The WKB approximation - Quantum mechanics 2 - Lecture 4

Contents General remarks The “classical” region Tunneling The connection formulas Literature1 General remarks2 The “classical” region3 Tunneling4 The connection formulas5 LiteratureIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureContents1 General remarks2 The “classical” region3 Tunneling4 The connection formulas5 LiteratureIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureWKB = Wentzel, Kramers, Brillouinin Holland it’s KWBin France it’s BKWin England it’s JWKB (for Jeffreys)Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureBasic idea:1 particle Epotential V (x) constantif E > V ⇒ ψ(x) = Ae ±ikx , k =A questionWhat’s the character of A and λ = 2π/k here?√2m(E − V )Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureBasic idea:1 particle Epotential V (x) constant√2m(E − V )if E > V ⇒ ψ(x) = Ae ±ikx , k =2 suppose V (x) not constant, but varies slowly wrt λIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureBasic idea:1 particle Epotential V (x) constant√2m(E − V )if E > V ⇒ ψ(x) = Ae ±ikx , k =2 suppose V (x) not constant, but varies slowly wrt λA questionWhat can we say about ψ, A and λ now?Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureBasic idea:1 particle Epotential V (x) constant√2m(E − V )if E > V ⇒ ψ(x) = Ae ±ikx , k =2 suppose V (x) not constant, but varies slowly wrt λA questionWhat can we say about ψ, A and λ now?We still have oscillating ψ, but with slowly changable A and λ.Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureBasic idea:1 particle Epotential V (x) constant√2m(E − V )if E > V ⇒ ψ(x) = Ae ±ikx , k =2 suppose V (x) not constant, but varies slowly wrt λ3 if E < V , the reasoning is analogousIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureBasic idea:1 particle Epotential V (x) constant√2m(E − V )if E > V ⇒ ψ(x) = Ae ±ikx , k =2 suppose V (x) not constant, but varies slowly wrt λ3 if E < V , the reasoning is analogousA questionWhat if E ≈ V ?Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureBasic idea:1 particle Epotential V (x) constant√2m(E − V )if E > V ⇒ ψ(x) = Ae ±ikx , k =2 suppose V (x) not constant, but varies slowly wrt λ3 if E < V , the reasoning is analogousA questionWhat if E ≈ V ? Turning pointsIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureContents1 General remarks2 The “classical” region3 Tunneling4 The connection formulas5 LiteratureIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureS.E.− 2p2∆ψ + V (x)ψ = Eψ ⇐⇒ ∆ψ = −2m ψ , p(x) = √ 2m [E − V (x)]2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureS.E.− 2p2∆ψ + V (x)ψ = Eψ ⇐⇒ ∆ψ = −2m ψ , p(x) = √ 2m [E − V (x)]2“Classical” region E > V (x) , p realIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureS.E.− 2p2∆ψ + V (x)ψ = Eψ ⇐⇒ ∆ψ = −2m ψ , p(x) = √ 2m [E − V (x)]2“Classical” region E > V (x) , p real ψ(x) = A(x)e iφ(x)A(x) and φ(x) realIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteraturePutting ψ(x) into S.E. gives two equations:]A ′′ = A[(φ ′ ) 2 − p2(1) 2(A 2 φ ′) ′= 0 (2)Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteraturePutting ψ(x) into S.E. gives two equations:]A ′′ = A[(φ ′ ) 2 − p2(1) 2(A 2 φ ′) ′= 0 (2)Solve (2)A =C √ φ′ , C ∈ RIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteraturePutting ψ(x) into S.E. gives two equations:]A ′′ = A[(φ ′ ) 2 − p2(1) 2(A 2 φ ′) ′= 0 (2)Solve (2)Solve (1)A = √ CAssumption: A varies slowly, C ∈ R φ′ ⇒ A ′′ ≈ 0φ(x) = ± 1 ∫p(x)dxIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureSolve (2)Solve (1)A = √ CAssumption: A varies slowly, C ∈ R φ′⇒ A ′′ ≈ 0φ(x) = ± 1 ∫p(x)dxResulting wavefunctionψ(x) ≈√ C e ± i ∫ p(x)dxp(x)Note: general solution is a linear combination of these.Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureSolve (1)Solve (2)A = √ CAssumption: A varies slowly, C ∈ R φ′⇒ A ′′ ≈ 0φ(x) = ± 1 ∫p(x)dxResulting wavefunctionψ(x) ≈√ C e ± i ∫ p(x)dxp(x)Note: general solution is a linear combination of these.Probability of finding a particle at x|ψ(x)| 2 ≈ |C|2p(x)Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: potential well with two vertical walls{ some function , 0 < x < aV (x) =∞ ,otherwiseIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: potential well with two vertical walls{ some function , 0 < x < aV (x) =∞ ,otherwiseAgain, assume E > V (x) =⇒ψ(x) ≈1√[C +e iφ(x) + C −e −iφ(x)] 1= √ [C 1 sin φ(x) + C 2 cos φ(x)]p(x) p(x)whereφ(x) = 1 ∫ x0p(x ′ )dx ′Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample (cont.)Boundary conditions: ψ(0) = 0, ψ(a) = 0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒∫ a0p(x)dx = nπIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample (cont.)Boundary conditions: ψ(0) = 0, ψ(a) = 0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒∫ a0p(x)dx = nπTake, for example, V (x) = 0 ⇒E n = n2 π 2 22ma 2We got an exact result...is this strange?Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample (cont.)Boundary conditions: ψ(0) = 0, ψ(a) = 0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒∫ a0p(x)dx = nπTake, for example, V (x) = 0 ⇒E n = n2 π 2 22ma 2We got an exact result...is this strange? No, since A = √ 2/a = const.Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureContents1 General remarks2 The “classical” region3 Tunneling4 The connection formulas5 LiteratureIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureNow, assume E < V :ψ(x) ≈C√ e ± 1 ∫ |p(x)|dx|p(x)|where p(x) is imaginary.Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureNow, assume E < V :ψ(x) ≈C√ e ± 1 ∫ |p(x)|dx|p(x)|where p(x) is imaginary.Consider the potential:{ some function , 0 < x < aV (x) =0 , otherwiseIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturex < 0ψ(x) = Ae ikx + Be −ikxIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturex < 0x > aψ(x) = Ae ikx + Be −ikxψ(x) = Fe ikxTransmission probability: T =|F |2|A| 2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturex < 0 0 ≤ x ≤ a x > aψ(x) = Ae ikx + Be −ikx ψ(x) ≈ √ C e 1 ∫ x0|p(x ′ )|dx ′|p(x)|+ √ D e − 1 ∫ x0|p(x ′ )|dx ′|p(x)|ψ(x) = Fe ikxTransmission probability: T =|F |2|A| 2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturex < 0 0 ≤ x ≤ a x > aψ(x) = Ae ikx + Be −ikx ψ(x) ≈ √ C e 1 ∫ x0|p(x ′ )|dx ′|p(x)|+ √ D e − 1 ∫ x0|p(x ′ )|dx ′|p(x)|ψ(x) = Fe ikxTransmission probability:|F |2T =|A| 2High, broad barrier 1st termgoes to 0Why?Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturex < 0 0 ≤ x ≤ a x > aψ(x) = Ae ikx + Be −ikx ψ(x) ≈ √ C e 1 ∫ x0|p(x ′ )|dx ′|p(x)|+ √ D e − 1 ∫ x0|p(x ′ )|dx ′|p(x)|ψ(x) = Fe ikxTransmission probability:|F |2∫ a0|p(x ′ )|dx ′T =|A| ∼ 2 2 e− High, broad barrier 1st termgoes to 0Why?T ≈ e −2γ ,γ = 1 ∫ a0|p(x)|dxIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Gamow’s theory of alpha decayfirst time that quantummechanics had beenapplied to nuclearphysicsIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Gamow’s theory of alpha decay (cont.)first time that quantummechanics had beenapplied to nuclearphysicsturning points:1 r 1 ↦−→ nucleus radius(6.63 fm for U 238 )2 r 2 ↦−→1 2Ze 2= E4πɛ 0 r 2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Gamow’s theory of alpha decay (cont.)γ = 1 ∫√r2( ) √ ∫ 1 2Ze2m22mE r2√r2− E dr = r 14πɛ 0 r 2 r 1r − 1drIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Gamow’s theory of alpha decay (cont.)γ = 1 ∫√r2( ) √ ∫ 1 2Ze2m22mE r2√r2− E dr = r 14πɛ 0 r 2 r 1r − 1drSubstituting r = r 2 sin 2 u gives√ [ ( √ )2mE πγ = r 2 2 − r1sin−1 − √ ]r 1(r 2 − r 1)r 2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Gamow’s theory of alpha decay (cont.)γ = 1 ∫√r2( ) √ ∫ 1 2Ze2m22mE r2√r2− E dr = r 14πɛ 0 r 2 r 1r − 1drSubstituting r = r 2 sin 2 u gives√ { [ √ ]2mE πγ = r 2 2 − r1sin−1 − √ }r 1(r 2 − r 1)r 2 } {{ }} {{ } √r 1 ≪r−−−→2 √ r1 r 2 −r 2 r 1 ≪r1 −−−→2 √ r1 r 2r1 /r 2} {{ }π2r 2 −2 √ r 1 r 2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Gamow’s theory of alpha decay (cont.)Substituting r = r 2 sin 2 u gives√ { [ √ ]2mE πγ = r 2 2 − r1sin−1 − √ }r 1(r 2 − r 1)r 2 } {{ }} {{ } √r 1 ≪r−−−→2 √ r1 r 2 −r 2 r 1 ≪r1 −−−→2 √ r1 r 2r1 /r 2} {{ }π2r 2 −2 √ r 1 r 2where√2mE[ π]γ ≈ 2 r2 − 2√ Z √r 1r 2 = K 1 √E − K 2 Zr1K 1 =K 2 =( ) √e2 π 2m= 1.980 MeV 1/24πɛ 0 ( ) e2 1/2 √ 4 m= 1.485 fm −1/24πɛ 0 Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Gamow’s theory of alpha decay (cont.)v average velocity2r 1/v average timebetween “collisions”with the nucleuspotential “wall”v/2r 1 averagefrequancy of “collisions”e −2γ “escape”probability(v/2r 1)e −2γ “escape” probability perunit timeLifetime:τ = 2r1v e2γIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Gamow’s theory of alpha decay (cont.)v average velocity2r 1/v average timebetween “collisions”with the nucleuspotential “wall”v/2r 1 averagefrequancy of “collisions”e −2γ “escape”probability(v/2r 1)e −2γ “escape” probability perunit timeLifetime:τ = 2r1v e2γ ⇒ ln τ ∼ 1 √EIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureHWSolve Problem 8.3 from Ref. [2].Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureContents1 General remarks2 The “classical” region3 Tunneling4 The connection formulas5 LiteratureIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureLet us repeat:⎧⎨ψ(x) ≈⎩√1[Be i ∫ 0 x p(x ′ )dx ′ + Ce − i ∫ 0p(x)√1De − 1p(x)x p(x ′ )dx ′] , if x < 0∫ x0|p(x ′ )|dx ′ , if x > 0Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureLet us repeat:⎧⎨ψ(x) ≈⎩√1[Be i ∫ 0 x p(x ′ )dx ′ + Ce − i ∫ 0p(x)√1De − 1p(x)x p(x ′ )dx ′] , if x < 0∫ x0|p(x ′ )|dx ′ , if x > 0Our mission: join these two solutions at the boundary.Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureLet us repeat:⎧⎨ψ(x) ≈⎩√1[Be i ∫ 0 x p(x ′ )dx ′ + Ce − i ∫ 0p(x)√1De − 1p(x)x p(x ′ )dx ′] , if x < 0∫ x0|p(x ′ )|dx ′ , if x > 0Our mission: join these two solutions at the boundary.A problemWhat happens with the w.f. whenE ≈ V ?Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureLet us repeat:⎧⎨ψ(x) ≈⎩√1[Be i ∫ 0 x p(x ′ )dx ′ + Ce − i ∫ 0p(x)√1De − 1p(x)x p(x ′ )dx ′] , if x < 0∫ x0|p(x ′ )|dx ′ , if x > 0Our mission: join these two solutions at the boundary.A problemWhat happens with the w.f. whenE ≈ V ?E ≈ V ⇒ p(x) → 0 ⇒ ψ → ∞ !Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureLet us repeat:⎧⎨ψ(x) ≈⎩√1[Be i ∫ 0 x p(x ′ )dx ′ + Ce − i ∫ 0p(x)√1De − 1p(x)x p(x ′ )dx ′] , if x < 0∫ x0|p(x ′ )|dx ′ , if x > 0Our mission: join these two solutions at the boundary.A problemWhat happens with the w.f. whenE ≈ V ?E ≈ V ⇒ p(x) → 0 ⇒ ψ → ∞ !A solutionConstruct a “patching”wavefunction ψ p.Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureApproximation: we linearize the potentialV (x) ≈ E + V ′ (0)xIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureApproximation: we linearize the potentialFrom S.E. we get V (x) ≈ E + V ′ (0)x[ ]d 2 1ψ p2m= zψdz 2 p , z = αx , α = V ′ 3(0)2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureApproximation: we linearize the potentialFrom S.E. we get V (x) ≈ E + V ′ (0)xd 2 ψ p= zψdz 2 p ,} {{ }z = αx , α =Airy’s equation[ ] 12m V ′ 3(0)2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureApproximation: we linearize the potentialFrom S.E. we get V (x) ≈ E + V ′ (0)xd 2 ψ p= zψdz 2 p ,} {{ }z = αx , α =Airy’s equation[ ] 12m V ′ 3(0)2ψ p = a Ai(αx)} {{ }+b Bi(αx)} {{ }Airy function Airy functionIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturea delicate double constraint has to be satisfiedIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturea delicate double constraint has to be satisfiedwe need WKB w.f. and ψ p for both overlap regions (OLR)Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturep(x) = √ 2m(E − V ) ≈ α 3 2√−xIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas Literaturep(x) ≈ α 3 √2 −xOLR 2 (x > 0)∫ x0|p(x ′ )|dx ′ ≈ 2 3 (αx) 3 2ψ WKBψ z≫0p≈≈D√α3/4x 1/4 e− 2 3 (αx)3/2a2 √ π(αx) 1/4 e− 2 3 (αx)3/2b+ √ e 3 2 (αx)3/2π(αx)1/4⇒ a = D√4πα , b = 0Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureOLR 2 (x > 0)∫ x0ψ WKBψ z≫0p√4πα , b = 0 OLR 1 (x < 0)|p(x ′ )|dx ′ ≈ 2 ∫3 (αx) 3 02xD√ 2 α3/4x 1/4 e− 3 (αx)3/2ψ WKB ≈a2 √ 2 π(αx) 1/4 e− 3 (αx)3/2b+ √ e 3 2 (αx)3/2 ψ z≪0π(αx)1/4 p ≈≈≈⇒ a = D⇒p(x ′ )dx ′ ≈ 2 3 (−αx) 3 21[√ Be i 2 3 (−αx)3/2α3/4(−x) 1/4+Ce −i 2 3 (−αx)3/2]a 1√ π(−αx)1/4 2i−e −iπ/4 e −i 2 3 (−αx)3/2]a2i √ π eiπ/4 = √ Bα−a2i √ π e−iπ/4 = √ Cα[e iπ/4 e i 2 3 (−αx)3/2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureThe connection formulasB = −ie iπ/4 · D ,C = ie −iπ/4 · DIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureThe connection formulasB = −ie iπ/4 · D ,C = ie −iπ/4 · DWKB w.f.⎧⎪⎨ψ(x) ≈⎪⎩[ ∫2D 1 x2√ sin p(x ′ )dx ′ + π ], if x < x 2p(x) x4[D√ exp − 1 ∫ x]|p(x ′ )|dx ′ , if x > x 2|p(x)| x 2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potentail well with one vertical wallIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potentail well with one vertical wallBoundary condition: ψ(0) = 0, gives for ψ WKB∫ x2(p(x)dx = n − 1 )π , n = 1, 2, 3, . . .40Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potentail well with one vertical wall (cont.)For instance, consider the “half-harmonic oscillator”:⎧1 ⎪⎨V (x) =2 mω2 x 2 , x > 0⎪⎩0 otherwiseIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potentail well with one vertical wall (cont.)For instance, consider the “half-harmonic oscillator”:⎧1 ⎪⎨V (x) =2 mω2 x 2 , x > 0⎪⎩0 otherwiseHere we have√p(x) = mω x2 2 − x 2Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potentail well with one vertical wall (cont.)For instance, consider the “half-harmonic oscillator”:⎧1 ⎪⎨V (x) =2 mω2 x 2 , x > 0⎪⎩0 otherwiseHere we haveSo√p(x) = mω x2 2 − x 2∫ x20p(x)dx = πE2ωIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potentail well with one vertical wall (cont.)For instance, consider the “half-harmonic oscillator”:⎧1 ⎪⎨V (x) =2 mω2 x 2 , x > 0⎪⎩0 otherwiseHere we haveSoComparisson now gives:E n =√p(x) = mω x2 2 − x 2∫ x20p(x)dx = πE2ω(2n − 1 ) ( 3ω =2 2 , 7 2 , 11 )2 , . . . ωIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potentail well with one vertical wall (cont.)For instance, consider the “half-harmonic oscillator”:⎧1 ⎪⎨V (x) =2 mω2 x 2 , x > 0⎪⎩0 otherwiseHere we haveSoComparisson now gives:E n =√p(x) = mω x2 2 − x 2∫ x2Compare this result with an exact one.0p(x)dx = πE2ω(2n − 1 ) ( 3ω =2 2 , 7 2 , 11 )2 , . . . ωIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potential well with no vertical wallsIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potential well with no vertical wallswe have seen the connection formulas for upward potential slopesfor downward slopes (analogous):⎧D ′√ exp|p(x)|⎪⎨ψ(x) ≈⎪⎩[− 1 ∫ x1x]|p(x ′ )|dx ′ , if x < x 1[2D ′ ∫ 1 x√ sin p(x ′ )dx ′ + π ], if x > x 1p(x) x 14Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potential well with no vertical wallswe want the w.f. in the “well”, i.e. where x 1 < x < x 2:ψ(x) ≈2D √p(x)sin θ 2(x) ,ψ(x) ≈ − 2D′ √p(x)sin θ 1(x) ,θ 2(x) = 1 θ 1(x) = − 1 ∫ x2x∫ xp(x ′ )dx ′ + π 4x 1p(x ′ )dx ′ − π 4Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potential well with no vertical wallssin θ 1 = sin θ 2 =⇒ θ 2 = θ 1 + nπ =⇒∫ x2(p(x)dx = n − 1 )π , n = 1, 2, 3, . . .x 12Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureExample: Potential well with no vertical wallssin θ 1 = sin θ 2 =⇒ θ 2 = θ 1 + nπ =⇒∫ x2(p(x)dx = n − 1 )π , n = 1, 2, 3, . . .x 12 0, two vertical walls 1/4, one vertical wallIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureConclusionsWKB advantagesgood for slowly changing w.f.good for short wavelengthsbest in the semi-classicalsystems (large n)one doesn’t even have to solvethe S.E.WKB disadvantagesbad for rapidly changing w.f.bad for long wavelengthsinappropriate for lower states(small n)constraint trade-off (sometimesnot possible)Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureContents1 General remarks2 The “classical” region3 Tunneling4 The connection formulas5 LiteratureIgor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region Tunneling The connection formulas LiteratureLiterature1 R. L. Liboff, Introductory Quantum Mechanics, Addison Wesley, SanFrancisco, 2003.2 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., PearsonEducation, Inc., Upper Saddle River, NJ, 2005.3 I. Supek, Teorijska fizika i struktura materije, II. dio, Školska knjiga,Zagreb, 1989.4 Y. Peleg, R. Pnini, E. Zaarur, Shaum’s Outline of Theory and Problems ofQuantum Mechanics, McGraw-Hill, 1998.Igor LukačevićThe WKB approximationUJJS, Dept. of Physics, Osijek