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Quantum Mechanics II WKB Alpha-Decay Gamows Simple Model Some Improvements Tristan Hbsch Department of Physics and Astronomy, Howard University, Washington DC http://physics1.howard.edu/~thubsch/ Q WKB M II The Story

  1. Quantum Mechanics II WKB Alpha-Decay 
 Gamow’s Simple Model 
 Some Improvements Tristan Hübsch Department of Physics and Astronomy, Howard University, Washington DC 
 http://physics1.howard.edu/~thubsch/

  2. Q 
 WKB M 
 II The Story So Far… Extra! b Focus&on&1)dimensional&physics: H ψ = E ψ q h 2 ⇥ ⇤ d 2 H = − ¯ 2 M b k ( x ) : = E � W ( x ) d x 2 + W ( x ) h 2 ¯ 2 M The&“standard&wave)functions”&are: q k ( x ) e + i R k ( x ) e � i R ⇥ ⇤ d x k ( x ) + d x k ( x ) A B where%E%>%W(x) WKB ( x ) = ψ p p κ ( x ) e � R κ ( x ) e + R d x κ ( x ) + d x κ ( x ) p C D where%E%<%W(x) WKB ( x ) = ψ p p � ⇤ Matching&conditions: Barrier to Left Barrier to Right The&lower&limit&in&the 
 ( ϑ ⇤ A + ϑ B ) 2 ( ϑ ⇤ A + ϑ B ) = = 1 C C integrals&in&the&exponents 
 2 ( ϑ A + ϑ ⇤ B ) ( ϑ A + ϑ ⇤ B ) = 1 = D D is&the&reference&point&in 
 2 ϑ C + ϑ ⇤ D 2 ϑ ⇤ D = 1 = ϑ C + 1 A A the&matching&condition 
 2 ϑ ⇤ C + ϑ D ϑ ⇤ C + 1 = 1 = B B 2 ϑ D speci?ication 2

  3. q ⇥ ⇤ Q 
 WKB M 
 κ ( x ) e � R κ ( x ) e + R d x κ ( x ) + d x κ ( x ) C D WKB ( x ) = ψ p p II R R d x k ( x ) + k ( x ) e + i d x k ( x ) k ( x ) e � i A B WKB ( x ) = ψ p p Applications Potential&well&w/transition&points& x *& = & a &and& x *& = & b & The&energy)quantization&relation&is:& Z b n ( 2 n + 3 4 ) π n ( 2 n + 1 2 ) π n ( 2 n + 1 ) π q 2 M = h 2 [ E � W ( x )] = 4 ) π = a d x ( 2 n + 7 ( 2 n + 3 2 ) π ( 2 n + 2 ) π ¯ W ( x ) W ( x ) W ( x ) “symmetric” “antisymmetric” E E E a b a b a b n Whenever& W ( x )&crosses& E &discontinuously& 4 ψ ( x ) = 0 = 4 ψ 0 ( x ) p � ⇤ Whenever& W ( x )&crosses& E &continuously Barrier to Left Barrier to Right = ( ϑ ⇤ A + ϑ B ) = 1 2 ( ϑ ⇤ A + ϑ B ) C C WKB connection formulae 2 ( ϑ A + ϑ ⇤ B ) ( ϑ A + ϑ ⇤ B ) = 1 = D D 2 ϑ C + ϑ ⇤ D 2 ϑ ⇤ D = 1 = ϑ C + 1 A A = 1 2 ϑ ⇤ C + ϑ D = ϑ ⇤ C + 1 3 B B 2 ϑ D

  4. Q 
 WKB: α -Decay M 
 II General Physics Facts Extra! One&typical&α)decay&starts&with&Uranium)234&(92& p &&&142& n ) Too&complicated:&234&3)vectors&(702&equations&of&motion) …with&C 2234 &=&27,261&pairwise&potential&terms …representing&the& strong &nuclear&forces&keeping&the&nucleus&stable …where&a&(2 p 2 n )&subset&form&a&subsystem …that&escapes&the&strongly&attractive&potential&of&230&other&nucleons 2 He ++ + A + 4 A + 4 4 A A α Z Y −− , Z + 2 X → or Z + 2 X Z Y. − − → Special&cases:& A – Z &=(# n )=&2,&8,&20,&28,&50,&82,&126&(“magic&numbers”) Lead)208&(# p =82&&&# n =126)&is&doubly&magical p ’s&and& n ’s&separately&form&“closed&shells” Polonium)212&→&α&+&Lead)208;&think&&Po)212&=&[Pb)208+α],&which&decays “Parent&nucleus”&=&[“Daughter&nucleus”+ α ]&→&“Daughter&nucleus”&+& α “Daughter&nucleus”⇒&(classical)&potential&in&which&(quantum)& α &moves 4

  5. Q 
 WKB: α -Decay M 
 II General Physics Facts Extra! The&“daughter&nucleus”&potential: is&attractive&0&≤& r, ≤&(R~1fm)&(strong&interactions),&where& W ( r )&<&0 away&from&the&daughter&nucleus,& r &=& R ∞ , ≫&( R ~1fm)&,& W ( r )&=&2 Ze′, 2 / r &(E&M) in)between,&R&≤&r&≪& R ∞ ,& W ( r )&provides&a&barrier For&min[ W ( r )]&≤& E &≤&0,& α &is&stably&bound For&0&≤& E &≤&max[ W ( r )],& α &is&unstably&bound&and&can&decay&(&&“un)decay”) &&there&exist&two&points&where& E α &=& W ( a )&=& W ( b ) Classically&allowed 
 & 0&≤& r &≤& a &&&&& b &≤& r &≤&∞ Classically&forbidden 
 W ( r ) ~ strong intermediate electrostatic & a &≤& r &≤& b & nuclear region interaction forces For&max[ W ( r )]&≤& E &≤&0 
 barrier a b E α &is&free&to&move 
 R r inside everywhere the well outside 5

  6. Q 
 WKB: α -Decay M 
 II Gamow’s Simple Model Extra! 1928&(only&2&years&after&Schrödinger’s&equation),&G.A.&Gamow: For&0&≤& r &≤& R ,& W ( r )&=&– V 0 &=& const .  � For& R &≤& r &<&∞,& W ( r )&=&2 Ze′, 2 / r & h ¯ r 2 + W ( ~ r ) y ( r , q , f ) = E y ( r , q , f ) , ~ � 2 m a W ( r ) A 0 e i R r B 0 e � i R r b d r k ( r ) + b d r k ( r ) , u III ( r ) = p p k ( r ) k ( r ) I II III u I ( r ) = A sin ( Kr + δ ) , E 0 R b r r C e � R r D R r 2 m α R d r κ ( r ) + R d r κ ( r ) , defined by setting u II ( r ) = e K = h 2 ( E + V 0 ) , p p p p κ ( r ) κ ( r ) ¯ ( 2 e 0 )( Ze 0 ) ! = E b � V 0 s s ⇣ 2 Ze 0 2 E � 2 Ze 0 2 2 m α 2 m α ⌘ ⇣ ⌘ κ ( r ) = � E k ( r ) = , . h 2 h 2 r r ¯ ¯ d 2 u ⇤ � ` ( ` + 1 )  2 m α � 6 ψ ( r , θ , φ ) = u ( r ) r P m ` ( cos θ ) e im φ , ⇥ d r 2 + E � W ( r ) u = 0, h 2 r 2 ¯  6

  7. Q 
 WKB: α -Decay W ( r ) M 
 I II III II E 0 R b r defined by setting Gamow’s Simple Model Extra! ( 2 e 0 )( Ze 0 ) ! = E b � V 0 @ r &=& R ,&I&↔&II: r ! R � u 0 r ! R + u 0 r ! R � u I ( r ) = r ! R + u II ( r ) , I ( r ) = II ( r ) . lim lim lim lim A sin ( KR ) = C + D AK cos ( KR ) = ( � C + D ) p κ R , , p κ R solve&for&C,&D: A ⇥ ⇤ C = κ R sin ( KR ) � K cos ( KR ) , 2 p κ R A ⇥ ⇤ D = κ R sin ( KR ) + K cos ( KR ) . 2 p κ R @& r &=& b ,&II&↔&III&(for& α )decay,&B′&=&0): C = ϑ ⇤ e σ A 0 , Z b Z b r r 2 Ze 0 2 2 m α σ = R d r κ R = R d r � E . 2 ϑ e � σ A 0 , D = 1 h 2 r ¯ ϑ = e i π /4 , 7

  8. Q 
 WKB: α -Decay M 
 II Gamow’s Simple Model Extra! So,&we&have: A C = ϑ ⇤ e σ A 0 , ⇥ ⇤ C = κ R sin ( KR ) � K cos ( KR ) , ⇥ ⇤ 2 p κ R I↔II II↔III A A ⇥ ⇤ D = 1 2 ϑ e � σ A 0 , ⇥ ⇤ D = κ R sin ( KR ) + K cos ( KR ) , 2 p κ R Dividing&one&by&the&other: ϑ 2 e � 2 σ = 2 κ R sin ( KR ) + K cos ( KR ) imaginary real that ϑ 2 = i , κ R sin ( KR ) � K cos ( KR ) …which&is&horribly&wrong! What&have&we&done&??? Math:&imposed&boundary&conditions&left&and&right&( u I (0)&=&0&&& B ′&=&0) Physics:&assumed&only& α )decay,&no& α )capture&(un)decay) 8

  9. Q 
 II WKB: α -Decay M 
 Gamow’s Simple Model Extra! Re)do,&not&assuming& B ′&=&0: ⇤ + ϑ ⇤ e σ A A 0 = ϑ e � σ A ⇥ ⇥ ⇤ κ R sin ( KR ) � K cos ( KR ) κ R sin ( KR ) + K cos ( KR ) , 4 p κ R 4 p κ R = ϑ e σ A cos ( KR ) h ⇤i κ R tan ( KR ) � K � 2 ie 2 σ ⇥ κ R tan ( KR ) + K h ⇤i , 4 p κ R ⇥ ⇤ e � A e A ⇤ + ϑ e σ A B 0 = ϑ ⇤ e � σ A ⇥ ⇥ ⇤ κ R sin ( KR ) � K cos ( KR ) κ R sin ( KR ) + K cos ( KR ) , 4 p κ R 4 p κ R = ϑ ⇤ e σ A cos ( KR ) h ⇤i κ R tan ( KR ) � K + 2 ie 2 σ ⇥ κ R tan ( KR ) + K . 4 p κ R This&implies&that&| A ′| 2 &=&| B ′| 2 &! For& B ′&=&0,&the&real&and&imaginary&parts&must&vanish&separately: setting K = 0 , i.e. , E = � V 0 . κ R tan ( KR ) � K = 0 κ R tan ( KR ) + K = 0, and 9

  10. Q 
 II WKB: α -Decay M 
 Gamow’s Simple Model Extra! Consider&the&amplitude&of&the& α )decay&component: A 0 = ϑ e σ A cos ( KR ) h ⇤i κ R tan ( KR ) � K � 2 ie 2 σ ⇥ κ R tan ( KR ) + K , 4 p κ R ⇤ e � σ A ⇥ e σ A ⇥ This&stems&from&the&wave)function&within&the&barrier ⇤ u II ( r ) = C e � σ b d r κ ( r ) + D e σ e � R r R r b d r κ ( r ) . e p p κ ( r ) κ ( r ) …where&the& C )term&is&exponentially&suppressed&by& e –σ . This &is&what&allows&the& approximation ,&which&produces A 0 ⇡ ϑ e σ AK cos ( KR ) . 2 p κ R and&is&the&oft)quoted&result&[Gamow,&1928] …and&which&turns&out&to&agree&with&experiments& very2well ! 10

  11. Q 
 II WKB: α -Decay M 
 Gamow’s Simple Model Extra! | | With&this& approximation , hK 2 | A 0 | e � 2 σ cos 2 ( KR ) h | A 0 | λ = 4 π ¯ ⇡ 4 π ¯ , m α m α κ R …and Z b r r 2 Ze 0 2 2 m α σ = R d r � E , h 2 r ¯ s Z b r 2 m α E b = R d r r � 1, h 2 ¯ s r r r  � 2 m α E R R 1 � R p ⇡ � � ⇡ = b arccos , b � h 2 b b ¯ r p m α E R 5 1 p p 8 m α E b R + 1 2 m α E R 3 / b + + . . . h σ ⇡ π 2 m α E b � p ¯ 2 3 b 3 10 2 R / b &≪&1 r r r 11

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