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2 IOP Concse Physcs A Concse Introducton to Quantum Mechancs Mark S Swanson Chapter Classcal mechancs and electromagnetsm The mechancs of moton formulated by Newton and contemporares n the 7th Century s the foundaton of contemporary physcs. It s mmensely successful, explanng and quantfyng a vast array of phenomena, rangng from planetary moton to the flow of fluds. By the latter part of the 9th Century the study of electrc and magnetc felds and ther ncluson n the framework of Newtonan physcs appeared to be near completon. Snce moton and electromagnetsm are central to the falures of Newtonan physcs n the atomc realm that eventually resulted n the creaton of quantum mechancs, t s useful to revew the essental deas, concepts, and the mplct assumptons of what s now referred to as classcal physcs. Many of the results presented n ths chapter are assumed to be famlar to the reader, but some concepts are more thoroughly developed.. Newtonan mechancs For smplcty, ntal consderaton s lmted to a sngle pont partcle of nertal mass m. It s possble to formulate Newtonan mechancs n terms of contnuous matter dstrbutons, referred to as contnuum mechancs, but for the purposes of ths monograph only pont partcles wll be consdered. Pont partcles are often used as an approxmaton to the consttuents of real matter n Newtonan mechancs. A true pont partcle would be nfntesmally small and ndvsble. A pont partcle s poston therefore concdes wth a pont n space at every gven moment of tme, parameterzed wth t, and can be descrbed mathematcally by a vector x from the choce of orgn to that pont. As the pont partcle moves ts poston s gven by the tme-dependent trajectory x(). t The components of the vector trajectory are denoted x (t), where the subscrpt runs over the dmensons of the space under consderaton. In one dmenson the subscrpt s dropped and the trajectory s wrtten x(t). do:0.088/ ch - ª Morgan & Claypool Publshers 08
3 A Concse Introducton to Quantum Mechancs As the pont partcle moves t has a velocty v()and t an acceleraton a(), t both vector quanttes, gven from the trajectory by d x( t) d x( t) v() t = x (), t a() t = x (). t (.) dt dt The defntons of (.) determne the velocty and acceleraton when the partcle s at the poston x()snce t the tme t s a parameter common to all three. As the partcle moves, t can be subject to a force, F( x, t), whch s also a vector wth the same number of components as x. The force F may depend explctly on the tme t as well as mplctly through the partcle s current poston x(). t Newton s second law of moton states that F = m a. Usng (.) allows Newton s second law to be wrtten d x( t) Fx (, t) = ma( t) = m. (.) dt Snce Newton s second law (.) has become a second order dfferental equaton, a unque soluton requres two boundary or ntal condtons for each spatal component to be specfed. For example, specfyng the partcle s ntal poston and velocty s suffcent for a unque soluton to (.). Once these boundary condtons are specfed, the trajectory of the partcle at all subsequent and prevous tmes s determned solely from F. The trajectory may be extremely senstve to the ntal condtons mposed, but at each moment the poston s unque and therefore entrely determnstc n nature. Determnsm s a cornerstone of Newtonan physcs and ts mechancal vew of the world. In what follows, the soluton of Newton s second law (.) s referred to as the classcal trajectory of the partcle and s desgnated xc () t.aconservatve force actng on a mass m partcle s a force that has no explct tme dependence and whose components are gven by F( x) = U( x)/ x, where U( x) s a scalar functon of poston known as the potental energy. The knetc energy s gven by K = m x x = mv v = mv v mv, where the sum s over the dmensons of the space under consderaton. The scalar product of two spatal vectors s gven by x y = xy, and can be shown to be the same as x y = x y cos θ, where θ s the angle between the two vectors. The combnaton of the knetc and potental energy, E = mv + U( x), s called the total mechancal energy. If all the forces on a partcle are conservatve, then the total mechancal energy E s tmendependent or conserved as the partcle moves along the classcal trajectory. Fndng ts tme dervatve va the chan rule and evaluatng t along the classcal trajectory gves de dt = + U( x) mx x = ( mx c F( xc)) x c = 0, (.3) x x= xc x= xc whch vanshes snce x c solves (.). The concept of energy s at the core of theoretcal physcs and plays a central role n formulatng quantum mechancs. -
4 A Concse Introducton to Quantum Mechancs However, t should be noted that energy s defned only up to an arbtrary constant, whch can be added to the defnton of U( x) wthout alterng (.3) or Newton s equaton of moton (.). In ts modern formulaton Newtonan mechancs s stated n terms of a varatonal or acton prncple. The varatonal verson of (.) starts by consderng an arbtrary trajectory x(), t assumed to be a dfferentable functon of t. Ths arbtrary trajectory s used to evaluate the classcal acton S[ x] for the tme nterval T, whch s gven by T S[ x] = d t( K( x, x, t) U( x, x, t)) d t L( x, x, t), (.4) 0 0 where both the knetc energy K and the potental U may depend explctly on x, ẋ, and t. The functon L( ẋ, x, t) s referred to as the Lagrangan densty. The value of the acton S[ x]depends on the trajectory functon x(), t and for that reason the acton s referred to as a functonal. The calculus of varatons determnes the form of the trajectory that extremzes the value of S, consstent wth boundary condtons, by generatng the dfferental equaton that ths trajectory must solve. Ths dfferental equaton s found by treatng x and ẋ as ndependent degrees of freedom and fndng the condtons under whch the acton s an extremum. The form (.4) for S has been chosen to ensure that the extremal trajectory and the classcal trajectory, xc () t, concde for smple Newtonan systems. Unquely specfyng the extremal trajectory requres two boundary condtons. As an example, Drchlet boundary condtons specfy xc(0) = xo and xc( T ) = xf. Other boundary condtons nvolvng dervatves, such as Neumann and Cauchy boundary condtons, are possble. A trajectory that s close to the extremal trajectory can be wrtten x() t = xc () t + δx() t. The functon δx()s t an nfntesmal devaton from the extremal trajectory that must satsfy the boundary condtons δx(0) = δx( T ) = 0. Snce δx s nfntesmal, a Taylor seres expanson around the extremal soluton need only retan terms of O( δx), gvng T + δ + δ = + δ L L L( xc x, xc x, t) L( xc, xc, t) x + δx x x = L( x, x, t) + d L δ x dt x c c c L d L + δx x dt x c c x= xc x= xc x= xc x= xc. (.5) Returnng (.5) to the acton (.4) shows that the total tme dervatve wll vansh snce δx(0) = δx( T ) = 0. The varaton of the acton around the extremal trajectory becomes T L( x, x ) d L( x, x ) δs[ xc] = S[ xc + δx] S[ xc] = d t δx( t) 0 x dt x = x xc. (.6) -3
5 A Concse Introducton to Quantum Mechancs Smlar to an extremum pont of a functon, an extremal trajectory must satsfy δ S[ xc ] = 0. Snce δx s arbtrary, the functon xc() t must therefore satsfy the Euler Lagrange equaton, gven by L( x, x ) d L( x, x ) x dt x x= xc = 0. (.7) Usng form (.4) n the Euler Lagrange equaton (.7) mmedately reproduces the usual form of Newton s second law of moton (.) for a conservatve force. It s worth notng that addng an arbtrary constant to L does not alter (.7). The acton formulaton allows the generalzaton of both Newtonan mechancs and the concepts of momentum and energy. For example, the momentum p that s canoncally conjugate to the poston coordnate x s defned as = L( x, x p ). (.8) x It s assumed that (.8) can be nverted to fnd the soluton for ẋ as a functon of p and x, denoted x ( p, x). Ths process smply ntroduces a new varable p that replaces the varable ẋ. Usng a Legendre transformaton, the generalzaton of the total mechancal energy, called the Hamltonan and denoted H( p, x),sdefned as H( p, x, t) = p x ( p, x) L( x ( p, x), x, t). (.9) For the case that the Lagrangan densty has no explct tme-dependence, t follows that the tme dervatve of the Hamltonan along the classcal trajectory s gven by dh = + L = px px t x x L x x p L d x x = x xc x= xc, (.0) where (.8) was used to cancel the second and thrd terms Once agan, ths vanshes along the classcal trajectory snce the combnaton of (.8) wth the Euler Lagrange equaton (.7) gves p = x= xc d L( x, x ) = L( x, x ) dt x x x= x c x= xc. (.) The freedom to add an arbtrary constant to the Lagrangan L reflects the earler observaton that the energy s defned only up to an arbtrary constant. Ths means that only dfferences n energy are relevant to moton n Newtonan physcs. For the case that L = mx x U( x), the famlar elementary results p = mx and H = p / m + U( x) are obtaned. Newton s law of moton then follows from (.) snce mac = mx c = p c = L ( x c, xc)/ xc= U( xc)/ xc= F. It s the canoncal momentum and the Hamltonan generalzaton of energy that are used n formulatng quantum mechancs. A second means of mplementng the varatonal prncple (.6) starts by usng the Hamltonan (.9) to express the Lagrangan densty as L = p x H( p, x, t), -4
6 A Concse Introducton to Quantum Mechancs where p = px (, x ). The varaton of the acton s then gven by varyng both the momentum and the poston accordng to x = xc + δx and p = pc + δp, where both δp and δx vansh at t = 0 and t = T. The result s δ = δ + δ T H S[ x t x p p δ x x H c] d c c 0 p = x = x x c x xc (.) = δ δ + T H H d t p x x p c c, 0 p = x = x x c x xc where the second step follows from an ntegraton by parts employng the boundary condtons on δx. Snce the two varatons are ndependent, (.) yelds H( pc, xc, t) = + H( pc, xc, t) xc 0 and pc = 0, (.3) p x c whch are referred to as Hamlton s equatons. Hamlton s equatons can be used to state the tme evoluton of a mechancal quantty O( p, x, t), referred to as an observable, along the classcal trajectory x c, do = O O + + = + t t x x O p p O O H O H c c. (.4) d t x p p x c c c c c c c Usng (.4) the Posson bracket of any two mechancal quanttes u( p, x, t) and v( p, x, t) s defned as u v u v { u, v}, (.5) x p p x c c c c so that (.4) can be wrtten as do/dt =O/ t + {O, H}. The Posson bracket has the property of antsymmetry, { u, v} = { v, u}, and t obeys the Jacob dentty, { u,{ v, w}} + { w,{ u, v}} + { v,{ w, u}} = 0, both of whch are easly demonstrated by drect substtuton usng the defnton (.5). Antsymmetry guarantees that { u, u} = 0. Notable Posson bracket results nclude { x, pj } = δj, where δ j s called the Kronecker delta. The Kronecker delta has the property that δ j = f = j and s zero otherwse. A second notable Posson bracket nvolves the orbtal angular momentum of a partcle, L = r p, where denotes the vector product. The Cartesan x and y components are gven by Lx = ypz zpy and Ly = zpx xpz. Substtutng these nto the Posson bracket gves { } L, L = xp yp = L, (.6) x y y x z where L z s the z-component of the orbtal angular momentum. Smlar results follow for the cyclc permutatons of the other components, { Ly, Lz} = Lx and { Lz, Lx} = Ly. An often overlooked but very mportant property of Newtonan mechancs s that t ncorporates Gallean relatvty. Ths means that the equatons of moton and the -5
7 A Concse Introducton to Quantum Mechancs acton are form nvarant under a Gallean transformaton. A Gallean transformaton relates the coordnates of a pont x and a tme t n the unprmed coordnate system to the coordnates of the same pont and tme n the prmed coordnate system, assumed to be movng at the constant velocty v relatve to the unprmed system. If the orgns concde at t = 0, these coordnate transformatons and ther nverses are gven by x = x vt x = x + vt, t = t t = t. (.7) For example, the orgn n the unprmed system occurs at x = 0, and ths pont s at the locaton x = vt n the prmed coordnate system. The transformatons of (.7) reflect the Newtonan belef n an absolute tme that s the same for all observers and s unaffected by relatve moton. In Newtonan physcs t s an mportant axom that a Gallean transformaton to a reference frame movng at a constant velocty wth respect to the frst reference frame should not alter the laws of physcs. Newton s law of moton s nvarant under a Gallean transformaton (.7) as long as v s constant, d/d v t = 0, snce the acceleraton transforms as a = d x d dx d x = v = = a. (.8) dt dt dt dt It eventually became necessary to replace the Gallean transformatons of (.7) wth the Lorentz transformatons of Ensten s specal relatvty, whch reduce to (.7) for veloctes small compared to that of lght. However, for the purposes of ths text the requrement for Ensten s specal relatvty wll be waved and form nvarance under Gallean transformatons wll be mplemented.. Lght and electromagnetsm One of the central goals for physcs n the 9th Century was understandng the dual phenomena of electrcty and magnetsm and ther relatonshp to lght. Because the nteractons between lght and matter were crtcal to probng atomc level phenomena, t s useful to revew the structure that was n place at the end of the 9th Century. Early n the 9th Century Coulomb s law was expermentally establshed. It states that the electrc force between two statonary electrc pont charges, q and Q, separated by the dstance r has the magntude n SI unts gven by KQq F =, (.9) r where K /4πϵ 0 s the Coulomb constant. It s attractve f Qq < 0 and repulsve otherwse. For two equal and opposte charges, q and q, expresson(.9) s assocated wth a potental energy of separaton gven by Kq Ur () =. (.0) r These two results underle the formal development of electromagnetc theory. The gradent operator s central to the formal statement of electromagnetsm and useful throughout physcs. The gradent operator s a vector dfferental operator. -6
8 A Concse Introducton to Quantum Mechancs In Cartesan coordnates t s wrtten n vector notaton as = e / x j j j, where the e j are the Cartesan coordnate unt vectors, so that = e + + x e e x y y z z. (.) The Cartesan unt vectors are orthonormal, whch s to say ther scalar product satsfes e ej = δj, where δ j s the Kronecker delta defned earler. Of partcular mportance n physcs s the quantty =, referred to as the Laplacan. In Cartesan coordnates the Laplacan takes the straghtforward form = + + x y z. (.) When the gradent s appled to a functon φ( x), the vector gven by φ( x) ponts n the drecton of maxmum ncrease for the functon at the pont x. In other coordnate systems the gradent and the Laplacan wll take a dfferent form. An llustratve example s the relatonshp between Cartesan coordnates ( x, y, z) and cylndrcal coordnates ( ρ, ϕ, z), gven by the expressons x = ρcos ϕ, y = ρsn ϕ, and z = z, along wth ther nverses, ρ = x + y, ϕ = arctan yx /, and z = z. It s mportant to remember that both the partal dervatves and the unt vectors change. Usng the chan rule wth these expressons gves ρ = cos ϕ + sn ϕ, = ρ sn ϕ + ρ cos ϕ, ϕ =. (.3) x y x y z z The unt vectors n cylndrcal coordnates n terms of Cartesan unt vectors can be read off from (.3) by usng the assocaton e. Ths gves e = cos ϕ e + sn ϕ e, e = sn ϕ e + cos ϕ e, e = e. (.4) ρ x y ϕ x y z z Ths correspondence works because the partal dervatves gve the rate of change n the drecton of the respectve unt vectors. The three unt vectors of (.4) are also orthonormal, but e ρ and e ϕ change orentaton at dfferent ponts around the z-axs, wth e ρ always perpendcular to the z-axs and e ϕ always tangent to crcles around the z-axs. The gradent n cylndrcal coordnates s then gven by = eρ + eϕ + e ρ ρϕ z. (.5) z Verfyng (.5) follows from substtutng (.3) and (.4) nto (.5) to show that t reproduces (.). In cylndrcal coordnates the Laplacan s gven by = + + ρ ρρ ρ ϕ + z. (.6) Ths follows by usng (.3) n(.6) to show that t becomes (.). Smlarly, sphercal coordnates ( r, θ, ϕ) are related to Cartesan coordnates by -7
9 A Concse Introducton to Quantum Mechancs x = r sn θ cos ϕ, y = r sn θ sn ϕ, and z = r cos θ. An dentcal treatment gves the gradent n sphercal coordnates, = + er eθ + eϕ, (.7) r r θ r sn θ ϕ whle n sphercal coordnates the Laplacan s gven by = + θ + sn +. (.8) r r r r sn θ θ θ r sn θ ϕ The gradent can also be used combned wth a general vector valued functon, A( x, t),todefne the dvergence of A( x, t) as A( x, t). Ths defnton appears n the dvergence theorem, whch states that n three spatal dmensons V 3 d x A( x, t) = d S A( x, t), (.9) S( V) where V s an arbtrary three-dmensonal volume and S(V) s the two-dmensonal surface S formng the boundary of the three-dmensonal volume V. The rght-hand sde of (.9) s sometmes referred to as the flux of the vector quantty represented by A through the surface S. If the dvergence of the vector functon A yelds a postve result for the left-hand sde of (.9), then the overall flux of the quantty A s drected outward through the surface boundng that volume. The dvergence theorem plays a central role n statng the conservaton laws n physcs. The two physcal quanttes of electromagnetsm are the electrc feld E( x, t) and the magnetc feld B( x, t), both of whch are vector valued functons of the poston vector x as well as the tme t. It follows that these felds are fundamentally dfferent from a pont partcle snce the electrc and magnetc felds are not localzed at a pont. Instead, they exst over a regon of space and may be created or dsappear as a result of nteracton wth charged matter. The charged matter s descrbed by a charge densty ρ( x, t), a charge per unt volume, and the vector charge current densty J( x, t), a rate of charge flow per unt area. It has been expermentally determned that electrc charge s a conserved quantty, and ths s stated mathematcally by the relaton J( x, t) + ρ( x, t) = 0. (.30) t Relaton (.30) can be combned wth the dvergence theorem to show that t V d V ρ( x, t) = d S J( x, t). (.3) S( V) Result (.3) states that the charge lost or ganed n a volume V of space per unt tme s dentcal to the charge flowng out of or nto that volume of space through ts surface S(V) per unt tme. It s worth notng that the classcal current relaton (.30) makes no statement regardng the physcal nature of the consttuents of electrc charge. The charged matter may be treated as a collecton of partcles or a -8
10 A Concse Introducton to Quantum Mechancs contnuous dstrbuton of mass. It was not untl 897 that Thomson dentfed the electron as a low mass pont-lke partcle carryng electrc charge. In effect, Thomson dentfed the frst elementary partcle. The electron contnues to be consdered elementary n that t cannot be subdvded nto smaller consttuents. Its electrc charge s therefore the fundamental unt of all observed electrc charges, whch are nteger multples of ts charge. At the tme, the electron was treated as a massve pont partcle and therefore assumed to be governed by Newtonan mechancs. There are four fundamental laws, referred to as Maxwell s equatons, that govern electrc and magnetc felds. In the absence of delectrc materals and usng Gaussan unts, they are gven n dfferental form by Gauss s law E = 4 πρ, (.3) π E 4 Ampere s law B = J, (.33) c t c + B Faraday s law E c t = 0, (.34) No magnetc monopoles B = 0, (.35) where c s the speed of lght. Equatons (.34) and (.35) are referred to as the homogeneous equatons and can be solved by ntroducng the vector and scalar potentals, A( x, t) and φ( x, t), such that E( x, t) = φ( x, t) A( x, t) and B( x, t) = A( x, t). (.36) c t Demonstratng that ths satsfes the homogeneous equatons uses the denttes φ =0 and ( A) = 0 for nonsngular potental functons. An mportant property of the E and B felds n (.36) s that they are left unchanged by the smultaneous redefntons A A + Λ and φ φ Λ, (.37) c t where Λ( x, t) s an arbtrary functon. Ths freedom to redefne the potentals s known as gauge nvarance. The two potentals are not unque and therefore not expermentally observable. They requre another equaton, referred to as a gauge condton, to specfy them unquely. A common and useful choce s the Lorentz condton, whch s gven by + φ A c t = 0. (.38) Another common choce s the Coulomb condton, A = 0. -9
11 A Concse Introducton to Quantum Mechancs Usng (.36) and (.38) wth the two nhomogeneous Maxwell equatons reduces them to the followng forms, 4π φ φ = 4πρ and A A = J, (.39) c t c t c where = /( ct) s referred to as the d Alembertan operator. In free space, where ρ and J vansh, relatons (.39) reduce to the wave equaton for the propagaton of electromagnetc felds. Ths electromagnetc wave s dentfed as lght. Settng φ = 0, asmple harmonc plane wave soluton to (.39) and(.38) takes the form A( x, t) = A cos( k x ωt + ϕ), (.40) 0 where the magntude A 0, the wave vector k, the angular frequency ω, and the phase ϕ are all constants. In order to satsfy (.38), whch becomes A = 0, the magntude A 0 must be transverse to the drecton of propagaton, gven by the wave vector k,so that k A0 = 0. The magntude k of the wave vector k and the angular frequency ω must also satsfy ck ω = 0 n order for ths soluton to satsfy (.39). Because t must be perpendcular to k the vector A 0 must le n the two-dmensonal plane that s perpendcular to the drecton of propagaton for the wave. There are therefore two ndependent polarzatons avalable to the lght wave of (.40). In the presence of electrc and magnetc felds the charged current undergoes a force per unt volume, referred to as the Lorentz force densty f, gven by f = ρe + J B. Ths result generalzes the Coulomb force (.9) to the case of c movng charges. Ths force causes the energy and momentum of charged matter to be changed by nteractng wth electromagnetc felds and waves. The electromagnetc wave must therefore transport energy and momentum n order to delver them to electrcal charges. In the case of electrc and magnetc felds, the energy per unt volume u n free space s gven by u = ( E + B ). (.4) 8 π Result (.4) can be demonstrated from elementary arguments by calculatng the work requred to charge a capactor and establsh a current n a solenod. The flow of energy n the electrc and magnetc felds s gven by the current densty known as the Poyntng vector S, c S = E B. (.4) 4π The defntons (.4) and (.4) can be combned wth Maxwell s equatons to gve the work energy relatonshp S + u/ t = E J. The rght-hand sde s the rate at whch mechancal work s done by the electrc feld on the charged matter current. The dvergence theorem shows that the Poyntng vector s the rate at whch electromagnetc energy s flowng through the surface of a volume of space, matchng the rate of change n electromagnetc energy plus the mechancal work done n the nteror of the volume. -0
12 A Concse Introducton to Quantum Mechancs The power W of an electromagnetc wave movng n the z-drecton s defned by the rate at whch energy crosses the surface element da z per unt tme. Snce the wave travels at speed c, n the tme nterval dt the wave wll flow nto the nfntesmal volume 3 d x = daz c dt. Ths gves 3 de = u d x = u daz c dt, so that W = d E/(dAz d t) = c u. Smlarly, the momentum densty π( x, t) of an electromagnetc wave can be found by consderng the force that an electromagnetc wave creates. If f z s the force densty delvered n the wave, then the force must be generated by a loss n the momentum of the wave, so that Newton s second law gves 3 3 fz d x = ( π/ t) d x = c( π/ t) daz dt. At the same tme, ths force must be gven by the gradent of the energy densty of the wave, whch acts as a potental. Therefore, fz d 3 x = ( u/ z) d 3 x = ( u/ z) daz d z = ( u/ t) daz dt, where the chan rule d z / z = d t / t s vald snce dz = c d t. Equatng the two forms for the force gves c π / t = u/ t. Therefore, the magntude of the momentum densty and the energy densty of an electromagnetc wave are related by c π ( x, t) = u( x, t), (.43) where the absence of energy has been chosen to correspond to the absence of a wave. For the reader famlar wth basc specal relatvty, result (.43) s consstent wth the relatvstc energy momentum expresson for a massless partcle, 4 E cp = mc = 0. The smple harmonc wave (.40) can be used n (.36), and ths gves B = A = ( k A0) sn( k x ωt + ϕ) B0sn( k x ωt + ϕ), ω = A A0 E = sn( k x ωt + ϕ) c t c E0 sn( k x ωt + ϕ), (.44) so that E0 = ωa 0/ c and B0 = k A0. Usng the dentty A0 ( k A0) = A0 k ( k A0) A0 and k A0 = 0 gves Averagng c ωa 0 S = E B = k sn ( k x ωt + ϕ). (.45) 4π 4π sn ( ωt) over a perod T = πω / yelds T sn ( ωt) dt sn ( ωt) = T 0. (.46) Combnng (.46) wth ω A0 = ce0 and ka0 = B0 n (.45) shows that the smple harmonc wave (.40) has the average energy ntensty ωk c S = A 0 = EB 0 0, (.47) 8π 8π wth S drected parallel to the wave vector k. Because E 0 and B 0 are smply the magntudes of the electrc and magnetc felds n the wave and satsfy E0 = B0,(.47) shows that the ntensty of a classcal lght wave s ndependent of ts frequency. An extremely mportant property of the free space versons of (.39) s that they defne a lnear equaton. Ths means that f φ and φ are any two solutons to the -
13 A Concse Introducton to Quantum Mechancs equaton φ = 0, then so s the lnear superposton, φ = aφ + bφ, where a and b are any two real constants. Ths property of electromagnetc waves n vacuum gves rse to one of the most mportant dstnctons between wave phenomena and pont partcle behavor, whch s the appearance of nterference when two lght waves overlap. These nterference patterns can be explaned usng the prncple of superposton, whch s vald snce lght waves n vacuum obey a lnear equaton. For example, f two lght waves wth equal ampltude and frequency but opposte drectons overlap n vacuum, they are smply superposed to fnd the resultant wave. The trgonometrc dentty cos( α ± β) = cos α cos β sn α sn β gves Ar( x, t) = A0cos( k x ωt) + A0cos( k x + ωt) (.48) = A0 cos( k x)cos( ωt). Ths s known as a standng wave snce ts nodes, those locatons where cos( k x) = 0, do not move. As another example, f two lght waves wth equal ampltudes, wavelengths, drectons, but opposte phases overlap n vacuum, the resultant wave s Ar = A0cos( k x ωt + ϕ) + A cos( k x ωt ϕ) 0.Asn (.45), the ntensty of the two overlappng waves s proportonal to the square of the resultant ampltude. Applyng the same trgonometrc dentty gves Ar = 4A0 ϕ cos cos ( k x ωt). (.49) For the case that the two waves are n phase and ϕ = nπ, where n s an nteger, the resultant wave wll have maxmum ntensty or brghtness, referred to as constructve nterference. For the case that the two waves are out of phase and ϕ = (n + ) π, where n s an nteger, the resultant wave wll have zero ntensty or brghtness, referred to as destructve nterference. It must be stressed that ths possblty s purely a property of wave phenomena and has no counterpart n the Newtonan mechancs of pont partcles. An extremely mportant example of (.49) occurs n the case of monochromatc coherent lght passng through two slts separated by the dstance d. The term coherent lght s taken to mean that the lght s exactly n phase as t passes through both slts. For such a case the lght s assumed to leave the two slts at all angles and the lght emanatng from the two slts wll therefore nterfere when t recombnes on a remote screen. If θ s the angle measured from a normal to the two-slt devce, then the path dfference between the two slts s d sn θ, so that the phase dfference between the lght from the two slts s ϕ = πd sn θ/ λ. As a result, the crteron for constructve nterference, ϕ = πn, yelds a formula for the angular poston of the brght frnges formed on the screen, d sn θ = nλ, wheren =,,3. (.50) Of course, the number of brght frnges formed, f any, depends on both the slt separaton d and the wavelength λ of the lght. A second mportant optcal phenomenon s referred to as dffracton, and occurs when coherent lght of wavelength λ s passed through a sngle slt of wdth a. -
14 A Concse Introducton to Quantum Mechancs The lght s spread subsequent to passng through the slt, and brght and dark bands are formed on the screen behnd the slt. The dark bands form at the angles θ that solve a sn θ = nλ, wheren =,,3. (.5) The orgn of ths formula s the nterference between lght passng through dfferent parts of the slt as opposed to dfferent slts n a gratng. It s mportant to note that the angular spread of the central n = peak, whch occurs at sn θ = λ/ a, s neglgble when λ s small compared to the slt wdth a. In such a case the lght wave passes through the slt wth no dscernble devaton, much as f the lght was a stream of Newtonan partcles passng through the slt. However, there s no real soluton to (.5) f λ > a. Detaled analyss shows that lght wll not pass through a slt consderably smaller than the wavelength. Because electromagnetc nteractons play such a central role n probng the atomc world, t wll be valuable to have the classcal descrpton of such phenomena for comparson wth experment. The nteracton of a partcle of mass m and electrc charge q wth the electromagnetc feld s descrbed by the Lagrangan densty q L = mx + A x q φ. (.5) c Applyng the Euler Lagrange equaton (.7) to(.5) for the th drecton yelds q c j φ + A xj( Aj ja) q mx = 0. c t (.53) Usng the forms (.36) for the electromagnetc potentals and the vector dentty x ( A A ) = ( x ( A )) j j j j, result (.53) becomes the Lorentz force law for a pont charge, F = qe + ( q/ c) v B. Under the gauge transformaton of (.37) the Lagrangan densty becomes L L + ( qc / ) ( Λ x +Λ ) = L + qd Λ/dt. The addtonal term does not contrbute to the varaton of the acton (.4) snce t s an exact dervatve, becomng ( qc / )[ Λ( x( T), T) Λ( x(0), 0)] after ntegraton. Therefore the acton s gauge nvarant, a hallmark of electromagnetsm. The Lagrangan dentfes the canoncal momentum of the partcle as p =L / x = mx + qa/ c, so that x = p/ m qa/ mc. Usng ths n the defnton of the Hamltonan for the charged partcle gves q H( p, x, t) = p x L( x, x ) = p A( x, t) + q φ( x, t). (.54) m c The potentals A( x, t) and φ( x, t) may be tme-dependent and therefore the Hamltonan (.54) may not be conserved..3 Propertes of Newtonan pont partcle solutons There are several aspects of Newtonan mechancs that wll play a crtcal role n ts eventual falure n the atomc realm. In order to recognze these falures t s worth -3
15 A Concse Introducton to Quantum Mechancs revewng brefly the predctons of Newtonan mechancs for two mportant physcal systems, the one-dmensonal harmonc oscllator and the nverse square force. The Newtonan behavor of these two systems can be understood from elementary arguments. In the case of the harmonc oscllator the potental energy s U( x) = kx, where k s real and postve. Equaton (.7) yelds mx = kx. Ths dfferental equaton has the soluton x = Asn( ωt + δ), where ω = km /. The values of the constants A and δ are determned by the ntal condtons, xo = Asn δ and vo = Aω cos δ. Ths system of two equatons has solutons for all values of x o and v o, gven by tan δ = xo/ ωvo and A = xo + ( vo/ ω ). The total mechancal energy s constant and gven by E = mx + kx = ka = kxo + mvo. (.55) The energy of (.55) can have any postve value snce x o and v o are arbtrary. In the case of the standard nverse square laws, such as the Coulomb or Newtonan gravtaton cases, a pont partcle wth mass m s assumed to be movng n the presence of a fxed attractve potental gven by U() r = β/, r where r s the radal poston of the pont partcle and λ s real and postve. Whle (.7) can be solved for both the bound and scatterng behavor of the pont partcle, t s possble to use very basc physcs to nvestgate the nature of crcular orbts. The condtons for a crcular orbt of radus R follow from the requrement for the orbtal tangental velocty v to satsfy the centrpetal force law, mv R β v β = ac = = R R mr, (.56) so that the knetc energy s mv = β/ R. Combnng ths wth the potental energy gves the total mechancal energy E of a crcular orbt wth radus R, β E = R. (.57) The negatve value for the total energy reflects the fact that the pont partcle s, by assumpton, bound and wll therefore requre an ncrease n energy to break free. However, once agan the value of E can be any of a contnuum of negatve values snce R s arbtrary. In addton, the orbtal angular momentum L of the pont partcle, gven by β L = mvr = m R = mβr, (.58) mr s also arbtrary snce R s arbtrary. There are several features of Newtonan mechancs that led to ts falure n the atomc realm of matter. Frst, t should be noted that Newtonan mechancs does not possess a natural length scale that would ndcate where and under what -4
16 A Concse Introducton to Quantum Mechancs crcumstances t would not be applcable. There s therefore no a pror reason that the prevous nverse square soluton s not applcable to both planetary moton and the orbt of electrons n an atom. Second, n Newtonan mechancs the possble energes of a partcle belong to a contnuous set of values as long as there s a soluton to (.7). Thrd, Newtonan mechancs places no restrctons on smultaneously specfyng both the ntal poston and momentum of the partcle to an arbtrary accuracy. In fact, t often uses these two quanttes to determne a unque trajectory for the partcle for the remander of tme as long as the subsequent net force actng on the partcle s known. As mentoned earler, ths concept of determnsm was consdered one of the phlosophcal cornerstones of physcs pror to the advent of quantum mechancs. Newtonan mechancs s remarkably successful n descrbng the dynamcal behavor of macroscopc objects. As a result, any alteratons made to t must be done wth extreme care to preserve these successes. Ths was but one of the many dffcultes physcsts faced when Newtonan mechancs faled n descrbng atomc phenomena, creatng the need for an alternatve form of mechancs n the atomc realm. -5
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