Quanum Mechanics Wave Funcion, Pobabiliy Densiy, Poagaos, Oeao, igen Value quaion, ecaion Value, Wave Packe
Aioms fo quanum mechanical desciion of single aicle We conside a aicle locaed in sace,y,z a ime z y z y In classical mechanics, aicle is well localized in sace:, : R 3 R In quanum mechanics aicle is descibed by a wave funcion: Ψ,, Ψ:R 3 R -- > C [eal, im] no obabiliy is consideed The obabiliy ha he aicle can be found a osiion a ime is: P, I Ψ, I Ψ*, Ψ, I comle conjugae : a+ib*a-ib The wave funcion is nomalized: 3 Ψ, d 1 Ω [ 0, ]
The wave funcion is assumed o ovide a qunaum mechanical comlee desciion of he behavio of a aicle of mass m wih he oenial enegy V,. The wave funcion is ineeed in saisical ems. We can imagine a vey lage numbe of idenical, indeendan nonovelaing egions of sace, each lage enough o conain all he hysically ineesing feaues of he moion. We have also o imagine a vey lage numbe of indeenden eeiions of he same moion in he egion of sace, fo each of which is efeed o he aicula oigin in ime. The numeical esul of he measuemens a a aicula ime of any hysically meaningful quaniy osiion, momenum o enegy will in geneal no be he same fo all he egions. Rahe, hee will be a disibuion of hese numbes ha can be descibed by a obabiliy funcion, e.g., he esul of a osiion deeminaion is o be egaded as unceain by an amoun of he ode of he linea dimensions of he wavefuncion. I is naual o egad waefuncion as a measue of he obabiliy of finding a aicle a a aicula osiion wih esec o he oigin of is egion
Time evoluion of a aicle Ω z y Ω z y The ime evoluion of a classical aicle is well conolled by Newon dynamics and he simle ajecoy: a +v+ 0 0 The ime evoluion of a quanum aicle can be descibed using a linea oagao φ, Fom he oagao heoy, he Schödinge equaion can be deived consisenly
Wave funcion fom Wave funcion of Ψ, ha eesens a aicle of comleely undeemined osiion avelling bu ecisely known momenum and kineic enegy would have fom of he hamonic funcion sink-ω, cosk-ω o e ik-ω and e -ik-ω Ψ In geneal case, he wave acke eesenaion is alied
Wave ackes Paicles of mae o quana of adiaion ae eesened by concenaed bunches of waves once hey ae localized in sace and ime. Sace-wave inde disibuion: k 0 π/λ 0 Ψ λ 0 Fouie Tansfom Of Ψ k k Poeies of Ψ: I can inefee wih iself I is lage in magniude whee he aicle o hoon is likely o be and small elsewhee I will be egaded as descibing he behavio of a single aicle o hoon no he saisical disibuion of a numbe of such quana!!!
Sace and ime ackes Sace ackes Tyical fom of Ψ fo a concenaed bunch of waves is shown in he evious figue a aicula ime Poeies: -aveage wavelengh λ 0, -aoimae eension The Fouie inegal analysis of Y wih esec of shows how Y may be buil u ou of coninuous hamonic waves of vaious lenghs. This is chaaceized by inoducion of he oagaion numbe k k is elaed o momenum: h/ πλhk/π >h/π Time ackes In analogous fashion, deendance of Ψ on he ime fo a oin same figue as eviously: -. Poeies: -avaage duaion of he wave -aoimae ime eension The Fouie ansfom shows how Ψ can be buil u ou of coninuous hamonic waves of vaious fequencies. hν >h/π The unceainy elaion fo osiion and momenum ime and enegy of a quanum mae o adiaion follows diecly fom he wave-acke desciion.
Oeaos dynamical vaiables-obsevables Dynamical vaiables: quaniies such as he coodinaes and he comonens of velociy, momenum and angula momenum of aicles, and funcions of hese quaniies The linea oeaos coesond o he dynamical vaiables, i.e., each dynamical vaiable ha elaes o he moion of he aicle can be eesened by a linea oeao. Oeao can be simly mulilicaion oeao e.g., o i may be a diffeenial oeao o fo he mmenum. ih When an obsevaion is made i coesonds o measuemen of dynamical vaiable eal numbe
Oeaos, eecaion value V m V z y m H + + + + h h i h h ih z y i,, amles: osiion:, y, z o momenum oal enegy: d O d OP O i 3 3,, *, Ψ Ψ h The eisance of he obabiliy densiy P, makes ossible o calculae he eecaion value fo aicula oeao: The hysical ineeaion of he wave funcion is elied on he comuaion of aveage o eecaion values and obabiliy funcions of oeaos ha eesen vaious hysical condiions
henfes s Theoem I is easonable o eec he moion of wave acke o agee wih he moion of he coesonding classical aicle. P, d 3 3 d 3 d P, d m P, d m m d d d d This calculaions agee wheneve he oenial enegy changes by negligible amoun ove he dimensions ove he acke. The velociy of he acke will be he ime ae of change of he eecaion value of he comonen of he osiion Moe in L. Schiff, Quanum Mechanics In simila fashion he he ime ae of change of he momenum of he aicle can be calculaedi henfes s heoem is an eamle of he coesondance incile.
Oeao-linea eigenvalue equaion Wih oeao, O, can be associaed a linea eigenvalue equaion: Oφ i o i φi Whee φ i is eigenfuncion and o i is eigenvalue vey eigenvalue is he only ossible esul of a ecise measuemen of he dynamical vaiable fo some sae of he sysem. This gives he hysical significance of he eigenalues. The se of eigenvalues of a eal dynamical vaiable coesonding o he aoiae oeao ae jus he ossible esuls of measuemens of ha dynamical vaiable. A eal dynamical vaiable whose eigensaes fom a comlee se is an obsevable.
Wave funcion-eansion in eigenfuncions All he eigenfuncions of any dynamical vaiable consiue a comlee se of funcions in he sense ha an abiay coninuous funcion can be eanded in ems of hem: amle: : The oal enegy oeao igenvalue equaion: ^ H ϕ ϕ Wave funcion eanded in eigenvalues: Ψ A ϕ The momenum oeao ih,, ih y z ih φ φ Ψ A φ The numbe of measuemens ha esul in he eigenvalue is ooional o he squae of he magniude of he coefficien a in he eansion of Ψ
A i coefficien negy Momenum Ψ A ϕ Ψ A φ A unique coefficien can be deemined by mulilying boh sides by φ Ε and iegaing ove he sace whee aicle is locaed [ he same can be eeaed fo φ* ] ϕ 3 3 * Ψ d A ϕ * ϕ d Aδ ' A '
Pobabiliy funcion and eecaion value The enegy momenum eigenvalues ae he only ossible esuls of ecise measuemen of oal enegy and he obabiliy of finding a aicula value when he aicle is descibed by he wave funcion Ψ is ooional o IA I 1 ' ] * [ * ' ' * ' ' * 3 3 3 3 3 Ψ Ψ Ψ Ψ Ψ d d d d d P A P ϕ ϕ ϕ ϕ We can comue he aveage o eecaion value of he enegy: P
Fom of momenum eigenfuncion φ C e[ i ], k / h φ k C e[ i k]
Moion of a wave acke in one dimension The minimum unceainy oduc is calculaed as a oo mean squae deviaion fom he mean eecaion value: h 1 ] [ 1 L. Schiff Quanum Mechanics Fom of he minimum acke: ] 4 e[ ] [ 1 4 1 h i + Ψ π
Change wih ime of a minimum acke Geneal eession fo he wave funcion of he wave acke: ] / 4 e[ 1 ] [ 1, 1 4 1 m i m i h h + + Ψ π I follows afe : ] e[, m k i A k k k ϕ h Ψ 0 and m k k k h h
The osiion obabiliy densiy Ψ,0 ] [ e[ ] [ 1, 1 m m + + Ψ h h π The equaion is of he same fom as ece ha ime conaining comonen. The cene of he acke emains a 0 while he beadh of he acke inceases as deas fom 0 in boh as and fuue diecions. The smalle he iniial unceainy in osiion, he lage he unceainy in momenum and he moe aidly he acke seads. The ime deendan comonen of he eession is he disance aveled by a classical aicle of momenum in he ime. 4 m m + + h
Aendi: Some emaks fom he agao heoy
Poeies of he oagao: Assume ha a ime 0 a aicle is a one oin 0 in sace, i.e., Ψ 0 ; 0 δ, 0. The sae of a sysem a ime, descibed by Ψ;, equies hen a knowledge of he sae a all sace oins a some inemediae ime. This is diffeen fom he classical siuaion whee he aicle follows a discee ah and, hence, a any inemediae ime he aicle needs only be known a one sace oin, namely he oin on he classical ah a ime 0.
The genealizaion of he comleeness oey o N - 1 inemediae oins > N-1 > N- > : : : > 1 > 0 is mloying a coninuum of inemediae imes [ 0 ; 1 ] yields an eession of he fom:
L is he Lagangian of he classical aicle. Howeve, in comlee disincion fom Classical Mechanics, eessions ae buil on acion inegals fo all ossible ahs, no only fo he classical ah. Siuaions which ae well descibed classically will be disinguished hough he oey ha he classical ah gives he dominan, acually ofen essenially eclusive, conibuion o he ah inegal. Howeve, fo micoscoic aicles like he elecon his is by no means he case, i.e., fo he elecon many ahs conibue and he acion inegals fo nonclassical ahs need o be known.
Poagao fo fee aicle: This oagao, accoding o allows us o edic he ime evoluion of any sae funcion ; of a fee aicle. Below we will aly his o a aicle a es and a aicle foming a so-called wave acke.
Comaision of Moving Wave Packe wih Classical Moion I is evealing o comae he obabiliy disibuions fo he iniial sae and fo he final sae, esecively. The cene of he disibuion moves in he diecion of he osiive -ais wih velociy vo o/m which idenifies o as he momenum of he aicle. The widh of he disibuion: inceases wih ime, coinciding a 0 wih he widh of he iniial disibuion This seading' of he wave funcion is a genuine quanum henomenon. Anohe ineesing obsevaion is ha he wave funcion conseves he hase faco e[io/h] of he oiginal wave funcion and ha he esecive hase faco is elaed wih he velociy of he classical aicle and of he cene of he disibuion. The consevaion of his faco is aiculaly siking fo he un-nomalized iniial wave funcion i.e., he saial deendence of he iniial sae emains invaian in ime. Howeve, a ime deenden hase faco aises which is elaed o he enegy of a aicle wih momenum o.
We had assumed above o 0. The case of abiay can be ecoveed. This yields: i.e., he saial as well as he emoal deendence of he wave funcion emains invaian in his case. One efes o he esecive saes as saionay saes. Such saes lay a cadinal ole in quanum mechanics.