JWKB QUANTIZATION CONDITION. exp ± i x. wiggly-variable k(x) Logical Structure of pages 6-11 to 6-14 (not covered in lecture):

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1 7-1 JWKB QUANTIZATION CONDITION Last time: ( ) i 3 1. V ( ) = α φ( p) = Nep Ep p 6m hα ψ( ) = Ai( z) zeroes of Ai, Ai tales of Ai (and Bi) asymptotic forms far from turning points 2. Semi-Classical Approimation for ψ() classical wavefunction p() = [( E V() )2m] modifies classical to make it QM wavefunction ψ() = p() ep ± i h p( ) d envelope 142 c 43 adjustale phase for wiggly-variale k() oundary conditions ψ without differential equation qualitative ehavior of integrals (stationary phase) dλ validity: <<1 valid not too near turning point. d [ One reason for using semi-classical wavefunctions is that we often need to evaluate integrals of the type ψ Oˆ ψ d. If Oˆ is a slow function of, the phase factor is i p j p i d ep [ pj( ) pi( )] d. Take to find sp... δis range aout sp.. over which d = 0 h phase changes y / 2. Integral is equal to I δ.] ±π ( sp..) Logical Structure of pages 6-11 to 6-14 (not covered in lecture): 1. ψ JWKB not valid (it lows up) near turning point can t match ψ s on either side of turning point. 2. Near a turning point, ± (E), every well-ehaved V() looks linear V() V ( + (E))+ dv ( + ) d =+ first term in a Taylor series. This makes it possile to use Airy functions for any V() near turning point.

2 asymptotic-airy functions have matched amplitudes (and phase) across validity gap straddling the turning point. 4. ψ JWKB for a linear V() is identical to asymptotic-airy! TODAY 1. Summary of regions of validity for Airy, a-airy, l-jwkb, JWKB on oth sides of turning point. This seems complicated, ut it leads to a result that will e eceptionally useful! 2. WKB quantization condition: energy levels without wavefunctions! 3. compute dn E /de (for o normalization can then convert to any other kind of normalization) 4. trivial solution of Harmonic Oscillator E v = hω (v+) v = 0, 1, 2 Non-lecture (from pages 6-12 to 6-14) classical ψ a AIRY =π 1/12 1/12 2mα h 2 a foridden ψ a AIRY = π 1/12 1/12 2mα 2 h 2 a classical ψ l JWKB = C a foridden ψ l JWKB = D a ( ) 1/4 sin 2 3 ( ) 1/4 ep 2 3 ( ) 1/4 sin 2 3 ( ) 1/4 ep 2 3 C, D, and φ are determined y matching. 2mα h 2 2mα h 2 a 2mα h 2 ( ) 3/2 + π 4 a a 2mα h 2 ( ) 3/2 ( ) 3/2 +φ a ( ) 3/2 These Airy functions are not normalized, ut each pair has correct relative amplitude on opposite sides of turning point. l-jwkb has same functional form as a-airy. This permits us to link pairs of JWKB functions across invalid region and then use JWKB to etend ψ() into regions further from turning point where linear approimation to V() is no longer valid.

3 7-3 I CLASSICAL II FORBIDDEN ψ() (E) + Regions of Validity Near Turning Point E = V( ± (E)) OK at t.p. ut not too far ψ AIRY OK at edges of Airy ψ a AIRY region LINEAR V() Common Validity of l-jwkb and a-airy ASYMPTOTIC AIRY LINEAR V() OK far enough from t.p. ψ JWKB EXACT ψ JWKB OK at onset of JWKB region ψ l JWKB LINEAR V() JWKB Common region of validity for ψ a-airy and ψ l-jwkb same functional form, specify amplitude and phase for ψ JWKB () valid far from turning point for eact V()!

4 Quantization of E in Aritrary Shaped Wells 7-4 E I V ( ) E α( a) V ( ) = E ( E) = a α > 0 II V ( ) E+ β( ) V ( ) = E ( E) = + + β > 0 III a (E) = a + (E) = Already know how to splice across I, II and II, III ut how do we match ψ s in a < < region? Region I I ψ JWKB () = C 2 p() e 1 h a p( )d < a (foridden region) (real, no oscillations) Note carefully that argument of ep goes to as, thus ψ I ( ) 0. Note also that (ψ I /C) increases monotonically as increases up to = a. When you are doing matching for the first time, it is very important to verify that the phase of ψ varies with in the way you want it to.

5 7-5 Region II () = Cp() sin 1 h p( )d + π 4 IIa ψ JWKB a The first zero is located at an accumulated phase of (3/4)π inside =a ecause (3/4 + 1/4)π = π and sin π = 0. It does not matter that ψ IIa is invalid near = a, = a < < Note that phase increases as increases - as it must. The π/4 is the etra phase required y the AIRY splice across I,II. It reflects the tunneling of ψ() into the foridden region. PHASE starts at π/4 in classical region and always increases as one moves (further into classical region) away from turning point. NEVER FORGET! Region III III ψ JWKB () = C 2 p() e 1 h p( )d > III ψ JWKB Note that phase advances (i.e. the phase integral gets more positive) as. decreases monotonically to 0 as +. Region II again II ψ JWKB () = Cp() sin 1 h p( )d + π 4 note: argument of sine starts at π/4 and increases as one goes from = inward. In other words, opposite to ψ IIa, the argument decreases from left to right! But ψ IIa () = ψ II () for all a < <! 2 ways to satisfy this requirement 1. sin(θ() { ) = sin[ θ() 1234 argument of ψ IIa ( ) argument of ψ II + (2n + 1)π] AND C = C [sin θ = sin( θ), sin(θ +(2n + 1)π) = sin θ, sin θ = sin( θ + (2n + 1)π)]

6 sin(θ()) = sin[ θ() +2nπ] if C = C now look at what the 2 cases require for the arguments 1. C = C 1 h a pd + π 4 ψ IIa θ() 1 + pd h ( a ) = 1 h pd + π 4 ψ II θ() + ( 2n + 1)π = ( 2n +1 )π π 4 π 4 p( )d = hπ 2n + 12 a + ( 2n +1 )π [ ] Quantization. 2. C = C get p( )d = hπ 2n 12 a [ ] comine the two: a ( ) p( )d = hπ n +1 2 n =0,1,2, C = C( 1) n WKB quantization condition. Most important result of this lecture. n is # of internal nodes ecause argument always starts at π/4 and increases inward to (n + 3/4)π at other turning point. inner t.p. outer t.p. for n = 0 sin(π/4) sin(3π/4) NO NODE! n = 1 sin(π/4) sin(7π/4) 1 node. etc. Node count tells what level it is. pd at aritrary E proe tells how many levels there are at E E proe!

7 7-7 Density of States dn de h dn de is the classical period of oscillation. n(e) = 2 p E ( )d 1 + (E) h (E) 2 dn de = 2 h p E( + ) d + de p E( ) d de + ut p E ( ± ) 0 dn de = 2 d 2mE V( h + [ de ( ) ) ] d + dp E de d (must take derivatives of limits of integration as well as integrand) dn de = 2 h [ ( ) ] d (2m) 2m E V( ) you show that, for harmonic oscillator V() = 1 2 k2 ω ( km) that dn de = 1 hω independent of E, thus period of h.o. is independent of E. Non-lecture for general o normalization + = LLL: can still use this to compute dn de ecause d + de =0 (even though p E( + ) 0). location of right hand turning point is independent of E. Can always use WKB quantization to compute density of o normalized ψ E s, provided that E > V() everywhere ecept the 2 turning points.

8 7-8 Use WKB to solve a few standard prolems. Since WKB is semi-classical, we epect it to work in the n limit. Could e some errors for a few of the lowest-n E n s. Harmonic Oscillator V() = k 2 /2 12 / 1 2 p ( ) = 2m E k 2 At turning points, V tp Eand p thus, at turning points ± =±[ 2En k] 1 2 ecause En = k± 2 ( )= ( tp)= [ ] + = 2E n k 12 / 0, [ ( )] d hπ( n + )= 2m E n k 2 2 [ ] = 2E n k Non-lecture: Dwight Integral Tale t [ a 2 2 ] I = 2mk 2 td = t 2 + a2 2 sin 1 ( / a) here t =0 at oth + and - ( ) [ 2E n k] 2E n k 2 [ 2E n k] ( ) 2E n I = 2mk 2 I = m k k [ ] d [ ] sin 1 1 sin 1 ( 1) E ( n ( π 2) ( π 2) )=π m k (k is force constant, not wave vector) E n V() use the nonlecture result: hπ ( n + )=π m k E n E n = h k m ω ( n + )

9 7-9 I suggest you apply WKB Quantization Condition to the following prolems: See Shankar pages Vee V() = a E n ( n + ) quartic V() = 4 E n ( n + ) l = 0, H atom V() = c 1 E n n 2 harmonic V() = 1 2 k2 E n ( n + ) What does this tell you aout the relationship etween the eponents m and α in V m m and E n n α? power of in V() power of n in E(n) / /3 Validity limits of WKB? splicing of ψ IIa,ψ II? d 2 V d 2 can t e too large near the splice region ψ JWKB is ad when dλ d > ~ 1 (λ changes y more than itself for = λ) near turning points and near the minimum of V() can t use WKB QC if there are more than 2 turning points near ottom of well d2 V dλ d 2 is not small and d > 1 (near oth turning points). However, most wells look harmonic near minimum and WKB gives eact result for harmonic oscillator - should e more OK than one has any right to epect. semi-classical: should e good in high-n limit. If eact E n has same form as WKB QC at low-n, WKB E n is valid for all n. H.O., Morse Oscillator