Physics 2D Lecture Slides Lecture 22: Feb 22nd 2005

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1 Physics D Lecture Slides Lecture : Feb d 005 Vivek Sharma UCSD Physics Itroducig the Schrodiger Equatio! (, t) (, t) #! " + U ( ) "(, t) = i!!" m!! t U() = characteristic Potetial of the system Differet potetial for differet forces Hece differet solutios for the Diff. eq. characteristic wavefuctios for a particular U()

2 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 -! #! ( ) m # #"( t) i! = E" ( t) # t + U ( )! ( ) = E! ( ) #(, t) =! ( ) "( 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 Eample of a Particle Iside a Bo With Ifiite Potetial (a) Electro placed betwee set of electrodes C & grids G eperieces o force i the regio betwee grids, which are held at Groud Potetial However i the regios betwee each C & G is a repellig electric field whose stregth depeds o the magitude of V (b) If V is small, the electro s potetial eergy vs has low slopig walls (c) If V is large, the walls become very high & steep becomig ifiitely high for V (d) The straight ifiite walls are a approimatio of such a situatio U= U() U=

3 A More Iterestig Potetial : Particle I a Bo U() Write the Form of Potetial: Ifiite Wall U(,t) =!; " 0, # L U(,t) = 0 ; 0 < X < L Classical Picture: Particle daces back ad forth Costat speed, cost KE Average <P> = 0 No restrictio o eergy value E=K+U = K+0 Particle ca ot eist outside bo Ca t get out because eeds to borrow ifiite eergy to overcome potetial of wall What happes whe the joker is subatomic i size?? Ψ() for Particle Iside D Bo with Ifiite Potetial Walls Iside the bo, o force " U=0 or costat (same thig) " -! d! ( ) m d + 0! ( ) = E! ( ) d! ( ) " = # k! ( ) ; k d d! ( ) or d + k! = $ me =! ( ) 0 figure out what! () solves this diff eq. I Geeral the solutio is! ( ) = A sik + B cosk (A,B are costats) Need to figure out values of A, B : How to do that? Apply BOUNDARY Coditios o the Physical Wavefuctio We said! ( ) must be cotiuous everywhere So match the wavefuctio just outside bo to the wavefuctio value just iside the bo " At = 0 "! ( = 0) = 0 & At = L "! ( = L) = 0 %! ( = 0) = B = 0 (Cotiuity coditio at =0) &! ( = L) = 0 " A Si kl = 0 (Cotiuity coditio at =L) & " kl = & " k =, =,,3,... ' L &! So what does this say about Eergy E? : E = ml Quatized (ot Cotiuous)! Why ca t the particle eist Outside the bo? E Coservatio X=0 X=L 3

4 Quatized Eergy levels of Particle i a Bo What About the Wave Fuctio Normalizatio? The particle's Eergy ad Wavefuctio are determied by a umber We will call! Quatum Number, just like i Bohr's Hydroge atom What about the wave fuctios correspodig to each of these eergy states? " # = Asi( k) = Asi( ) for 0< < L L = 0 for % 0, % L Normalized Coditio : L L * " '# d = A ' Si L 0 0 = # ( ) Use Si $ = & Cos$ L A ( " ) = & cos( ) ad sice cos $ = si$ ' * +, L ' - 0 A L A L =. = So # = " si( k) = si ( )...What does this look like? L L L 4

5 Wave Fuctios : Shapes Deped o Quatum # Wave Fuctio Probability P(): Where the particle likely to be Zero Prob Where i The World is Carme Sa Diego? We ca oly guess the probability of fidig the particle somewhere i For = (groud state) particle most likely at = L/ For = (first ecited state) particle most likely at L/4, 3L/4 Prob. Vaishes at = L/ & L How does the particle get from just before =L/ to just after?» QUIT thikig this way, particles do t have trajectories» Just probabilities of beig somewhere Classically, where is particle most likely to be? Equal prob. of beig aywhere iside the Bo NOT SO says Quatum Mechaics! 5

6 Remember Sesame Street? This particle i the bo is brought to you by the letter Its the Big Boss Quatum Number How to Calculate the QM prob of Fidig Particle i Some regio i Space Cosider = state of the particle L 3L Ask : What is P ( # # )? 4 4 3L 3L 3L $ % " ' ( L L ) * L 4 4 4!! P = d = si d =. ( & cos ) d L L L L 3 L / 4 + L, + L!, $! 3L! L % P = si si. si. L - & = & ' & ( /. 0 - /! L. 0! ) L 4 L 4 * L / 4 P = & ( & & ) = %! Classically 50% (equal prob over half the bo size) Substatial differece betwee Classical & Quatum predictios 6

7 Whe The Classical & Quatum Pictures Merge: But oe issue is irrecocilable: Quatum Mechaically the particle ca ot have E = 0 This is a cosequece of the Ucertaity Priciple The particle moves aroud with KE iversely proportioal to the Legth Of the D Bo Fiite Potetial Barrier There are o Ifiite Potetials i the real world Imagie the cost of as battery with ifiite potetial diff Will cost ifiite $ sum + ot available at Radio Shack Imagie a realistic potetial : Large U compared to KE but ot ifiite Regio I Regio II Regio III U E=KE X=0 X=L Classical Picture : A boud particle (o escape) i 0<<L Quatum Mechaical Picture : Use ΔE.Δt h/π Particle ca leak out of the Bo of fiite potetial P( >L) 0 7

8 Fiite Potetial Well -! d " ( ) + U" ( ) = E " ( ) m d d " ( ) m $ = ( U # E) " ( ) d! =! " ( );! = m(u-e)! $ Geeral Solutios : " ( ) = Ae + Be Require fiiteess of " ( ) $ " ( ) = Ae +!...<0 (regio I) +! #! #! " ( ) = Ae...>L (regio III) Agai, coefficiets A & B come from matchig coditios at the edge of the walls ( =0, L) But ote that wave f at " ( ) at ( =0, L) % 0!! (why?) d" ( ) Further require Cotiuity of " ( ) ad d These lead to rather differet wave fuctios Fiite Potetial Well: Particle ca Burrow Outside Bo 8

9 Fiite Potetial Well: Particle ca Burrow Outside Bo Particle ca be outside the bo but oly for a time Δt h/ ΔE ΔE = Eergy particle eeds to borrow to Get outside ΔE = U-E + KE The Ciderella act (of violatig E Coservatio cat last very log Particle must hurry back (cat be caught with its had iside the cookie-jar)! Peetratio Legth! = = " m(u-e) If U>>E # If U $ % #! $ 0 Tiy peetratio Fiite Potetial Well: Particle ca Burrow Outside Bo! Peetratio Legth! = = " m(u-e) If U>>E # Tiy peetratio If U $ % #! $ 0!! E =, =,,3, 4... m( L + " ) Whe E=U the solutios blow up # Limits to umber of boud states(e < U ) Whe E>U, particle is ot boud ad ca get either reflected or trasmitted across the potetial "barrier" 9

10 Simple Harmoic Oscillator: Quatum ad Classical Sprig with Force Cost k X=0 m a Stable Ustable b U() c Stable Particle of mass m withi a potetial U()! du( ) F()= - d! du( ) F(=a) = - = 0, d!! F(=b) = 0, F(=c)=0...But... look at the Curvature:! U! U > 0 (stable), < 0 (ustabl e)!! Stable Equilibrium: Geeral Form : U() =U(a)+ k(! a) Rescale " U( ) = k(! a) Motio of a Classical Oscillator (ideal) Ball origially displaced from its equilibirium positio, motio cofied betwee =0 & =A k U()= k = m# ;# = = Ag. Freq m E = ka " Chagig A chages E E ca take ay value & if A $ 0, E $ 0 Ma. KE at = 0, KE= 0 at = ± A 0

11 Quatum Picture: Harmoic Oscillator Fid the Groud state Wave Fuctio! () Fid the Groud state Eergy E whe U()= m" Time Depedet Schrodiger Eq: -! #! ( ) + m # m"! ( ) = E! ( ) $ d! ( ) m = ( E % " )! ( ) = 0 What! () solves this? d! m Two guesses about the simplest Wavefuctio:.! () should be symmetric about.! () & 0 as & ' d! () +! () should be cotiuous & = cotiuous d % 0 0 My guess:! () = C e ( ; Need to fid C &( : What does this wavefuctio & PDF look like? Quatum Picture: Harmoic Oscillator P() = C 0e "! " () = C 0 # e! C C 0 How to Get C 0 & α?? Try pluggig i the wave-fuctio ito the time-idepedet Schr. Eq.

12 Time Idepedet Sch. Eq & The Harmoic Oscillator % " ( ) m Master Equatio is : m E %! ( ) $! d" $! Sice " ( ) = C0e, = C0( $! ) e, d = [ # $ ] " ( ) d " ( ) d( $! ) $! $! $! = C 0 e + C0( $! ) e = C0[4! $!] e d d m & C0[ 4! $ ] = [ # ]! $! $!! e m $ E C 0e Match the coeff of ad the Costat terms o LHS & RHS m m# & 4! = or = m #!!! m & the other match gives! =, substituig E! &! E=!# =hf!!!!...( Plack's Oscillators) What about C 0? We lear about that from the Normalizatio cod. +$ +$ % m!! - # 0( ) = = - 0 %$ %$ Sice SHO: Normalizatio Coditio d C e d +$ % a - %$ e d = " (dot memorize this) a m! Idetifyig a= ad usig the idetity above! 4 & m! ' ( C0 = ) + "! *, Hece the Complete NORMALIZED wave fuctio is : & m! ' # 0() = ) + "! *, 4 e has eergy E = hf % m!! Groud State Wavefuctio Plack's Oscillators were electros tied by the "sprig" of the mutually attractive Coulomb Force

Quatum Oscillator I Pictures E = KE + U ( ) > 0 for =0 Quatum Mechaical Prob for

U() U C 0 -A +A -A +A Classically particle most likely to be at the turig poit

Quatum Pictures of SHO compared Limits of classical vibratio : Turig Poits (do o

13 Quatum Oscillator I Pictures E = KE + U ( ) > 0 for =0 Quatum Mechaical Prob for particle To live outside classical turig poits Is fiite! U() U C 0 -A +A -A +A Classically particle most likely to be at the turig poit (velocity=0) Quatum Mechaically, particle most likely to be at = 0 for =0 Classical & Quatum Pictures of SHO compared Limits of classical vibratio : Turig Poits (do o Board) Quatum Probability for particle outside classical turig poits P( >A) =6%!! Do it o the board (see Eample problems i book) 3

14 Ecited States of The Quatum Oscillator 0 H ()=4 # 3 3 )=8 # m! #! " ( ) = C H ( ) e ; H ( ) = Hermite Polyomials with H ()= H ()= H ( H ()=(-) e ad E # d e d = ( + )!! = ( + ) hf Agai =0,,,3... $ Quatum # Ecited States of The Quatum Oscillator Groud State Eergy >0 always 4

Physics 2D Lecture Slides Lecture 25: Mar 2 nd

Physics 2D Lecture Slides Lecture 25: Mar 2 nd 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 Simple Harmoic Oscillator: Quatum ad Classical