Physics 2D Lecture Slides Lecture 25: Mar 2 nd

Transcription

1 Cofirmed: D Fial Eam: Thursday 8 th March :3-:3 PM WH 5 Course Review 4 th March am WH 5 (TBC) Physics D ecture Slides ecture 5: Mar d Vivek Sharma UCSD Physics

2 Simple Harmoic Oscillator: Quatum ad Classical Sprig with Force Cost k X= m Quatum Picture: Harmoic Oscillator P() = C e α ψ () = C e α C C How to Get C & α?? Try pluggig i the wave-fuctio ito the time-idepedet Schr. Eq.

3 Quatum Harmoic Oscillator I Pictures E = KE + U ( ) = ω > for = U() Quatum Mechaical Prob for particle to live outside classical turig poits Is fiite! U C -A +A -A +A Classically particle most likely to be at the turig poit (velocity=) Quatum Mechaically, particle most likely to be at = for = Classical & Quatum Pictures of Harmoic Oscillator compared imits of classical vibratio Turig Poits Classical oscillator : at = ± A, chages all KE ito potetial eergy of sprig Total eergy E ( = ±A) = KE ( = ±A) + U ( = ±A) = + A For Quatum Oscillator : Total Eergy E = ω; comparig classical ad quatum eergies ω = A A = ; Classical oscillator boud withi -A A= m ω Caot veture outside = ±A because it has o KE left But due to Ucertaity priciple, Quatum Probability for particle outside classical turig poits P( >A) >!!

4 Quatum Oscillator I The Classically Forbidde Territory Calculate probability of Quatum oscillator where a Classical oscillator ca't dare be! 4 -A Calculate P( X >A = ) ; P( X >A)= ψ ( ) d + ψ ( ) d Sice ψ ( ) = π P( X >A)= ψ ( ) = π A A e Chage variable: z = ad write A = z P( X >A)= e dz = Error F=erfc()=.57 π P( X >A) =6%!!! arge probability to go o to the "other side"! - A is symmetric about = d d e C -A A Ecited States of The Quatum Oscillator ψ ( ) = CH( ) e ; H with H()=4 3 3 )=8 ( ) = Hermite Polyomials H()= H ()= H( H()=(-) e ad E = ( + ) ω = ( + ) hf Agai =,,,3... Quatum # de d

5 Ecited States of The Quatum Oscillator As Æ classical ad quatum probabilities become similar Groud State Eergy > always Measuremet Epectatio: Statistics esso Esemble & probable outcome of a sigle measuremet or the average outcome of a large # of measuremets i i < >= = i i i i = N = P( )d P( )d For a geeral F f() < f ( ) >= f ( ) ψ i = i N i = * ( ) f ( )ψ ( )d Sharpess of A Distr: Scatter aroud average P( )d σ= ( i ) N σ = ( ) ( ) σ = small Sharp distr. Ucertaity X = σ

6 - Particle i the Bo, =, fid <> &? π ψ ()= si π π <>= si si d π = si d, chage variable = π <>= θsi θ, π <>= π π θ π θdθ - π use si θ = ( cos θ ) θ cosθ dθ π <>= (same result as from graphig ψ ( )) = π Similarly < >= s π i ( d ) = 3 π use udv=uv- 3 π 4 X= % of, Particle ot sharply cofied i Bo ad X= < > < > = =.8 vdu Epectatio Values & Operators: More Formally Observable: Ay particle property that ca be measured X,P, KE, E or some combiatio of them,e,g: How to calculate the probable value of these quatities for a QM state? Operator: Associates a operator with each observable Usig these Operators, oe calculates the average value of that Observable The Operator acts o the Wavefuctio (Operad) & etracts ifo about the Observable i a straightforward way gets Epectatio value for that observable + * ˆ * < Q >= Ψ (, t) [ Q] Ψ ( td, ) Q is the observable, [ Qˆ ] is the operator & < Q > is the Epectatio value d Eam p les: [X] =, [P] = i d [P] - [K] = m m [E] = = i t

7 Operators Æ Iformatio Etractors = d Mometum Operator i d gives the value of average mometum i the followig way: [p] or pˆ = + <p> = ψ * ()[ p ]ψ ( ) d = Plug & play form - + ψ - * = dψ () d i d Similerly : ˆ = - = d gives the value of average K E [K] or K m d + + = d ψ ( ) <K> = ψ * ()[ K ]ψ ( ) d = ψ * () d m d - - Similerly + <U> = ψ * ( )[U ( )]ψ ( ) d : plug i the U() f for that case - + a d <E> = ψ ()[ K + U ( )]ψ ( ) d = * - + * ψ () - = d ψ ( ) + U ( ) d m d Hamiltoia Operator [H] = [K] +[U] The Eergy Operator [E] = i= iforms you of the averag e eergy t

8 [H] & [E] Operators [H] is a fuctio of [E] is a fuctio of t.they are really differet operators But they produce idetical results whe applied to ay solutio of the time-depedet Schrodiger Eq. [H]Ψ(,t) = [E] Ψ(,t) + U(, t) Ψ (, t) = i Ψ(, t) m t Thik of S. Eq as a epressio for Eergy coservatio for a Quatum system Where do Operators come from? A touchy-feely aswer Eample :[ p] The mometum Etractor (operator): Cosider as a eample: Free Particle Wavefuctio Ψ i(k-wt) (,t) = Ae ; k = π h, λ = k = λ p p p i( -wt) Ψ(,t) p i( -wt) rewrite Ψ (,t) = Ae ; = i Ae = i Ψ(,t) = p Ψ(,t) i p p Ψ(,t) So it is ot ureasoable to associate [p]= with observable p i

9 Eample : Average Mometum of particle i bo Give the symmetry of the D bo, we argued last time that <p> = : ow some iglorious math to prove it! Be lazy, whe you ca get away with a symmetry argumet to solve a problem..do it & avoid the evil itegratio & algebra..but be sure! π * π ψ ( ) = si( ) & ψ ( ) = si( ) + * * d < p>= ψ [ p] ψd= ψ ψd id π π π < p>= si( )cos( ) d i π Sice sia cosa d = si a...here a = a = π < p>= si ( i = sice Si () = Si ( π ) = = We kew THAT before doig ay math! Quiz : What is the <p> for the Quatum Oscillator i its symmetric groud state Quiz : What is the <p> for the Quatum Oscillator i its asymmetric first ecited state But what about the <KE> of the Particle i Bo? p < p >= so what about epectatio value of K=? m < K >= because < p>= ; clearly ot, sice we showed E=KE Why? What gives? π Because p = ± me =± ; " ± " is the key! The AVERAGE p =, sice particle is movig back & forth p <p > <KE> = < > ; ot! m m Be careful whe beig "lazy" Quiz: what about <KE> of a quatum Oscillator? Does similar logic apply??

10 Schrodiger Eq: Statioary State Form Recall whe potetial does ot deped o time eplicitly U(,t) =U() oly we used separatio of,t variables to simplify Ψ(,t) = ψ() φ(t) & broke S. Eq. ito two: oe with oly ad aother with t oly - ψ ( ) m φ() t i = Eφ() t t + U( ) ψ( ) = E ψ( ) Ψ ( t, ) =ψ ( ) φ( t) How to put Humpty-Dumpty back together? e.g to say how to go from a epressio of ψ() Ψ(,t) which describes time-evolutio of the overall wave fuctio Schrodiger Eq: Statioary State Form d d f( t) Sice [ l f( t) ] = dt f( t) dt φ() t φ( t) E ie I i = Eφ( t), rewrite as = = t φ( t) t i ad itegrate both sides w.r.t. time t = t t t φ() t ie d φ( t) ie dt = dt dt = φ() t t φ() t dt t= ie l φ( t) l φ() = t, ow epoetiate both sides ie t φ( t) = φ() e ; φ() = costat= iitial coditio = (e.g) ie t φ( t) = e & T hus Ψ(,t)= ψ () e where E = eergy of system ie t

11 Schrodiger Eq: Statioary State Form * * + ie ie ie ie t t t t * Pt (, ) =ΨΨ= ψ ( ) e ψ( ) e = ψ ( ) ψ( e ) = ψ( ) I such cases, P(,t) is INDEPENDENT of time. These are called "statioary" states because Prob is idepedet of time Eamples : Particle i a bo (why?) : Quatum Oscillator (why?) Total eergy of the system depeds o the spatial orietatio of the system : charteristic of the potetial U(,t)! C