L is the Lagrangian. We begin with the fundamental principle of classical mechanics. For a particle of mass m moving in a force field described by a potential V(r), we may write: 27
- 28. An element of Hamiltonian action is given by Ldt. To go from Hamilton’s principle of stationary action to the principle of least action, the total energy, E, of the system must be assumed to be constant. The energy of the (free) particle is given by: 2017 MRT Note the striking similarity between δ ∫pds=0 and δ ∫ds/λ=0. If we let p=h/λ then the principle of least action δ ∫pds=0 is identical to δ ∫ds/λ=0. Here h is a pure constant of proportionality identified earlier as Planck’s constant. )( 2 2 rV m p E += The corresponding momentum is: )]([2 rVEmp −= In geometrical optics, the basic principle is Fermat’s (1601-1665) principle of least time: 0=∫ B A phv sd δ For wave motion, we may write vph =λν where vph is the phase velocity (speed) of a mono- chromatic wave of frequency ν and wavelength λ. On substituting vph =λν into δ ∫ds/vph = 0, we get: 0=∫ B A sd λ δ 28
- 29. Ever since Maxwell, light had been understood to be a wave of electric and magnetic waves, but after Einstein and Compton, it became clear that it is also manifested in a particle, the photon. So, is it possible that something like the electron (that had always been regarded as a particle) could also be manifested as some sort of wave? This was first suggested in 1923 by Louis de Broglie as a doctoral student in Paris. 2017 MRT Any kind of wave of frequency ν and wave number k has a spacetime dependence exp(ik•r−iωt), where ω=2πν. Lorentz invariance requires that [ω,k] transform as a four- vector, just like the momentum four-vector [E,p]. For light, according to Einstein, the energy of a photon is E=hν =hω/2π, and its momentum has magnitude |p|=E/c=hν /c= h/λ=h|k|/2π, so de Broglie was led to suggest that in general a particle of any mass is associatedwithawave havingthefour-vector[ω,k]equalto2π/h timesthefour-vector[E,p]: The rays of a monochromatic wave represented by δ ∫ds/λ coincide exactly with the trajectories of the particle with fixed energy, E, determined by δ ∫pds because of p=h/λ. In order to establish a complete description of the motion of a particle by the motion of a wave, we must: 1. Find a suitable wave representation of a single particle, and; 2. Establish the kinematical equivalence of a ray and a particle trajectory. A localized wave whose amplitude is zero everywhere except in a small region in space, a wave packet, will satisfy Condition (1). E hh π2 ω π2 == andpk 29
- 30. A monochromatic plane wave in one dimension (say x) may be represented by: 2017 MRT where F(kx) is some smooth function with peak at an argument ko. where kx is the x-component of the propagation vector, k, for |k|=2π/λ and ω=ω(k)=2πν. The superposition of a group of plane waves of nearly the same wavelength and frequency that interfere destructively everywhere except in a small region results in a wave packet. In the one-dimensional case, such a wave packet may be represented in Fourier analysis by: and therefore |ΨΨΨΨ(x,t)|≅|ΨΨΨΨ([x−ω′(ko)t],0). The wave packet that was concentrated at time t=0 near x=0 is evidently concentrated at time t near x=ω′(ko)t, so it moves with speed vx =dω/dkx =dE/dpx=c2px /E in agreement with velocity within special relativity! ∫ ∞ ∞− − = x tkxki x kdkFtx xx ])(ω[ e)( π2 1 ),(ΨΨΨΨ )ω( e)(),( txki k x ktx − = ϕϕ A maximum (constructive interference) will occur when kx x−ω(kx)t=0 since the sum over the oscillating exponential function for different kx values would result, on the average, in a flat pattern (destructive interference). Suppose also that the wave ∫F(kx)exp(ikx x)dkx at t=0 is peaked at x=0. By expanding ω(kx) around ko, we have: ∫ ′−′−− ≅ x tkxki x kkkti kdkFtx x ])(ω[])(ω)(ω[ oooo e)(e),(ΨΨΨΨ 30
- 31. We now assume that the form of ΨΨΨΨ(0,0) is (see Figure): Expanding ω(k) in a Taylor series about ko, we get: 0])ωω()[( oo =∆−−∆− txkki x A wave packet is a localized wave whose amplitude is zero everywhere except in a small region in space and is a suitable wave representation of a single particle. kx ΨΨΨΨ(0,0) For a maximum, we require that: 2o k k ∆ +2o k k ∆ − ∆k ok Neglecting d2ω/dkx 2 and higher-order terms in the above expansion, we find that ∆x/∆t =(ω−ωo)/(kx −ko) becomes: 2017 MRT ∫ ∆+ ∆− = )2( )2( o o )( π2 1 )0,0( kk kk xkdkFΨΨΨΨ At a later time ∆t and distance ∆x, the form of ΨΨΨΨ is: ∫ ∫ ∆+ ∆− ∆−−∆−∆−∆ ∆+ ∆− ∆−∆ = =∆∆ )2( )2( ])ωω()[()ω( )2( )2( )ω( o o oooo o o e)(e π2 1 e)( π2 1 ),( kk kk x txkkitxki kk kk x txki kdkF kdkFtx x x ΨΨΨΨ or: o oωω kkt x x − − = ∆ ∆ K+ ⋅−+ ⋅−+= == oo ωω 2 2 2 o ωω oo ω )( ω )(ωω xx kd d kk kd d kk xtg x kd d t x v kd d t x ω lim ω 0 oωω = ∆ ∆ = = ∆ ∆ →∆ = or where vg is the group velocity of the wavepacket. 31
- 32. The establishment of the kinematical equivalence of a ray and a particle trajectory (i.e., Condition (2) above) can be accomplished by requiring that the group velocity,vg, of the wave packet equal the velocity of the particle,vp. The velocity of the particle,vp, is given by (using the momentum p=h/λ for the particle): )1( )( )( )( λd hEd hpd hEd pd Ed vp === Now, since: )( 2 2 xV m p E += from which we find (and using de Broglie’s relation): λ h xVEmp =−= )]([2 The group velocity of the wave, vg, is given by (using kx =2π/λ): )1( )π2ω( )π2( ωω λλ d d d d kd d v x g === Since we require that vg=vp, we must have: ν== π2 ω h E thus revealing Planck’s quantization of energy condition (N.B., light speed c=λv): λν /chhE == 2017 MRT 32
- 33. Now we must find an appropriate mathematical formulation for the wave aspect of massive objects. In addition, we must furnish a physical interpretation for the fundamental mathematical entities that will enter our formalism. Obviously the de Broglie-Einstein relations will play a basic role in our work, but they in themselves are not sufficient to provide an adequate theory. We shall need to define some function of space and time to represent the oscillating variable of the wave theory, and this variable will have to be provided with some physical meaning. To carry out this program, let us review in some detail what we know about some aspects of electromagnetic waves and the quantum behavior of electromagnetic fields. 33 2017 MRT )(ε 2 o rEcc ==Φ E In classical electromagnetic theory, the intensity (i.e., the energy delivered to a unit area of beam cross-section per unit time) of a beam of electromagnetic radiation is given by: where E is the energy density of the radiation field of the beam and E(r) is the electric field vector. The very fact that this radiation is characterized in classical theory by an energy density indicates an important distinction between waves (or fields) and particles in classical physics. Evidently, we must think of the energy of the waves as being spread throughout the region of space where the wave field in nonvanishing: the energy is not well localized!
- 34. On the other hand, a particle, whose classical idealization is a mass point without geometric extension (i.e., being of infinitesimally small size) is characterized by a perfectly localized energy. In effect, the energy density of a particle can be written in terms of a three-dimensional δ -function: 34 2017 MRT )( o 3 ParticleParticle rr −−−−δU=E where UParticle is the energy of the mass point and ro is its location in space. Correspondingly, when we make the transition from the wave theory to the particle theory of electromagnetic radiation we might think of each photon as having well- localized energy (and momentum) that it can deliver instantaneously to a well-localized particle such as an electron. Here it should be noticed that the frequency ν and wave number k of the corresponding waves, according to the uncertainty principle, must be completely undefined. If we think of a photon as a radiation pulse with no spatial extension ∆ x1 =∆ x2 =∆ x3 =0, then the wave number spectrum must have infinite breadth, ∆ k1 =∆ k2 =∆ k3 =∞. Thus the phrase “a photon with frequency ν” seems to be self-contradictory; it bids us to think of the photon simultaneously as a wave and as a particle. More accurately we should say “a photon that gave up about hν of energy upon interacting with a massive particle”, keeping in mind that the photon in this context is quite well localized, but not completely so; its energy density is sharply peaked near ro but is not a δ -function. We could speak also of “a photon in a beam of radiation having dominant frequency ν.”
- 35. Although the wave and particle notion, in their idealized forms, cannot be applied simultaneously, it certainly is possible to have waveforms whose spatial extensions is so small that they have essentially particle-like features! This, of course, is the view that we must take in a quantum-mechanical world. The famous ‘wave-particle duality’, then, refers to descriptions of two extreme types of behavior for a single physical phenomenon, and arises from our classical prejudices about the structure of matter rather that from any duality in nature. It has more to do with the limitations of our direct sense of perception and the ways in which we describe our perception than with the fundamental properties of matter. Einstein, de Broglie and their successors have provided us with a new prejudice that gives, within a single framework, a more comprehensive representation of physical phenomena than did the old ones. 35 2017 MRT ννννν hE Λ=Λ=Φ Bearing the above in mind, we may think of a beam of electromagnetic radiation as a beam of photons, the mean energy per photon being hν if ν is the frequency of the radiation. Thus the energy intensity of the beam must be: with Eν =hν and where Λν is the photon flux – the number of photons passing through a unit area of beam cross section per unit time, that is: tA N ∆∆ ∆ =Λ ν ν where ∆Nν is the number of photons passing through cross-sectional area ∆A in time ∆ t.
- 36. Equating the intensities obtained from this particle view and from the wave view, we have: 36 2017 MRT )(ε 2 o rEννν ν ch =Λ=Φ or, using Λν =∆Nν /∆A∆t above: tA h ctAN ∆∆=∆∆Λ=∆ ν ν νν )( ε 2 o rE must be the number of photons in the volume having cross-sectional area ∆A and length c∆t. Since ∆A and ∆t may be chosen arbitrarily, the number ∆Nν is not necessarily an integer. Thus, either photons are indivisible particles and fractions of photons may exist, or the interpretation of ∆Nν as the actual number of photons in the volume ∆Ac∆t should not be taken too literally. The former alternative would render the photon concept meaningless, so we choose the latter. Instead of taking ∆Nν to mean the actual number of photons in the volume ∆Ac∆t, we may take it to mean the number of particles probably to be found in ∆Ac∆t. Thus: ν ν h )( ε 2 o rE may be interpreted as the density of the probability number of photons in the monochromatic beam represented by the electric field vector Eν (r).
- 37. The total number of photons in the beam then is: 37 2017 MRT ∫ rrE rE d)( )( 2 2 ν ν the integral being taken over all of space (represented here by dr≡d3r), and the probability that a single photon would be found in volume element dr centered at r is: ∫∫ == rrEr rE d h d h N )( ε)(ε 2o 2 o ν ν ν νν this is called the probability density for a single photon. This indicates that the magnitude |Eν (r)| of the oscillating physical variable representing electromagnetic waves is proportional to the amplitude of the probability density in the sense that the square of [∫ Eν 2(r)dr]−1/2|Eν (r)| is the probability density for photons in the beam. This suggestion that the quantum interpretation of the oscillating physical variable should be that it represents a probability density amplitude was contributed by Max Born (1882-1970) in 1926, although he introduced the idea in reference to matter fields rather than electromagnetic fields.
- 38. With this view, we are able to weld together the concepts of particle and wave, replacing such statements as “a particle lies within volume element ∆V centered at r” with, for example, “it is 90% probable that a particle will be found in volume element ∆V centered at r”, and also replacing phrases like “the wave with frequency ν” with corresponding probabilistic statements such as “the wave whose probability of having frequency between νo −∆ν and νo +∆ν is ⅔”. A particle-like system corresponds to a wave field whose probability amplitude is vanishingly small except within a ‘small’ region of space-time, whereas wave-like systems correspond to wave fields with probability amplitudes that are vanishingly small except within a ‘small’ volume of wave number- frequency space. 38 2017 MRT A succinct statement of this situation is found in Niels Bohr’s principle of complementarity: At the quantum level, the most general physical properties of any system may be expressed in complimentary pairs of variables, each of which can be better defined only at the expense of a corresponding loss in the degree of definition of the other. Let us consider this principle from a slightly different perspective from that which we adopted above. If we set up an experiment expressly designed to investigate the wave aspects of a system, we must expect that the particle aspects will not be clearly observable in the experiment; an apparatus designed to give good definition of variables that characterize waves must give correspondingly poor definition of the variables that characterize particles.
- 39. For example, if a light beam contains a large number of reasonably well-localized photons, then the addition or deletion of a few of these will not affect the behavior of the beam; the mean energy and momentum per photon is negligible compared with the energy and momentum of the entire beam. If the beam has N>>1 photons, then the fractional uncertainty in the number of photons may be expected to be on the order of 1/N<<1 is the beam is not interacting strongly with its environment. Also, the fractional uncertainty in energy and momentum will be on order of: 39 2017 MRT 1 1 1 1 <<== ∆ <<== ∆ NhN h NhN h E E k k p p and ν ν since |p|=E/c=hν /c=h/λ=h|k|/2π and when ν is the mean frequency of the beam and k is the corresponding wave vector. Therefore, since |p|=h/λ, the uncertainty principle implies: π4π4π4π4 λλ >>= ∆ = ∆ ≥∆ N hh p p pp r the spatial resolution of the beam is poor in the sense that the beam must contain many wavelengths of radiation. A natural measure of the spatial resolution is λ/4π|∆r|, which is small when the resolution is poor and is much larger than unity when the resolution is very good. This, in the situation described here, the photons, being many, tend to fill out a rather large region of space compared with thee beam’s mean wavelength, and the wavelength may be quite we defined. A large number of photons corresponds to wave-like behavior, which, for a light beam, is its classical behavior.
- 40. However, if a mechanism is available for the light beam to interact with its environment in a small region of space (e.g., is an electron or a crystal lattice, with dimensions that are small compared with a typical wavelength, lies within the spatial region covered by the beam), then single photons may be absorbed from the beam through particle-like phenomena, which the remainder of the beam persists in its wave-like properties. A single observation would reveal either the particle-like photon absorption, through the subsequent motion of an electron, say, or the wave-like behavior of the beam, but not both. 40 2017 MRT As was mentioned above, wave motion is what we expect classically for light. The condition under which we gain a correspondence between the newer quantum picture of the system and the older classical picture is that the number of particle quanta must be very large, N>>1. A similar condition yields correspondence between quantum and classical views for massive particles, except that in this case N must be the number of wave quanta with well-defined energy and momentum (and hence frequencies and wave vectors). Classical phenomena appear when the number of quanta with nonclassical properties becomes very large. Just as, for special relativity, the correspondence between ‘old’ and ‘new’ views is obtained when v/c<<1, the quantum-classical correspondence is obtained for N>>1 when N is the number of nonclassical quanta in the system.
- 41. Now we investigate a close relationship to rotations: angular momentum operators. +− + = −= = z yx yx z y x z y x R z y x ϕϕ ϕϕ ϕϕ ϕϕ ϕ cossin sincos 100 0cossin 0sincos )(3 but instead of a finite rotation ϕ about this z-axis, let the angle ϕ be an infinitesimal one ε such that to first order in the approximations sinε ≅ε and cosε ≅1−ε2/2=1 we can get: −= +− + = +− + = → z y x z yx yx z yx yx z y x 100 01 01 cossin sincos ε ε ε ε ϕϕ ϕϕ εϕ or 2017 MRT rr )(3 εR= with − = −= − 100 01 01 )( 100 01 01 )( 1 33 ε ε εε ε ε RR and Angular Momentum Consider once more the first coordinate rotation we developped earlier (i.e., a rotation Rzk about the z-axis – or generally about the 3-axis – by a finite angle of rotation ϕ ):ˆ 41
- 42. The operator D(3,ε) associated with the coordinate transformation R3(ε) gives: ])([)(),3()()( 1 3 rrr εε − == RfffR DD Tofirstorderinε (saya1°angle)we writethe D(3,ε)f(r) functionusingaTaylorexpansion: We define the azimutal angular momentum operator as(i.e.,Lz =xpy− ypx withp=‒ih∇∇∇∇/2π): 2017 MRT Since R3(ε) is an orthogonal matrix, R3 −1(ε) is simply the transpose of R3(ε) so that: ],,[)(1 3 zxyyxR εεε +−=− r KK + ∂ ∂ − ∂ ∂ +=+ ∂ ∂ ⋅+ ∂ ∂ + ∂ ∂ −+= )()(0),,()(),3( rrr f x y y xf z f y f x x f yzyxff εεεεD )( π2 )(),3( r1r 2 22 2 fL h i fLpypx x y y x zzxy +=⇒=−⇒ ∂ ∂ −⋅− ∂ ∂ −⋅ εεD h ihihi hi ππππ ππππππππ ππππ in which the exponential operator is to be understood as the power series: for a finite rotation through an infinitesimal angleε and it is merelynecessary to subdivide ε in Nε (many1°angles) in such a way thatε remains constant as N→∞ and ε →0. Then: ε εε εεε zL h iN z N N N L h i π2 00 e π2 lim),3(lim),3( = +== → ∞→ → ∞→ 1DD εεεε εε xyxyz py h i px h i ypxp h i zz L h i L h i L h i π2π2 )( π22π2 eee... π2 !2 1π2 e −− =≡+ ++= 1 42
- 43. ∂ ∂ + ∂ ∂ −−= ∂ ∂ + − ∂ ∂ + = ∂ ∂ − ∂ ∂ = ϕ ϕθ θ ϕ ϕθ sincotcos π2π2π2 2222 i h yx zy yx x i h z x x z i h Ly The z-component of the angular momentum operator is: This relation may be transformed to spherical coordinates by noting that r =√(x2 +y2 +z2), tanθ =√(x2 +y2)/z and tanϕ =y/x and ∂/∂x=(∂r/∂x)∂/∂r +(∂θ/∂x)∂/∂θ +(∂ϕ/∂x)∂/∂ϕ: θϕθϕθ ∂ ∂+ − ∂ ∂ = ∂ ∂ ∂ ∂ + − ∂ ∂ + + ∂ ∂ = ∂ ∂ ∂ ∂ + − ∂ ∂ + + ∂ ∂ = ∂ ∂ 2 22 2222222222 r yx rr z zyx x yxr zy rr y yyx y yxr zx rr x x and, Hence The other components Lx and Ly of the angular momentum operator may be found in spherical coordinates in a similar way: ∂ ∂ − ∂ ∂ = x y y x i h Lz π2 ϕϕ ∂ ∂ = ∂ ∂ + + = i h x y yx x i h Lz π2 1 π2 2 2 22 2 ∂ ∂ + ∂ ∂ −= ∂ ∂ + − ∂ ∂ + − = ∂ ∂ − ∂ ∂ = ϕ ϕθ θ ϕ ϕθ coscotsin π2π2π2 2222 i h yx zx yx y i h y z z y i h Lx and: 2017 MRT 43
- 44. We havebeenfollowingmostlyuptonowSakurai’s prose on the development of angular momenumbutwe will be mostly interested in the symmetry view advocated by Weinberg. According to Weinberg: operators that commute with the total energy (or Hamiltonian H) can be used to characterize physical states. As such, now try to consider those observables along with the energy that may be used to characterize physical states. One such observable is the angular momentum L=r××××p. It will now become an operator! kjj k kj k jj kk j k jj k x xx xx xx x x xx x δψδψψψ ψ ψ = ∂ ∂ ⇒= ∂ ∂ − ∂ ∂ ⇒ ∂ ∂ = ∂ ∂ − ∂ ∂ ,)()( ∇∇∇∇××××∇∇∇∇×××× rrL i hh i π2π2 ≡−= an ‘imaginary’ quantity and where r is the operator that multiplies a wave function with its argument. Written in terms of Cartesian components, this operator is: 2017 MRT To show that L commutes with the other observables (e.g., the Hamiltonian), let us first consider the commutator of Li with either the position coordinates, xj, or their differentials, ∂/∂xj. Recall that: ∑ ∂ ∂ = kj k jjkii x x i h L ε π2 Making the quantum-defined substitution for the momentum p with −ih∇∇∇∇/2π, this means that in quantum mechanics we will define an angular momentum operator as: 44
- 45. Sincethecomponentsof r commutewitheach other(i.e.,[xi ,xj]=0⇒xi xj = xj xi), we find: To evaluate the commutator of L with the gradient operator, we need only re-write [∂/∂xk,xj]=δkj as [xm,∂/∂xj]=−δjm so that, since the components of the gradient commute with each other: ∑−= k kjkiji L i h LL ε π2 ],[ 2017 MRT ∑−=−= ∂ ∂ + ∂ ∂ −= ∂ ∂ − ∂ ∂ = k kk L i h L i h x x hi x x hi i h x x x xL i h LL 123 2 1 1 2 3 1 1 3121 π2π2π2π2π2 , π2 ],[ ε Tocheck[Li,Lj]=−(h/2πi)Σkεijk Lk wheniandjarenotequal,considerthecasei=1and j=2: Both [Li,xj]=−(h/2πi)Σkεijk xk and [Li,∂/∂xj]=−(h/2πi)Σkεijk ∂/∂xk can be written in the form: This is obviously the case if i and j are equal, because εijk vanishes if two of its indices are equal. and likewise for [L2,L3] and [L3,L1]. Now back to Sakurai’s prose… ∑ ∂ ∂ −= ∂ ∂ k k jki j i xi h x L ε π2 , ∑∑ −== k kkji m mjmiji x i h x i h xL εε π2π2 ],[ 45
- 46. 2017 MRT If an operator commutes with Jz, it is invariant under a rotation about the z-axis. Generalizing this result to a rotation about an arbitrary axis, it may be said that an operator which commutes with all the components of J is invariant under rotation. The development leading to D(R)= D(3,ε)=exp(2πiLzε /h) may be repeated for any rotation axis whose direction is specified by a unit vector n or unit momentum p=p/|p|:ˆ ε εεε nL nnLn ˆ π2 e),ˆ(ˆ π2 1),ˆ( • =⇔•+= h i d h i d DD As an extension, it is possible to define a general angular momentum operator J (which includes L as a special case) as a vector operator that satisfies: ε ε nJ n ˆ π2 e),ˆ( • = h i RD and whose components Jx, Jy, and Jz are Hermitian. It follows at once from the latter property that D(R) is a unitary operator. We will soon show how this commutation rule comes about for J. where dε and ε are the infinitesimal and finite angles of rotation, respectively. The connection between rotation and orbital angular momentum is contained in D(n,ε)= exp(2πiL••••nε /h). ˆ ˆ Also, by considering two rotations in succession and then reversing the order it may be shown that the commutation rules J××××J=iJ follows from D(n,ε)=exp(2πiJ••••nε /h).ˆ ˆ ˆ 46
- 47. Now we investigate the commutation properties of these matrices. Looking at the products of rotations alternatively (and ignoring within any and all product terms higher than ε2) we show that the outcomes are not identical: −− − −− = −− − −− = 2 2 2 1 2 2 1 12 2 2 2 1 2 2 1 21 1 1 1 )()( 1 1 1 )()( εεε εε εε εε εεε εε εε εε 0000 0000 2222 2222 εεεε εεεε RRRR and 2017 MRT A rotationR1 about the 1-axis (or x-axis in Cartesian coordinates) followed by a rotation R2 about the 2-axis (or y-axis) does not equate to a rotation R2 about the 2-axis followed by a rotation R1 about the 1-axis (only if we consider an infinitesimal angleε ). This is not a ‘quantum’ effect per se but the result of infinitesimal rotations and plane mechanics. In retrospect, we have the three rotation matrices: R3(ϕ), R1(ϕ) and R2(ϕ): −= − = − = 100 0cossin 0sincos )( cos0sin 010 sin0cos )( cossin0 sincos0 001 )( 321 ϕϕ ϕϕ ϕ ϕϕ ϕϕ ϕ ϕϕ ϕϕϕ RRR and, and with the approximations (where ε is infinitesimal) sinϕ ≅ε and cosϕ ≅1− ε2/2,we get: −− − = − −− = −− −= 100 01 01 )( 10 010 01 )( 10 10 001 )( 2 2 1 2 2 1 3 2 2 1 2 2 1 2 2 2 1 2 2 1 1 εε εε ε εε εε ε εε εεε RRR &, 47
- 48. And now, the commutator (i.e., the square brackets) of these two rotation matrices is: 1−= =−≡ )( 000 00 00 )()()()(],[ 3122121 22222222 2222 −−−− εεεεεεεε εεεε RRRRRRR εεεε Also, 1= Rany(0) where ‘any’ stands for any rotation axis. )0()()()()()( any31221 RRRRRR −=− 2222 εεεεεεεε The problem now is to identify an operator that will generate these rotations! 2017 MRT )e),ˆ()(()( /ˆπ2 hi RR RR ε ε nJ n • === DDD withΨΨΨΨΨΨΨΨ where |ΨΨΨΨ〉R and |ΨΨΨΨ〉 stand for the kets of the rotated and original system, respectively. Let us look at the effect on a ket state |ΨΨΨΨ〉 representing an observable a subject to a rotation R. Because rotations affect physical systems, the state ket corresponding to a rotated system is expected to look different from the state corresponding to the original un-rotated system. Given a rotation operator characterized by a 3×3 orthogonal matrix R, we associate an operator D(R) in the appropriate ket space such that: The ‘braket’ convention |ΨΨΨΨ〉 is due to Paul Dirac (1902-1984). In quantum mechanics the state of a physical system is identified with a ray in a complex Hilbert space, HHHH, or, equivalently, by a point in the projective Hilbert space of the system. The ket can be viewed as a column vector and written out in coordinates as |ΨΨΨΨ〉=[ ao a1 a2 …]T. 48
- 49. Phase factors (i.e., ‘phases’ now on for short) can be chosen so that the effect of a symmetry transformation on any state vector |ΨΨΨΨ〉 is a transformation |ΨΨΨΨ〉→U|ΨΨΨΨ〉, with U a linear (vs anti-linear) operator satisfying the unitarity condition, i.e., |〈UΦΦΦΦ|UΨΨΨΨ〉|=|〈ΦΦΦΦ|ΨΨΨΨ〉|. There is a special class of symmetries represented by linear unitary operators – those for which U can be arbitrarily close to 1. Any such symmetry operator can conveniently be written as: 2017 MRT If U1 and U2 both represent symmetry transformations, then so does their product (i.e., U1U2). This property, together with the existence of inverses (i.e., U1 −1 and U2 −1) and a trivial transformation 1, means that the set of all operators representing symmetry transformations forms a group. )()( 2 εεε OTiU ++= 1 where ε is an arbitrary real infinitesimal number and T is an (ε-independent)operator. 49 Is this the only way that symmetry transformations can act on physical states (e.g., our ket state |ΨΨΨΨ〉 above subject to a rotation R)? In quantum mechanics a physical state is not represented by a specific individual normalized vector in Hilbert (1862-1943) space, but by a ‘ray’, the whole class of normalized state vectors that differ from one another only by phase factors (i.e., numerical factors with modulus unity). To represent a symmetry, such as a transformation of rays must preserve transition probabilities (i.e., if |ΦΦΦΦ〉 and |ΨΨΨΨ〉 are state vectors belonging to the rays representing two different physical states, and a symmetry transformation takes these two rays into two other rays containing the state vectors |ΦΦΦΦ〉 and |ΨΨΨΨ〉 then we must have |〈ΦΦΦΦ|ΨΨΨΨ〉|2=|〈ΦΦΦΦ|ΨΨΨΨ〉|2).
- 50. Bydefinition,a rotationisalineartransformationxi →ΣjRij xj of a 3D systemrepresented by (e.g., using Cartesian coordinates) xj that leave invariant the scalar product x•y=Σi xi yi: ∑∑ ∑∑ = i ii i k kki j jji yxyRxR with sums over i, j, k, &c. running over the values 1, 2, 3. By equating coefficients of xj yk on both sides of the equation, we find the fundamental condition for a rotation: or in matrix notation RTR=1 where RT denotes the transpose of a matrix, [RT]ji=Rij, and1 is here the unit matrix, [1]jk=δjk where δij is the Kronecker delta, i.e., δij =1 when i = j and δij =0 wheni ≠ j.Note that not all transformations xi →ΣjRij xj with Rij satisfying RTR=1, are rotations.Taking the determinant of RTR=1(i.e.,detRTR=1) and using the fact that the de- terminant of a product of matrices is the product of the determinants(i.e.,detRTR=detRT⊗ detR=1) and that the determinant of the transpose of a matrix equals the determinant of the matrix (i.e., detRT =detR) we see that detRTR=[detR]2=1, so detR can only be +1or −1: kj i kiji RR δ=∑ − + = InversionsSpace Rotations 1 1 det R The rotations form the special orthogonal group in three dimensions, or SO(3), where ‘orthogonal’ means that it consists of real 3×3 matrices satisfying the relation ΣiRij Rik =δjk above, and ‘special’ indicates that these matrices have unit determinant. 2017 MRT 50
- 51. Like other symmetry transformations, a rotation R induces on the Hilbert space HHHH of physical states a unitary transformation – in this case |ΨΨΨΨ〉=U(R)|ΨΨΨΨ〉. If we perform a rotation R1 and then a rotation R2, physical states undergo the transformation |ΨΨΨΨ〉= U(R2)U(R1)|ΨΨΨΨ〉, but this must be the same as if we had performed a rotationR2R1,so: Rotations (unlike inversions) can be infinitesimal. In this case: so ωT =−ω (the components are ω21=−ω12, &c.) or in other words ω is antisymmetric: For example, when it is acting on the general three-dimensional operator V representing a ‘vector’ observable (such as a cartesian coordinate vector r≡ri =[x y z]T or a general momentum vector p≡pi), U(R) must induce a rotation (on each of its components): )()()( 1212 RRURURU = )ω(ω 2 OR jijiji ++= δ with ω infinitesimal. The condition RTR=1 gives here: )ω(ωω)]ω(ω)][ω(ω[ 222 OOO +++=++++ TT 111 jiji ωω −= which in essence provides the definition of a three-tensor Vi (or similarity transformation). 2017 MRT ( )3,2,1)()( 332211 3 1 1 =++== ∑= − iVRVRVRVRRUVRU iii j jjii 51
- 52. For such infinitesimal rotations, the unitary operator U(R) must take the form: )ω(ω π )ω( 2 3 1 3 1 OJ h i U j i jiji ++→+ ∑∑= = 11 with Jij = −Jji a set of Hermitian operators. 2017 MRT ∑∑∑ == = −+→+ 3 1 3 1 3 1 2π ω π )ω,( k kk j i jiji P h i J h i U εε 11 If we now add to this the infinitesimal space translations ε, the unitary operator becomes: Finally, adding to this the infinitesimal time translationτ, the unitary operator becomes: E h i P h i J h i U k kk j i jiji τετε 2π2π ω π );ω,( 3 1 3 1 3 1 +−+→+ ∑∑∑ == = 11 ∑= −→ 3 1 )( k kk KiU ξζ 1 To this set, we can add the infinitesimal boost ξ, the unitary operator for this is: 52
- 53. We can find the transformation rules for Jij and their commutators with each other. As an application of U(R2)U(R1)=U(R2 R1) we have: )ω1()ω(])ω1([])ω1[(][)()ω1()( 111111 RRURRRRURRURURURUURU −−−−−− +=+=+=+=+ for any ωij = −ωji, and any rotation R, unrelated to ω. To first order in ω, we then have: ∑∑∑ == −− l ll l ll kji kjijki k kk ji jiji JRRJRRURUJRU ω)ω()()(ω 11 in which we have used RTR=1, which gives R‒1 =RT. Equating the coefficients of ωij on both sides of this equation then gives the transformation rule of the operator Jij : ∑=− l ll k kjkiji JRRRUJRU )()( 1 which indicates that Jij is a tensor, i.e., its components transform as a tensor! We can take this a step further, and let R itself be an infinitesimal rotation, of the form R→1+ω, with ωij = −ωji infinitesimal. Then, to first order in ω,U(R‒1)Jij U(R)=ΣklRik Rjl Jkl gives: ∑∑∑∑ +=+= l ll l lll l ll ij k jkki k kkijjki k kkji JJJJJ h i ωω)ωω(]ω,[ π δδ Equating the coefficients of ωkl on both sides of this equation gives the commutation rule of the Js: 2017 MRTjijikjjkijkikji JJJJJJ h i lllll δδδδ −++−=],[ π2 53
- 54. In three dimensions it is very convenient to express Jij in terms of a three-component operator J, defined by: 123312231 JJJJJJ ≡≡≡ and, or more compactly: ∑∑ == k kkjiji ji jikjik JJJJ εε 2 1 2 1 and )ω( π )ω1( 2 O h i U +•+→+ Jω1 where ωk ≡½Σij εijk ωij. The rotation here is by an infinitesimal angle |ωωωω| around an axis in the direction of ωωωω. Note also that, J is itself a three-vector. ∑= k kkjiji J hi JJ ε π2 ],[ 2017 MRT For instance, [J1,J2]=ihJ3/2π or, using (2πi/h)[Jij, Jkl], we get [J23,J31]=−ihJ21/2π. The operator U(1+ω)=1+(πi/h)Σijωij Jij for infinitesimal rotations then takes the form: The commutation relation (2πi/h)[Jij ,Jkl] =−δ il Jkj +δik Jl j +δjk Jil −δ jl Jik takes the form: with, e.g., J1= Jx, J2= Jy and J3= Jz, and where εijk is the totally antisymmetric coefficient, defined by εijk ≡+1(ε123 =ε231 =ε312 =+1)when i,j,k are even permutations of 1,2,3 andεijk ≡ −1 (ε213 =ε321 =ε132 =−1) when i, j, k are odd permutations of 1, 2, 3 and then 0 otherwize. 54
- 55. The equationjust obtained(i.e., [Ji, Jj]=ihΣkεijk Jk /2π) is the same commutation relation asthecommutationrelation[Li, Lj]=ihΣkεijk Lk/2π satisfiedbytheorbitalangularmomentum operator L, but derived here (à la Weinberg) from the assumptions of rotational symmetry, with no assumtions regarding coordinates or momenta. This commutation relation will be the basis of the treatment of angular momentum to come afterwards. and since L is a vector we must have: Consider that in general, the angular momentum J of any particle is the sum of its orbital angular momentum and the spin angular momentum. Thus the total angular momentum J of a particle may be different from its orbital angular momentum L. ∑= k kkjiji J hi LJ ε π2 ],[ 2017 MRT ∑= k kkjiji J hi LL ε π2 ],[ A system containing several particles has a total angular momentum given by the sum of the orbital angular momenta Ln and spins Sn of the individual particles: ∑∑ += n n n nn SLJ As we saw earlier, direct calculation shows that in the case of a particle in a central potentialtheoperatorL≡r××××p satisfies thesame commutation relation[Ji, Jj]=ihΣkεijk Jk /2π as J: 55
- 56. Let us consider a wave vector k in the z-direction z, and limit ourselves to rotations that leave z invariant. )ω2( π )ωω( π )ωω( π )]ωωω()ωωω()ωωω[( π )ωωω( π ω π )ω1( 3 1 3 1 3 1 yxyx yxyxyxyxyxyxxyxy zzzzzyzyzxzxyzyzyyyyyxyxxzxzxyxyxxxx j jzjzjyjyjxjx j i jiji J h i JJ h i JJ h i JJJJJJJJJ h i JJJ h i J h i U += +−⋅−+=++= +++++++++= +++=+=+ ∑∑∑ == = 1 11 1 11 ˆ ˆ − == − 100 0cossin 0sincos )]([)( 1 3 ϕϕ ϕϕ ϕϕ RR ji Now for infinitesimal ϕ, Rij =δij +ωij, where the non-vanishing elements of ωij are ωxy= −ωyx=−ϕ, so according to U(1+ω)=1+(πi/h)Σijωij Jij and Jij =½Σk εijk Jk (i.e., Jxy =−Jyx =Jz ) and finally we have for U (and U−1): zJ h i U ϕϕ 2π )( −= 1 2017 MRT According to the tensor relation U−1(R)Vi U(R)=Σj RijVj, under a rotation represented by an orthogonal matrix Rij. The ‘active’ rotations that leave z invariant have the form:ˆ 56
- 57. Introducing Dirac’s h as Planck’sconstant h divided by 2π wedefine an operatorS≡J−−−−L so that J=L++++S then by substracting [Li, Lj]=ihΣkεijk Lk from [Ji, Lj]=ihΣkεijk Lk, we find: Thus S acts as a new kind of angular momentum, and may be thought of as an internal property of a particle, called the spin. The total angular momentum is thus J (=L++++S). 0],[ =ji LS ∑= k kijkji SiSS εh],[ The total angular momentum J is the sum of the angular momentum vector L= r ×××× p and (intrinsic) spin angular momentum vector S =(h/2)σσσσ. The z-component of L is also shown – parallel to the z-axis. The particle thus has an (intrinsic) spin angular momentum S= (h/2)σσσσ over and above its orbital angular momentum L= (r ×××× p)i = ihSjkεijk xk∂j (∂j≡∂/∂xj) where we inserted r→xk and p=−ih∇∇∇∇→−ih∂j. y x z z ϕ L Hilbert Space (i) Quantum Realm (h) S J whereas, in spherical coordinates, and explicitly showing the azimutal ϕ-dependence, we get : ∂ ∂ − ∂ ∂ = ∂ ∂ − ∂ ∂ == x y y x iy x x yiLL z h h3 Lz 2017 MRT Strong Magnetic Field (Bz ) ϕϕ ∂ ∂ = ∂ ∂ −== i iLL z h h3 ϕ∂ ∂ = i Lz h Decomposing this relation to highlight the angular momentum components: L3=(r××××p)3=ihSjkε3jk xk∂j=ih[(ε311x1∂1+ε312x2∂1+ε313x3∂1) +(ε321x1∂2+ε322x2∂2+ε323x3∂2)+(ε331x1∂3+ε332x2∂3 +ε333x3∂3)]= ih[(0+1⋅x2 ∂1+0)+(−1⋅x1∂2+0+0)+(0+0+0)]= ih(x2∂1−x1∂2). So, in these Cartesian coordinates, we have: From [Si, Lj]=0, J=L+S, [Li, Lj]=ihΣkεijk Lk, and [Ji, Jj]=ihΣkεijk Jk we then have: 57
- 58. Since each Li commutes with r, it must act only on the direction of the argument r, not its length. That is, in polar coordinates defined by θϕθϕθ cossinsincossin 321 rxrxrx === and, Also in polar coordinates: ∂ ∂ + ∂ ∂ ∂ ∂ −= ϕθθ θ θθ 2 2 2 22 sin 1 sin sin 1 hL 2017 MRT 3 2 1 1 2 21 cossinsinsin L i x x x x x r x r x x i i i h = ∂ ∂ + ∂ ∂ −= ∂ ∂ + ∂ ∂ −= ∂ ∂ ∂ ∂ = ∂ ∂ ∑ ϕθϕθ ϕϕ As an example of how to these are derived, let us calculate L3, which will be of special importance for us later on. Note that: justifying the formula in −ih∂/∂ϕ for L3. The operators Li act only on θ and ϕ. From the definition Li =−ihΣjkεijk xj ∂/∂xk of these operators, we can work out their explicit form in polar coordinates: ϕϕ ϕθ θ ϕ ϕ ϕθ θ ϕ ∂ ∂ = ∂ ∂ + ∂ ∂ −= ∂ ∂ + ∂ ∂ = hhh iLiLiL 321 sincotcoscoscotsin and, 58
- 59. When we can neglect spin-angular momentum (i.e.,when S=0), the angular momen- tum J for a single particle is then the same as orbital angular momentum L, which is: Consider a spinless particle subjected to a spherical symmetri- cal potential. The wave function is known to be separable in sphe- rical coordinate and the energy-eigenfunction can be written as: prL ×= where r is a vector whose components ri are x, y, z (or x1, x2, x3) and p=−ih∇∇∇∇ is the linear momentum operator. It is possible to write 〈r|L2 |l,ml〉 using L2 =Lz 2 +½(L+ L− +L− L+) and 〈r|Lz |l,ml〉=−ih(∂/∂ϕ)〈r|l,ml〉 and〈r|L± |l,ml〉=−ihe±iϕ (±i∂/∂θ −cotθ ∂/∂ϕ )〈r|l,ml〉, we get: where the position vector r is specified by the spherical coordinates r, θ, and ϕ , and n stands for some quantum number other than l and ml , e.g., the radial quantum number for bound- state problems or the energy for a free-particle spherical wave. When the Hamiltonian is spherically symmetrical, H commutes with Lz and L2, and the energy eigenkets are expressed to be eigenkets of L2 and Lz also. Because Lk (with k=1,2,3) satisfy the angular-momentum commutation relations, the eigenvalues of L2 and Lz are expected to be l(l+1)h2, and ml h=[−lh,(−l+1)h,…, (l −1)h,lh]. ll lhl mm ,sin sin 1 sin 1 , 2 2 2 22 rLr ∂ ∂ ∂ ∂ + ∂ ∂ −= θ θ θθϕθ Relation between the rectangular coordinates (x, y, z) and spherical polar coordinates (r,θ ,ϕ ) of an electron (with charge e−) with respect to the nucleus (with charge Ze+) at the origin. 2017 MRT ),()(),,(,, ϕθϕθψ l l lllll m nmn YrRrmn ==r),()( 2 e ),()( ),,(,, 12 1 o o ϕθ ϕθ ϕθψ l l l l l l l l ll lll m n rr n m n mn YrL r r N YrR rmn ⋅ = ⋅= = + −− − r hhll 6)1( =+=L hl 2+=m Lz hl 2−=m hl +=m hl −=m 0=lm L ϕ y x r e− Ze+ θ zLz nˆ L p ∇∇∇∇×××× ×××× hir prL −= = 59
- 60. Quantum Behavior Quantum mechanics departs from classical mechanics primarily at the atomic and sub-atomic scales, the so-called quantum realm. In special cases some quantum mechanical processes are macroscopic, but these emerge only at extremely low or extremely high energies or temperatures. In order to peer into that quantum realm, let us start with the macroscopic world and consider a machine gun aiming randomly at a wall with two slits and shooting a stream of bullets. In this experiment, we find that bullets always arrive in lumps and is always measured as one whole bullet. We call the probability P12 because the bullets may have come either through hole 1 (P1) or through hole 2 (P2): “Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of what happens at the atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen.” Richard Feynman, Feynman Lectures on Physics, Vol. III (1964), P. 1. 2112 PPP += Quantum mechanics provides a mathematical description of the dual particle-like and wave-like behavior and interaction of matter and energy. Sir Isaac Newton (1643-1727) thought that light was made up of particles, but then it was discovered by Christiaan Huygens (1629-1695) that light behaves like a wave. In the beginning of the twentieth century it was found that light did indeed ‘sometimes’ behave as like a particle! P12 will end up large near the middle of the graph but gets small at either edge (see Figure). An experiment with bullets using a machine gun. Block slit 2 and P1 results – block slit 1 and P2 results. Keep both slits 1 and 2 open and P12 shows that the probabilities add up. 1P 2P 12P 2112 PPP += WALL ARMOR PLATE MACHINE GUN 2 1 • 2017 MRT 60
- 61. The measured intensity is proportional to the mean squared hei- ght, h2, or, when using complex numbers, to the absolute value squared |h1|2 for hole 1 and similarly |h2 |2 for hole 2. When both holes are open, the wave heights add to give the intensity |h1+h2|2: where δ is the phase difference between h1 and h2. In terms of intensities, we could write: Quantummechanics is based upon the concept that subatomic particles can have both wave-likeandparticle-likeproperties.This phenomenonis known as wave-particle duality. de Broglie proposedinhis1924 PhD thesis that subatomic particles(such as electrons) are associated with waves – “With every particle of matter with rest mass and velocity,a real wave must be ‘associated’ and related to the momentum by the equation”: δcoshh2hhhh 21 2 2 2 1 2 21 ++=+ δcos2 212112 IIIII ++= where the last term on the right side is the ‘interference’ term that provides the crests and troughs (see Figure). The result is quite different than that of P12 =P1+P2. If we expand |h1 + h2 |2 we see that: 2 2112 2 22 2 11 hhh,h +=== III and − = → −=== << energytotal theisE VEm h c v vm h p h p h cv )(2 1 o 2 oo λ γ λ whereλ is the wavelength,h is the Planck constant,p is the relativisticmomentum(p=γpo), mo is the rest mass, v is the velocity, c is the speed of light (vacuum) and V the potential. 2017 MRT An experiment with waves using a shallow trough of water. Block slit 2 and I1 results – block slit 1 and I2 results. Keep both slits 1 and 2 open and I12 shows an interference pattern! 1 2 •WAVE SOURCE 2 22 2 11 hh == II & 1I 2I 12I hh 2 2112 +=I WALL ABSORBER 61
- 62. RULE #2: When a particle can reach a given state (say point b) by two possible paths, the total probability amplitude for the process is the sum of the probability amplitudes for the two paths considered separately (see Figure – top): RULE #3: When a particle goes by some particular route through slit d, the amplitude for that route can be written as the product of the amplitude to go part way with the amplitude to go the rest of the way (see Figure – bottom): RULE #1: The probability P(b, a) that a particle will arrive at b, when let out at the source a, can be represented quantitatively by the absolute square of a complex number called the probability amplitude K(b, a) – in this case, the “amplitude that a particle from a will arrive at b” is: abababKabKabK 2211),(),(),( 2Path1Path +≡+= addbadKdbKabK 1Route2Route 11),(),(),( 1Path ≡×= A more complicated experiment showing an arbitrary path composed of two possible routes for the electron to take in order to reach the detector located at point b. 22 ),(),( ababKabP ≡= Restoring the braket convention and including all alternative paths: ∑ ∑= = = 2,1 ,,i edcj aiijjbab •ELECTRON GUN 2 1 a b Path 2 Path 1 2 2Path 2 1Path ),( ),( KabP KabP = = or 1Path),( abP ),( abP 2 ),( 2Path 1Path ),( ),( abK abK +=abP 2Path),( abP WALL PHOTO- GRAPHIC PLATE 2017 MRTWALL PHOTO- GRAPHIC PLATE •ELECTRON GUN 1 a d b Route 2 WALL c e aeeb accbab aeKebK acKcbKabK 22 ... 11 ),(),( ... ),(),(),( 2Path 1Path + ++ += + ++ += addb adKdbK 11 ),(),( 1Path or 2 Route 1 An experiment with an electron incident on two slits results in two possible paths in order to reach a detector located at point b. Assume we have an event consisting in the registration of an electron in a detector at some position along a photographic plate. Assume further that the electron can pass through either of two slits – hence two alternative paths for the event to occur. We can complicate things further by adding another wall with three slits and as a result include these individual steps that the electron must route through along it’s way to the detector. 62
- 63. Postulates POSTULATE #1: All the physically relevant information about a physical system at a given instant of time is derivable from the knowledge of the state function ΨΨΨΨ of the system. This state function is represented by a ray in a complex Hilbert space, H . In physics, a vector in Hilbert space H is denoted by |ΨΨΨΨ 〉, and the inner product (ΦΦΦΦ,ΨΨΨΨ) is written as 〈 ΦΦΦΦ | ΨΨΨΨ 〉 where | ΨΨΨΨ 〉 and 〈 ΦΦΦΦ | are called ket and bra vectors, respectively. This latter ‘bra-ket’ notation is due to the Nobel Laureat English physicist Paul A. M. Dirac. The vectors | ΦΦΦΦ 〉 and | ΨΨΨΨ 〉 in the complex Hilbert space H. The normalization condition for the state vectors is 〈ΨΨΨΨ|ΨΨΨΨ〉=1 where 〈ΦΦΦΦ | ΨΨΨΨ〉=〈ΨΨΨΨ | ΦΦΦΦ〉∗ denotes the scalar product of the state vectors. If |ΨΨΨΨ〉is a vectorwhich correspondsto a physicallyrealizablestate,then|ΨΨΨΨ〉 and a constant multiple of |ΨΨΨΨ〉 (e.g., ξ|ΨΨΨΨ〉) both represent this state. The normalization condition (i.e., so that probabilities add up to unity) is expressed as 〈ΨΨΨΨ|ΨΨΨΨ〉=1, whereas 〈ΦΦΦΦ|ΨΨΨΨ〉=〈ΨΨΨΨ|ΦΦΦΦ〉∗ is the scalar product of the vectors |ΦΦΦΦ〉 and |ΨΨΨΨ〉 (e.g., it is similar to the vector dot product u•v). For a Euclidian vector space Vn to be complete, it is required that each Cauchy sequence in Vn: ||ψi − ψj ||→ 0 (for i and j → 0) converges to a limit in Vn. A complete and infinite-dimensional complex Euclidian space is called a Hilbert space, H. ii Aψφ = where A is a linear operator. Using the Dirac notation, we write: ΨΨΨΨΦΦΦΦ A= For linear operators A and B, it is required that: ΨΨΨΨΨΨΨΨ ΨΨΨΨΨΨΨΨ ΨΨΨΨΨΨΨΨΨΨΨΨ ΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦ AkkA BABA BABA AAA = = +=+ +=+ )()( )( )( where k is a scalar. Theket vectors |ΦΦΦΦ〉 and |ΨΨΨΨ〉 are arbitrary in Vn. 2017 MRT |ΦΦΦΦ〉 |ΨΨΨΨ〉 〈ΦΦΦΦ|ΨΨΨΨ〉 〈ΨΨΨΨ|ΨΨΨΨ〉=1 Vn A Bˆ ˆ H A linear operator on a Euclidian vector space Vn is a procedure for obtaining a unique vector, φ i, in Vn for each ψ i in Vn. For example: 63
- 64. 1=′′ =′′′ ′′=′ ∑′ ′′′ a aa aa aa aaaA δ where the symbol δa′a" is to be understood as the Kronecker symbol if the eigen- values a′ and a" (i.e.,δa′a" =1 if a′=a" , δa′a" =0 if a′≠a" ) lie in the discrete spectrum and as the Dirac δ function δ (a′−a"), if either or both lie in the continuous spectrum. POSTULATE #2: To every measurable (i.e., observable) property, α, of a system corresponds a self-adjoint operator A = A* with a complete set of orthogonal eigenfunctions |Φi〉 or say |a′〉 (according to Sakurai) and real eigenvalues a′: Following the words of wisdom of Dirac: “A measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured.” In other words: Before a measurement of the observable α is made, the system is assumed to be represented by some linear combination (e.g., this is analogous to a vactor expansion V=Σiêi(êi••••V) of a vector V in – real – Euclidian space): So, when the measurement is performed, the system is ‘thrown into’ one of the eigenstates, say |a′〉 of observable α. Again, in other words: ∑∑∑ ′′′ ′ ′′≡′′== aaa a aaaac αααα 'aA → tmeasuremenOperator α 2017 MRT 64
- 65. aaaA ′′=′ The importance of properly solving eigenvalue problems in quantum mechanics is at the base of various academic problems. In general, it is assumed that associated with each linear operator is a set of functions and corresponding numbers such that: where A is a linear operator, |a′〉 (also represented as ui) are called eigenfunctions, and the a′ are known as eigenvalues. The matrix form of the eigenvalue problem is: 0)( =′′− aIaA In solving the general matrix equation BX=C for X, we obtain (trust me): X=BcTC/|B| where BcT is the cofactor transpose – or adjoint – matrix. If C is a null (i.e., zero) matrix and |B|≠0, then the solution, X, is a trivial one. For C equal to a null matrix, a necessary condition for a nontrivial solution of BX=0 is that |B|=0. Hence the condition for a nontrivial solution of (A−a′I)|a′〉=0 is that the determinant of A−a′I be equal to zero: 0)det( =′−≡′− IaAIaA which is called the secular (or characteristic) equation of A. The eigenvalues are just the roots of the equation obtained by expanding the determinant |A−a′I|=0. That is: 0 21 22221 11211 = ′− ′− ′− =′− aaaa aaaa aaaa IaA nnnn n n L MMM L L 2017 MRT 65
- 66. As an example of how to use secular equations, we shall compute the eigenvalues, eigenvectors and a new diagonalizing matrix, P, for the following Pauli spin matrix: = 01 10 xσ In quantum mechanics the operator for the x-component of spin is: xxS σ 2 h = which gives the equation to be solved as σx|S x〉=a′|S x〉. The secular equation for σx is: 101)( 2 ±=′⇒=−′−=′− aaIaxσ thus the x-component of spin, S x, has the eigenvalues ±h/2. To determine the eigenvectors that correspond to these eigenvalues, we put them back one at a time, into the equation σx|S x〉=a′|S x〉. For +1 we have: a==⇒ = 21 2 1 2 1 01 10 ξξ ξ ξ ξ ξ so an eigenvector belonging to a′=+1 has the form: ( )0≠ a a a 2017 MRT 66
- 67. Similarly, an eigenvector belonging to a′=−1 has the form: ( )0≠ − b b b 2017 MRT Thus: 02 ≠−= − = abP ba ba P and Therefore the inverse of the diagonalizing matrix, P−1, exists; it is given explicitely by: − −− −==− aa bb ab P P P 2 11 c1 T Now the similarity relation applied to σx implies that: ′ ′ = − + = − = − − = − − =− 2 1 1 0 0 10 01 20 02 2 1 2 1 01 10 2 1 a a ab ab abba ba aa bb abba ba aa bb ab PP xσ This means that since σx can be diagonalized by a similarity transformation P−1σx P, then the set of eigenvectors of σx spans the whole vector space (i.e., it is omnipo- tent!) as do also its counterparts σy and σz which is left as an Exercise to the reader. 67
- 68. POSTULATE #3: If a measurement is performed on a system to determine the value of the observable α, the probability of finding the system, described by the state vector |ΨΨΨΨ〉, to have α with the value a′ (according to Sakurai) is given by |〈a′|ΨΨΨΨ〉|2.In other words|〈a′|ΨΨΨΨ〉|2 is the probability amplitude of observing the value a′. As stated previously, a measurement on a system will, in general, perturb the system and, thus, alter the state vector of the system. If as a result of a measurement on a system we find that the observable α has the value a′ the (unnormalized) vector describing the system after the measurement is |a′〉〈a′|ΨΨΨΨ〉. An immediate repetition of the measurement will thus again yield the value a′ for the observable α. A measurement of the property α thus channels the system into a state which is an eigenfunction of the operator A. However, only the probability of finding the system in a particular eigenstate is theoretically predictable given the state vector |ΨΨΨΨ〉 of the system. If this state vector is known, measurements then allow the verification of the predicted probabilities. It is usually the case that several independent measurements must be made on the system to determine its state. It is therefore assumed in all of the realm of computational Quantum Mechanics that it is always possible to perform a complete set of compatible independent measurements, i.e., measurements which do not perturb the values of the other observables previously determined. 2017 MRT 68
- 69. So, again: A measurement usually changes the state! We do postulate, however, that the probability for jumping into some particular state |a′〉 is given by: provided that |α〉 is normalized. This equation is a definition; however, it agrees with our intuitive notion of average measured value because it can be written as: We define the expectation value of an operator A taken with respect to state |α〉 as: 2 αaa ′=′foryProbabilit αα AA ≡ where it is very important not to confuse eigenvalues with expectation values. According to Weinberg’s convention, a set of vectors |Φi〉 that are orthogonal and also normalized so that Σi 〈Φi |Φi 〉=1 is said to be orthonormal. For a complete orthonormal set of n basis vectors |Φi〉, we get: 2017 MRT aa aaA aaAaaA a a a ′′ ′×′= ′′′′′′= ∑ ∑∑ ′ ′ ′′ obtainingforyprobabilitvaluemeasured 2 α ααExpectation: Measured: nn n i ii ΦΦ++ΦΦ+ΦΦ=ΦΦ= ∑= ΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨ K2211 1 69
- 70. When we change our point of view in a certain way, the laws of physics do not change! e.g., travelling inside an airplane from Ottawa, Canada to Nadi, Fiji should not change the laws of physics observed in the airplane as compared to the ground (Earth). Thus symmetry transformations must leave all |〈 ΦΦΦΦ|ΨΨΨΨ〉|2 invariant! According to Weinberg, one way to satisfy this condition is: Suppose that a symmetry transformation takes general state vectors |ΨΨΨΨ〉 into other state vectors U|ΨΨΨΨ〉, where U is a linear operator satisfying the condition of unitarity that for any two state vectors |ΦΦΦΦ〉 and |ΨΨΨΨ〉, we have: In particular, symmetry transformations must not change transition probabilities: so the condition of unitarity may also be expressed as an operator relation: 2017 MRT In technical terms, moving or rotating our laboratory should not change the laws observed in the laboratory. Such special ways of changing our point of view are called symmetry transformations. This definition does not mean that a symmetry transformation does not change physical states, but only that the new states after a symmetry transformation will be observed to satisfy the same laws of nature as the old states. 2 ) ΨΨΨΨΨΨΨΨ iiP Φ=Φ→( ΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦ =UU Recall that the adjoint of an operator U is defined so that (i.e., 〈ΨΨΨΨ| A†ΨΨΨΨ〉=〈 A ΨΨΨΨ|ΨΨΨΨ〉): ΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦ UUUU † = 1=UU † 70
- 71. We limit ourselves to symmetry transformations that, like rotations and translations, have inverses, which undo the effect of the transformation (e.g., the symmetry transformation of rotating around some axis by an angle θ has an inverse symmetry transformation, in which one rotates around the same axis by an angle −θ.) If a symmetry transformation is represented by a linear unitary operator that takes any |ΨΨΨΨ〉 into U|ΨΨΨΨ〉, then its inverse must be represented by a left-inverse operator U−1 that takes U|ΨΨΨΨ〉 into|ΨΨΨΨ〉, so that the application of the left-inverse operator U−1 to U is unity: The same must be true for U−1 itself, so it has a left-inverse (U−1)−1 for which (U−1)−1U−1 = 1. Multiplying this on the right with U and using U−1U=1 then gives: Acting on U †U=1 on the right with U−1, we see that the inverse of a unitary operator is its adjoint: 2017 MRT so by applying U−1U=1 to U−1, we see that the left-inverse of U is also a right-inverse: 1≡− UU 1 †1 UU ≡− 1=−1 UU UU =−− 11 )( The operator 1 represents a trivial symmetry transformation (i.e., it does nothing to state vectors). It is of course unitary as well as linear. It’s also an identity operator. 71
- 72. With U(ε) =1 +iε T +O(ε 2) defined earlier, the unitary condition U−1(ε )U(ε ) =1 gives: Thus Hermitian operators arise naturally in the presence of infinitesimal symmetries. The operator T appearing in the unitary operator relation U(θ ) =1 +iθ T +O(θ 2) is known as the generator of the symmetry. As we shall see, many if not all of the operators representing observables in quantum mechanics are the generators of symmetries. 2017 MRT If we take ε =θ/N, where θ is some finite N-independent parameter, and then carry out the symmetry transformation N times and let N go to infinity, we find a transformation represented by the operator: UAUA 1− → 11111 =+++−=++++ )]()][([)]([)]([ 22†2†2 εεεεεεεε OTiOTiOTiOTi or, to first order in ε : †2† )( TTOTiTi =⇒=++− 11 εεε )(elim 2 2 θ θθ θ θ UT N iTiTi Ti N N =→ +=+ ++ ∞→ 11 K Under a symmetry transformation |ΨΨΨΨ〉→U|ΨΨΨΨ〉, the expectation value of any observable A is subjected to the transformation 〈ΨΨΨΨ| AΨΨΨΨ〉→〈UΨΨΨΨ|UAΨΨΨΨ〉=〈ΨΨΨΨ|U−1AUΨΨΨΨ〉, so we can find the transformation properties of expectation values by subjecting observables to transformations called similarity transformations: 72
- 73. POSTULATE #4: The position R (or the generalized coordinate q in some texts) and total momentum P ‘operators’ of a particle obey the following commutation rules: jiijjiji iRPPRPR δh=−=],[ A necessary and sufficient conditions for two measurements to be compatible (or simultaneously performable) is that the operators corresponding to the properties being measured (e.g., position comonents Ri or momentum components Pi) commute: A maximal set of observables which all commute with one another defines a complete set of commuting operators. Here is the algebra surrounding these commutators: 0],[ 0],[ = = ji ji PP RR where i=√−1, the imaginary number, and h is Planck’s constant h divided by 2π. )(0]],[,[]],[,[]],[,[ ],[],[],[ ],[],[],[ )(0],[ ],[],[ 0],[ cyclical numberaisif =++ += +=+ = −= = BACACBCBA CABCBABCA CBCACBA kkA ABBA AA 2017 MRT 73