WKB Alpha-Decay Gamows Simple Model Some Improvements Tristan - - PowerPoint PPT Presentation

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WKB Alpha-Decay Gamows Simple Model Some Improvements Tristan - - PowerPoint PPT Presentation

Nov 23, 2023 373 likes •532 views

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

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SLIDE 1

Tristan Hübsch

Department of Physics and Astronomy, Howard University, Washington DC
 http://physics1.howard.edu/~thubsch/

Quantum Mechanics II

WKB

Alpha-Decay
 Gamow’s Simple Model
 Some Improvements

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SLIDE 2

Q
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Extra! The Story So Far…

WKB

Focus&on&1)dimensional&physics: The&“standard&wave)functions”&are: Matching&conditions:

The&lower&limit&in&the
 integrals&in&the&exponents
 is&the&reference&point&in
 the&matching&condition
 speci?ication

2

b Hψ = Eψ b H = − ¯ h2 2M d2 dx2 + W(x) k(x) := q

2M ¯ h2

⇥ E W(x) ⇤ ψ

WKB(x) =

A

p

k(x) e+iR dx k(x) + B

p

k(x) eiR dx k(x)

q ⇥ ⇤ ψ

WKB(x) =

C

p

κ(x) eR dx κ(x) + D

p

κ(x) e+R dx κ(x)

where%E%>%W(x) where%E%<%W(x)

p

⇤

Barrier to Left Barrier to Right

C

= (ϑ⇤A + ϑB)

C

= 1

2(ϑ⇤A + ϑB)

D

= 1

2(ϑA + ϑ⇤B)

D

= (ϑA + ϑ⇤B)

A

= 1

2ϑC + ϑ⇤D

A

=

ϑC + 1

2ϑ⇤D

B

= 1

2ϑ⇤C + ϑD

B

=

ϑ⇤C + 1

2ϑD
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SLIDE 3

Q
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 II

=

n(2n + 3

4)π

(2n + 7

4)π =

n(2n + 1)π

(2n + 2)π

E a b W(x) E a b W(x) E a b W(x)

Z b

a dx

q

2M ¯ h2 [EW(x)] =

n(2n + 1

2)π

(2n + 3

2)π

Applications

WKB

Potential&well&w/transition&points&x*&=&a&and&x*&=&b& The&energy)quantization&relation&is:& Whenever&W(x)&crosses&E&discontinuously& Whenever&W(x)&crosses&E&continuously

3

“symmetric” “antisymmetric”

n

4ψ(x) = 0 = 4ψ0(x)

WKB connection formulae

p ⇤ Barrier to Left Barrier to Right C = (ϑ⇤A + ϑB) C = 1 2(ϑ⇤A + ϑB) D = 1 2(ϑA + ϑ⇤B) D = (ϑA + ϑ⇤B) A = 1 2ϑC + ϑ⇤D A = ϑC + 1 2ϑ⇤D B = 1 2ϑ⇤C + ϑD B = ϑ⇤C + 1 2ϑD

ψ

WKB(x) = A

p

k(x) e+i R dx k(x) + B

p

k(x) ei R dx k(x)

q ⇥ ⇤ ψ

WKB(x) = C

p

κ(x) eR dx κ(x) + D

p

κ(x) e+R dx κ(x)
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SLIDE 4

Q
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General Physics Facts

WKB: α-Decay

One&typical&α)decay&starts&with&Uranium)234&(92&p&&&142&n)

Too&complicated:&234&3)vectors&(702&equations&of&motion) …with&C2234&=&27,261&pairwise&potential&terms …representing&the&strong&nuclear&forces&keeping&the&nucleus&stable …where&a&(2p2n)&subset&form&a&subsystem …that&escapes&the&strongly&attractive&potential&of&230&other&nucleons 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

Extra!

A+4 Z+2X → 4 2He++ + A ZY−−,
  • r
A+4 Z+2X α

− − →

A ZY.
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SLIDE 5

Q
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General Physics Facts

WKB: α-Decay

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)&=&2Ze′,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&Eα&=&W(a)&=&W(b) Classically&allowed
 & 0&≤&r&≤&a&&&&&b&≤&r&≤&∞ Classically&forbidden
 & a&≤&r&≤&b&

For&max[W(r)]&≤&E&≤&0
 α&is&free&to&move
 everywhere

5

Extra!

W( ~ r) strong nuclear forces inside the well a R intermediate region barrier b electrostatic interaction E r
  • utside
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SLIDE 6

Q
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Gamow’s Simple Model

WKB: α-Decay

1928&(only&2&years&after&Schrödinger’s&equation),&G.A.&Gamow:

For&0&≤&r&≤&R,&W(r)&=&–V0&=&const. For&R&≤&r&<&∞,&W(r)&=&2Ze′,2/r&

6

Extra!

W(r)

V0

I R II b defined by setting

(2e0)(Ze0) b !

= E

E r III

  • ¯

h 2ma

~ r2 + W( ~

r)

  • y(r, q, f) = Ey(r, q, f),
6 ψ(r, θ, φ) = u(r) r Pm ` (cos θ)eimφ,

 d2u dr2 + 2mα ¯ h2 ⇥ E W(r) ⇤ `(` + 1) r2

  • u = 0,
uI(r) = A sin(Kr + δ), K = r 2mα ¯ h2 (E + V0), uII(r) = C p κ(r) e R r R dr κ(r) + D p κ(r) e R r R dr κ(r), uIII(r) = A0 p k(r) ei R r b dr k(r) + B0 p k(r) ei R r b dr k(r), p κ(r) = s 2mα ¯ h2 ⇣2Ze02 r E ⌘ , p k(r) = s 2mα ¯ h2 ⇣ E 2Ze02 r ⌘ .
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SLIDE 7

Q
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Gamow’s Simple Model

WKB: α-Decay

@r&=&R,&I&↔&II: solve&for&C,&D: @&r&=&b,&II&↔&III&(for&α)decay,&B′&=&0):

7

Extra!

W(r) V0 I R II b defined by setting (2e0)(Ze0) b ! = E E r III

lim

r!R uI(r) =

lim

r!R+ uII(r),

lim

r!R u0 I(r) =

lim

r!R+ u0 II(r).

A sin(KR) = C + D

pκR

, AK cos(KR) = (C + D)pκR, C = A 2pκR ⇥ κR sin(KR) K cos(KR) ⇤ , D = A 2pκR ⇥ κR sin(KR) + K cos(KR) ⇤ . C = ϑ⇤eσA0, D = 1

2ϑeσA0,

ϑ = eiπ/4, σ =

Z b

R dr κR =

r 2mα ¯ h2

Z b

R dr

r 2Ze02 r

E.

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SLIDE 8

Q
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 II real imaginary

WKB: α-Decay

So,&we&have: Dividing&one&by&the&other: …which&is&horribly&wrong! What&have&we&done&???

Math:&imposed&boundary&conditions&left&and&right&(uI(0)&=&0&&&B′&=&0) Physics:&assumed&only&α)decay,&no&α)capture&(un)decay)

8

Extra!

C = A 2pκR ⇥ κR sin(KR) K cos(KR) ⇤ , A ⇥ ⇤ C = ϑ⇤eσA0, ⇥ ⇤ D = A 2pκR ⇥ κR sin(KR) + K cos(KR) ⇤ , D = 1

2ϑeσA0,

ϑ2e2σ = 2κR sin(KR) + K cos(KR) κR sin(KR) K cos(KR)

Gamow’s Simple Model

that ϑ2 = i,

I↔II II↔III

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SLIDE 9

Q
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 II WKB: α-Decay Re)do,&not&assuming&B′&=&0: This&implies&that&|A′|2&=&|B′|2&! For&B′&=&0,&the&real&and&imaginary&parts&must&vanish&separately:

9

Extra! Gamow’s Simple Model

A0 = ϑeσA 4pκR ⇥ κR sin(KR) K cos(KR) ⇤ + ϑ⇤eσA 4pκR ⇥ κR sin(KR) + K cos(KR) ⇤ ,

= ϑeσA cos(KR)

4pκR h κR tan(KR) K 2ie2σ⇥ κR tan(KR) + K ⇤i ,

⇤e A

e A h ⇥ ⇤i B0 = ϑ⇤eσA 4pκR ⇥ κR sin(KR) K cos(KR) ⇤ + ϑeσA 4pκR ⇥ κR sin(KR) + K cos(KR) ⇤ ,

= ϑ⇤eσA cos(KR)

4pκR h κR tan(KR) K + 2ie2σ⇥ κR tan(KR) + K ⇤i .

κR tan(KR) K = 0 and κR tan(KR) + K = 0, setting K = 0, i.e., E = V0.

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SLIDE 10

Q
 M
 II WKB: α-Decay Consider&the&amplitude&of&the&α)decay&component: This&stems&from&the&wave)function&within&the&barrier …where&the&C)term&is&exponentially&suppressed&by&e–σ. This&is&what&allows&the&approximation,&which&produces and&is&the&oft)quoted&result&[Gamow,&1928] …and&which&turns&out&to&agree&with&experiments&very2well!

10

Extra! Gamow’s Simple Model

A0 = ϑeσA cos(KR) 4pκR h κR tan(KR) K 2ie2σ⇥ κR tan(KR) + K ⇤i ,

⇤eσA⇥

⇤ eσA ⇥ uII(r) = C eσ p κ(r) e R r

b dr κ(r) + D eσ

p κ(r) e

R r

b dr κ(r).

A0 ⇡ ϑeσAK cos(KR) 2pκR .

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SLIDE 11

Q
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 II WKB: α-Decay With&this&approximation, …and

11

Extra! Gamow’s Simple Model

| |

λ = 4π¯ h|A0| mα

⇡ 4π¯

hK2|A0|e2σ cos2(KR) mακR , σ = r 2mα ¯ h2

Z b

R dr

r 2Ze02 r

E, =

s 2mαE ¯ h2

Z b

R dr

r b r 1,

=

s 2mαE ¯ h2 b  arccos r R b r R b r 1 R b

  • ,

⇡

  • p
  • ⇡

¯ h σ ⇡ π

2

p

2mαE b p 8mαE b R + 1

3

p 2mαE R3/b + 1 10

p

2 r mαE R5 b3

+ . . .

r r r

R/b&≪&1

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SLIDE 12

Q
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Gamow’s Model—Exactly

WKB: α-Decay

Return&to&the&radial&equation: The&Bessel&equation.&Use&jℓ(Kr)—not&nℓ(Kr)—&for&0&≤&r&≤&R. For&r&>&R,&substitute Set&β2&=&–2mα,E/ħ,&to&obtain The&con?luent&hypergeometric&equation. In&fact,&the&Bessel&equation&is&a(nother)&special&case&of&the&CHEq.

12

Extra!

d2R dr2 + 2 r dR dr +  K2 `(` + 1) r2

  • R = 0,

r R R(r) = r`eβr f (r). ⇥

r f 00 + ⇥ 2(`+1) 2βr ⇤ f 0 ⇥(2β(`+1) + 4Ze02mα/¯ h2) + (β2 + 2mαE/¯ h2)r ⇤ f = 0.

  • z f 00(z) +

⇥ 2(`+1) z ⇤ f 0(z) ⇥(`+1) + 2Ze02mα/β¯ h2) ⇤ f (z) = 0. Rout(r) = r`eβrh B 1F1 `+1+w

2`+2 ; 2βr

+ C r2`1 1F1 w`

2` ; 2βr

i .

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SLIDE 13

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  • utgoing

incoming

WKB: α-Decay

Using&the&asymptotic&behavior&of&the&con?luent&hypergeometric: For&a&purely&outgoing&wave,&we&need&F&≈&0,&so&that Note&that&w&=&2Ze′2mα/βħ2&is&imaginary

Use:&x,ik&=&e,ik,ln(x)&=&cos[k,ln(x)]&+&i&sin[k,ln(x)]… Use&∆R(r)&=&0&=&∆R′(r)&&matching&conditions&@&r&=&R Setting&F&≈&0&leaves&only&2&constants&+&E,&w/2+1&constraints

13

Extra! Gamow’s Model—Exactly

Rout(r) ⇠ Fr(w+1) eβr + Grw1 e+βr

F = B ⇣eiπ 2β ⌘w+`+1 Γ(2`+2) Γ(`+1+w) + C ⇣eiπ 2β ⌘w` Γ(`+1+w) sin[π(`+w)] Γ(2`+1) sin[2`π] , G = B(2β)w`1 Γ(2`+2) Γ(`+1+w) + C(2β)w` Γ(`+1w) sin[π(`w)] Γ(2`+1) sin[2`π] .

so β2 = 2mαE/¯ h < 0; incoming wave. To determine

⇡

C ⇡ B (2`+1)

(2β)2`+1

Γ2(2`+1) sin[2`π] Γ2(`+1+w) sin[π(`+w)],

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SLIDE 14

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Improvements

Modify&the&model:

They&induce&higher&order&corrections,
 improving&the&precision&but&not&the


  • verall&behavior
15

WKB: α-Decay

W(r) V0 I R II b defined by setting (2e0)(Ze0) b ! = E E r III V(r) V0 $ a R b E r V(r) a R b E r σ = r 2mα ¯ h2  Z R a dr q 1 2mαωr2 E + Z b R dr r 2Ze02 r E
  • ,
= r 2mα ¯ h2 " R 2 q 1 2mαω2R2 E a 2 q 1 2mαω2a2 E + E ωp2mα ln ✓ amαω2 + ωp2mα q 1 2mαω2a2 E Rmαω2 + ωp2mα q 1 2mαω2R2 E ◆# + s 2mαE ¯ h2 b ✓ arccos r R b r R b r 1 R b ◆ ,
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SLIDE 15

Tristan Hübsch

Department of Physics and Astronomy, Howard University, Washington DC
 http://physics1.howard.edu/~thubsch/

Quantum Mechanics II

Now, go forth and

calculate!!