The WKB approximation - UNIOS

Contents General remarks The “classical” region

Example: potential well with two vertica

V (x) = some function , 0 < x < a
∞ , otherwise

Again, assume E > V (x) =⇒

ψ(x) ≈ 1 C+eiφ(x) + C−e−iφ(x)
p(x )

where 1
φ(x) =

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al walls

1
= [C1 sin φ(x) + C2 cos φ(x)]

p(x )

x

p(x )dx

0

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example (cont.)

Boundary conditions: ψ(0) = 0, ψ(a) =

a

p(x )d

0

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒
dx = nπ

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example (cont.)
Boundary conditions: ψ(0) = 0, ψ(a) =

a

p(x )d

0

Take, for example, V (x) = 0 ⇒
n

En =

We got an exact result...is this strange?

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒
dx = nπ

n2π2 2
2ma2
?

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example (cont.)
Boundary conditions: ψ(0) = 0, ψ(a) =

a

p(x )d

0

Take, for example, V (x) = 0 ⇒
n

En =

We got an exact result...is this strange?

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒
dx = nπ

n2π2 2
2ma2
? No, since A = 2/a = const.

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Contents

1 General remarks
2 The “classical” region
3 Tunneling
4 The connection formulas
5 Literature

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Now, assume E < V :
ψ(x) ≈ C
|p(x

where p(x) is imaginary.

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

e ± 1 |p(x)|dx
x )|

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Now, assume E < V :

ψ(x) ≈ C
|p(x

where p(x) is imaginary.

Consider the potential:

V (x) = some function , 0 < x < a
0 , otherwise

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

e ± 1 |p(x)|dx
x )|

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

x <0
ψ(x ) = Aeikx + Be−ikx

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

x <0
ψ(x ) = Aeikx + Be−ikx

|F |2
Transmission probability: T =

|A|2

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

x >a
ψ(x ) = Feikx

2
2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

x <0 0≤x ≤a

ψ(x ) = Aeikx + Be−ikx ψ(x) ≈ √
+√D e

|p(x )|

|F |2
Transmission probability: T =

|A|2

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

a x>a

√C e1 x |p(x )|dx ψ(x ) = Feikx
0

|p(x )|

e− 1 x |p(x )|dx
0

2
2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

x <0 0≤x ≤a

ψ(x ) = Aeikx + Be−ikx ψ(x) ≈ √
+√D e

|p(x )|

Transmission probability:
|F |2

T = |A|2

High, broad barrier 1st term
goes to 0
Why?

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

a x>a

√C e1 x |p(x )|dx ψ(x ) = Feikx
0

|p(x)|

e− 1 x |p(x )|dx
0

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

x <0 0≤x ≤a

ψ(x ) = Aeikx + Be−ikx ψ(x) ≈ √
+√D e

|p(x )|

Transmission probability:

T = |F |2 ∼ e− 2 a |p(x )|dx
0

|A|2

High, broad barrier 1st term
goes to 0
Why?

T ≈ e−2γ , γ =

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

a x>a

√C e1 x |p(x )|dx ψ(x ) = Feikx
0

|p(x )|

e− 1 x |p(x )|dx
0

1 a
=
|p(x )|dx

0

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Gamow’s theory of alpha deca

first time that quantum
mechanics had been
applied to nuclear
physics

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ay

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Gamow’s theory of alpha deca

first time that quantum
mechanics had been
applied to nuclear
physics
turning points:

1 r1 −→ nucleus radius
(6.63 fm for U238)

2 r2 −→
1 2Ze2
=E

4π 0 r2

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ay (cont.)

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Gamow’s theory of alpha deca

1 r2 1 2Ze2 − E
γ = 2m
r1 4π 0 r2

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ay (cont.) r2 r2 − 1dr
√ r1 r

E dr = 2mE

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Gamow’s theory of alpha deca

1 r2 1 2Ze2 − E
γ = 2m
r1 4π 0 r2

Substituting r = r2 sin2 u gives

√2mE π − sin−
r2 2
γ=

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ay (cont.) r2 r2 − 1dr
√ r1 r
2mE

E dr =

−1 r1 − r1(r2 − r1)
r2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Gamow’s theory of alpha deca

1 r2 1 2Ze2 − E
γ = 2m
r1 4π 0 r2

Substituting r = r2 sin2 u gives

√ 2mE π−
2
γ= r2 sin−1

−r−1 −→r2 √

π r2 −2√r1 r2
2

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ay (cont.) r2 r2 − 1dr
√ r1 r

E dr = 2mE

r1 − r1(r2 − r1)
r2
√ −r−1 −→r2 √r1 r2
√ r1 r2 −r12

r1 /r2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Gamow’s theory of alpha deca

Substituting r = r2 sin2 u gives

√ 2mE π−
2
γ= r2 sin−1

−r−1 −→r2 √

π r2 −2√r1 r2
2

√ π r2 − √
γ ≈ 2mE 2 2 r1

where

K1 = e2 √
K2 = 4π 0 π

e2 1/2
4π 0

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ay (cont.)

r1 − r1(r2 − r1)
r2
√ −r−1 −→r2 √
√ r1 r2 −r12 r1 r2

r1 /r2

1 r2 = K1 √Z √
E − K2 Zr1

√
2m = 1.980 MeV1/2
√
4 m = 1.485 fm−1/2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Gamow’s theory of alpha deca

v average velocity
2r1/v average time
between “collisions”
with the nucleus
potential “wall”
v /2r1 average
frequancy of “collisions”
e−2γ “escape”
probability
(v /2r1)e−2γ
“escape” probability per
unit time

Lifetime:
τ = 2r1 e2γ
v

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ay (cont.)

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Gamow’s theory of alpha deca

v average velocity
2r1/v average time
between “collisions”
with the nucleus
potential “wall”
v /2r1 average
frequancy of “collisions”
e−2γ “escape”
probability
(v /2r1)e−2γ
“escape” probability per
unit time

Lifetime:
τ = 2r1 e2γ ⇒ ln τ ∼ √1
vE

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ay (cont.)

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

HW
Solve Problem 8.3 from Ref. [2].

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Contents

1 General remarks
2 The “classical” region
3 Tunneling
4 The connection formulas
5 Literature

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Let us repeat:

 √1 i 0 p(x )dx
x
 Be
ψ(x) ≈ p(x )

 √1 De− 1 x |p(x )|dx
0

p(x )

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

+ Ce− i 0 p(x )dx , if x < 0
x

x , if x > 0

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Let us repeat:

 √1 i 0 p(x )dx
x
 Be
ψ(x) ≈ p(x )

 √1 De− 1 x |p(x )|dx
0

p(x )

Our mission: join these two solutions at

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

+ Ce− i 0 p(x )dx , if x < 0
x

x , if x > 0

t the boundary.

UJJS, Dept. of Physics, Osijek