Chapter 4 Postulates & Geeral Priciples of Quatu Mechaics Backgroud: We have bee usig quite a few of these postulates already without realizig it. Now it is tie to forally itroduce the. State of a Syste & the Wavefuctio - dyaical variables i classical echaics -- e.g. positio, x, oetu, p, eergy, E -- these variables are also kow as observables - classical echaics: -- ay state is specified by (x, y, z) with oetu (px, py, pz) at ay tie, t -- Newto s laws of otio rule i classical systes: d x F( xyz,, ) d y F( xyz,, ) d z F( xyz,, ) x y z dt dt dt -- these equatios alog with the iitial coditios, e.g. x(0), describe the particle s positio as a fuctio of tie - QM equivalet or Postulate 1 -- The state of a quatu echaical syste is copletely specified by a fuctio Ψ(x) that depeds upo the coordiate of the particle ad o tie. This fuctio, called the wave fuctio or the state fuctio, has the iportat property that Ψ (r,t)ψ(r,t) is the probability that the particle lies i the volue eleet dydz located at r at tie t. -- this postulate ca easily be exteded to ulti-diesios for siplicity all postulates will be itroduced i 1D -- for ore tha oe particle: Ψ (x1, x) Ψ(x1,x)1 - We wat oralized wave fuctios -- the total probability of fidig a particle soewhere is: -- for 3D: -- i geeral: ( x, yz, ) ( xyzdydz,, ) 1 -- oralizatio factor: d 1 where represets all coordiates 1 ( x ) 1 If d N thethe oralized N - We ust have well behaved ad ot aughty wavefuctios ad obey the followig: -- ψ ust be cotiuous -- ψ ust be cotiuous -- ψ ψ ust be sigle-valued -- ( x ) ust be fiite otherwise we caot oralize ψ QM operators represet CM variables - Postulate : for every observable i classical echaics there is a correspodig liear operator i quatu echaics.
-- i.e. x i CM correspods to ˆx i QM -- see Table 4.1 o p. 148 -- Agular Moetu operator, LorLˆ ˆ ˆ ˆ r xi yj zk p p ˆ ˆ ˆ ˆ ˆ ˆ xi pyj pzk vxi vyj vzk iˆ ˆj kˆ L r p x y z yp ˆ ˆ ˆ z zpy i zpx xpz j xpy ypx k p p p x y z --- Classically: Lx y z Ly z x Lz x y z y x z y x --- Quatu Mechaically: Lx iy z Ly iz x Lz i x y z y x z y x Observables ust eigevalues of QM operators - suppose we have states, 1,,,, which are eigefuctios of  ad possess the sae eigevalue, a, or Aˆ 1 a ˆ ˆ 1 A a A a -- a liear cobiatio of these states will also be a eigefuctio of  c11 c c AˆAˆ c1 1cc ˆ ˆ ˆ c1a1caca ca ca ca a c c c a 1 1 1 1 - Postulate 3: I ay easureet of the observable associated with the operator, Â, the oly values that will be observed are the eigevalues a which satisfy the eigevalue equatio:  a. No other values will be observed. -- Exaple: 1D PIB d x Ĥ sice V(x) = 0 i the box; si a a h We have show that this results i E Ĥ E 8a I which the eergy ad the wavefuctio are depedet o - Postulate 4: If a syste is i a state described by a oralized wavefuctio, ψ,
the the average value of the observable correspodig to  is give by A Aˆ d -- we have bee callig this the average/ea but it is ore cooly referred to as the expectatio value -- A Aˆ d a d a d C Coutators ad aipulatig operators - if we have two operators which eed to be applied to a fuctio, we do so sequetially fro right to left -- ABf ˆ ˆ A ˆ Bf ˆ d d -- e.g. xp ˆˆ x xi i x - coutator: is a set of operators i which order of applicatio is egligible -- atheatically: ABf ˆ ˆ BAf ˆˆ -- e.g. let kietic operator ad the oetu operator are coutator or they are said to coute 3 3 ˆ d d i d K ˆ xpx ( ) i x 3 3 3 ˆ d d i d pk ˆ x x i 3 Kˆ ˆ ( ) ˆ ˆ xpx x pxkx - o-coutator is a set of operators that do ot coute -- atheatically: ABf ˆ ˆ BAf ˆˆ -- e.g. the positio ad oetu operator d d pˆ ˆ xx i x i x d d i ix ix Clearly this ot what we obtaied for xp ˆˆ x - otatio, otatio, otatio -- sice coutators coute we ca say that: ABf ˆ ˆ AˆBf ˆ ABf ˆ ˆ BAf ˆˆ 0 ˆ f where 0ˆ is the zero operator -- aside fro the zero operator we also have the Idetity operator, Î -- oce agai i order to avoid writig we have a short hadfor: AB ˆ ˆ BA ˆˆ Aˆ, Bˆ - back to Heiseberg
d d pˆ, ˆ x xi ix ix i px ˆ ˆ ˆˆ ˆ x xpx ii Aother way to represet the Ucertaity priciple is: 1 1 ˆ 1 ( ) ˆ, ˆ p x x px x i I i - whe operators do ot coute this eas that we caot accurately easure the both at the sae tie - the opposite is also true if they do coute the we ca easure the both at the sae tie QM Operators Must be Heritia - Heritia Operators: operators which whe applied to oe of their eigefuctios are real regardless of whether the operator ad the eigefuctio is coplex -- atheatically: Aˆ A ˆ a a but sice a is real the a a ikx pˆ x where e -- exaple: ikx ikx ikx i e iike k e x real eigvalue - order of operatio is key - your book also defies Heritia as ˆ ˆ f ( xagx ) ( ) gxaf ( ) ( x ) - if you reverse the order you ust take the cojugate ad reverse the operator - Postulate : To every observable i classical echaics there correspods a liear, Heritia operator i quatu echaics - Applicatio of Dirac f f d f Afd ˆ A ˆ ad if Heritia: A ˆ A ˆ Eigefuctios of Quatu Mechaical Operators are Orthogoal 1 i j - usig the Kroeecker delta: ij 0 i j - for two oralized eigestates, ψ ad ψ: d If Two Operators Coute, They have a Mutual Set of Eigefuctios - I the text they prove if is a eigefuctio of both A & B ad f x c the A, B f x 0 - This eas that coutators share the sae set of eigefuctios which ca be easured siultaeously (see previous exaples)
The Probability of Obtaiig a Certai Value of a Observable i a Measureet is Give by a Fourier Coefficiet f x c - For a set of coplete fuctios we ca write a Fourier expasio x f x x c x c x x c c - The Fourier coefficiets are give by c - We use a siilar equatio to fid our average eergy Hˆ x Hˆ c x c Hˆ x ulitply by c x ad itegrate over ˆ ˆ cce xx cce E c x c H x c c x H x - So the probability of observig E Tie-Depedet Schrödiger Equatio - Postulate 5: The wavefuctio or state fuctio of a syste evolves i tie accordig to the tie-depedet Schrödiger equatio ˆ ( x, t) H( x, t) i t - usually the Hailtoia does ot have tie depedet ters, ad so we ca use our separatio of variables techique or ( x, t) f( t) 1 ˆ i df( t) H sice Hˆ E f( t) dt 1 i df( t) i df( t) i df( t) E E or Ef( t) f ( t) dt f ( t) dt dt - ow we eed to itegrate our expressio i df() t i iet Edt Et l f () t f () t e f() t - therefore our wavefuctio becoes: iet it ( xt, ) ( xe ) or( xt, ) ( xe ) usig Eh - I ost cheical systes we are dealig with a set of states so: iet i t ( x, t) e e - For a syste which is a eigestate of Hˆ E, the c iet iet ( x, t) ( x, t) e e - Therefore, ( x ) are statioary state wavefuctios ad the ability to deterie expectatio values, etc. for a tie depedet wavefuctio is the sae as that for
the idepedet for Quatu Mechaics Ca Describe the Two-Slit Experiet SKIP