Introduction to the quantum theory of matter and Schrödinger s equation

Transcription

1 Introduction to th quantum thory of mattr and Schrödingr s quation Th quantum thory of mattr assums that mattr has two naturs: a particl natur and a wa natur. Th particl natur is dscribd by classical physics (Nwton s laws), and th wa natur is dscribd by quantum physics. This is similar to light, which has both a wa and particl natur. Th wa natur is dscribd by th classical Maxwll s quations, whras th particl natur is dscribd by th quantum physics. Rlatiistic rlationships btwn mass, nrgy, and momntum On of th cntrpics of Einstins thory of spcial rlatiity is that th mass of a particl incrass as its locity incrass: m = ( ) 1 / c whr is th rst mass of th particl. To dri th quation for th kintic nrgy T of a particl in trms of its locity, w start by applying Nwton s scond law of motion: th forc F acting on a particl quals th tim-driati of th particl momntum p: F = dp dt = d(m) dt Th kintic nrgy T of a particl can thn b found by: T = Fds = dp dt ds = ds dt dp = d(m) intgrating by parts, this bcoms or T = m "# $ %& 0 ( md = m "# $ %& 0 = " m + c 1 / c # $ T = mc c 0 ( ) (mattr) % & = = 0 ( 0 d 1 ( / c) = m + c For small locitis, th kintic nrgy T bcoms: [1] 1 ( / c) c T = mc c = c m 1 ( / c) o c " 1 (for << c), [] Einstin intrprtd th trms c and mc as th rst nrgy and total rlatiistic nrgy of th particl, rspctily. Hnc w ha Einstin s famous statmnt: 1

2 E = mc (mattr) [3] Anothr forf this xprssion can by found by squaring both sids and rplacing m with its locity-dpndnt formula: E = 1 / c ( ) c4, which can b rarrangd to rad: E = c 4 + c ( ) 1 / c but th last trm is simply c p, so w ha E = c 4 + c p Th Dual Natur of Light (mattr) [4] Wa (classical) natur Th classical thory of light is that it is a wa, dscribd in trms of lctric and magntic filds. Maxwll s quations dscrib th rlationship btwn ths filds and thir sourcs (chargs and currnts). Th wa quation lctromagntic was in fr spac is for th lctric fild E (t,x) is: E x = 1 E c. (light was) t [5] which admits plan-wa solutions at a frquncy ν (in Hz) of th form: E(t,x) = E + j(t+kx) + E - j(tkx) (light was) Hr, ω =πν is th radian frquncy, k = / c is th wanumbr of th mdium, and E + and E - ar arbitrary constants. Th E + and E - trms rprsnt was propagating in th + and dirctions, rspctily. Both was rpat thmsls spatially or a distanc = " / k = c / #, calld a walngth. Anothr common way to rprsnt plan was is in th frquncy (phasor) domain, whr tim is supprssd: E(x) = E + jkx + E - jkx (light was) (Not hr, th physics tim-conntion jt is usd, unlik th jt conntion typically usd in nginring). [7] [6]

3 Particl (quantum) natur Th quantum thory of light, proposd by Einstin, stats that light consists of nrgy packts, calld photons, that tral at th spd of light. Photons ha momntum, but no rst mass. This thory was abl to xplain th puzzling rsults of photolctric xprimnts. Einstin proposd that th nrgy of a photon is dtrmind by its frquncy ν, E = h ν (photons), [8] whr h is Plank s constant. Einstin also proposd that photons ha momntum. To s how this could b so, w first rmmbr (quation 4) th rlatiistic rlationship btwn nrgy E and momntum p of a mass particl is: E = c 4 + c p Howr, sinc th rst mass of a photon is zro, this would imply from [4] that for photons: p = E c = h c = h " (photons) [9] Ths assumptions about th photon nrgy and momntum wr alidatd by Compton s famous Scattring xprimnt, whrby lctrons wr bombardd with x-ray photons. 3

4 In this xprimnt, photons of walngth λ bombard an lctron initially at rst. Aftr th collision, th lctron would rcoil at som angl φ and a nw, downshiftd, photon would b mittd at an angl θ. Equating th lctron+photon nrgy and momntum bfor and aftr th collision, th following rlationship can b drid btwn th walngth shift Δλ and th angl shift θ of th photon: " = h m c (1# cos$). [10] This rlation was rifid xprimntally by Compton in 193. Wa (quantum) Natur of Mattr dbrogli proposd that just as light has both a particl and wa natur, so too might mattr ha both naturs. This would man that mattr has a walngth and frquncy. d Brogli proposd that a mass particl has frquncy ν, and its rlationship to its kintic plus potntial nrgy E is th sam as for a light wa (quation 8): E = h (mattr) [11] dbrogli also proposd that mass particls ha walngth λ, and, whn th particl is fr (i.., not in a potntial fild), this walngth is rlatd to th particl s momntum p just th sam as for a photon (quation 9): p = h (mattr). [1] (Not hr that dbrogli did not say that p is qual to ithr E/c or hν/c, as in th cas of photons, sinc th quations for nonrlatiistic particls should not ha c in thm.) This prdiction of walngth was alidatd xprimntally by th Daisson-Grmr xprimnt, whr lctrons incidnt on a nickl sht scattrd off at an angl that corrspondd to th Bragg scattring of X-rays off of crystals. 4

5 Th classical and quantum thoris of light and mattr can now b summarizd: Classical Thory Quantum Thory Light Natur: wa Dscription: Maxwll s quations = c / " Natur: corpuscular Dscription: E = h p = E c = h c = h " Mattr Natur: corpuscular Dscription: E = 1 m (whn <<c) p = m Natur: wa Dscription: E = h = h p " h m# (whn <<c) Th Schrödingr Wa Equation Onc th concpts of mass frquncy and walngth wr accptd, th nxt qustion was: what sort of wa quation would a mass wa satisfy? Th lctromagntic wa quation cannot work for mass was, sinc th rlationship btwn walngth and frquncy for mass was is not = c / ", as it is for light was. As a starting point, w xpct a mass wafunction to ha th sam gnral form (quation 6) as for a light wa: (t,x) = + j("t+kx) + - j("t"kx), [13] xcpt that th radian frquncy = "# and th wanumbr k = / " will ha a diffrnt rlationship than thy do for light was. Nxt, considr an lctron traling in a forc-fr nironmnt at a constant, nonrlatiistic, spd <<c. W will lt E = 1 m rprsnt th particl s nrgy abo its rst nrgy, and p = m its momntum. Using quations 11 and 1, th frquncy and walngth of th lctron ar: = E / h and = h m = h me. [15] Substituting quation 14 into quation 15, w find that th rlationship btwn th frquncy ν and walngth λ of a mass wa is: = h (mass was) m" [16] A wafunction that has ths tim and position charactristics is: 5

6 jx (x,t) = + me " je t " jx me " je + " t [17] whr + and - ar constants and = h /. Th first and scond trms in this xprssion rprsnt forward and backward traling lctrons, rspctily. A wa quation that yilds this solution is: " # m "x = j "# "t. [18] This wa quation was proposd by Schrödingr in 196. Substituting th wa function into th right-hand sid of Schrödingr s quation yilds: j " t = j $ & t %& = j # je jx " + $ & %& me # je t # jx me # je + " # t jx " + ) () me # je t # jx me # je + " # t ) () = E" This mans that th Schrödingr quation for a singl nrgy particl can b writtn as: m 1 # " " #x = E Sinc w ha assumd that th particl is traling in a forc-fr nironmnt, its only nrgy E abo th particl s rst nrgy is kintic nrgy, so th trm 1 # " m " #x rprsnts th kintic nrgy of th particl. If th particl is also in a potntial fild with potntial V(x), th total nrgy E bcoms th suf th kintic plus th potntial nrgy, so Schrödingr s quation bcoms: m 1 # " " #x + V(x) = E which is Schrödingr quation for particls with constant nrgy E, oftn calld th timindpndnt Schrödingr quation. Finally, particls with uncrtain nrgy can b xprssd as th sum particls of constant nrgy particls, ach with a diffrnt nrgy. This is oftn calld th suprposition principl of quantum mchanics. Using [13], such a particl can b xprssd as: (x, t) = # jx p(e) me " je t de [0] whr p(e) is th probability that th particl has nrgy btwn E and E+dE [19] 6