The WKB approximation - UNIOS

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

A problem
What happens with the w.f. when
E ≈ V?

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

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

A problem
What happens with the w.f. when
E ≈ V?

E ≈ V ⇒ p(x) → 0 ⇒ ψ → ∞ !

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

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

A problem
What happens with the w.f. when
E ≈ V?

E ≈ V ⇒ p(x) → 0 ⇒ ψ → ∞ !

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.

A solution

Construct a “patching”
wavefunction ψp.

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Approximation: we linearize the potentia
V (x) ≈ E

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al
+ V (0)x

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Approximation: we linearize the potentia

V (x) ≈ E

From S.E. we get

d2ψp = zψp , z = αx
dz 2

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al
+ V (0)x

1

2m 3
x , α = V (0)

2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Approximation: we linearize the potentia

V (x) ≈ E

From S.E. we get

d2ψp = zψp , z = αx
dz2

Airy’s equation

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al
+ V (0)x

1

2m 3
x , α = V (0)

2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Approximation: we linearize the potentia

V (x) ≈ E

From S.E. we get

d2ψp = zψp , z = αx
dz2

Airy’s equation

ψp = a Ai(αx)

Airy functio

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al
+ V (0)x

1

2m 3
x , α = V (0)

2

+b Bi(αx)

on Airy function

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

a delicate double constraint has to

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

be satisfied

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

a delicate double constraint has to
we need WKB w.f. and ψp for both

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

be satisfied
h overlap regions (OLR)

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

p(x) = 2m(E −

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

3√
− V ) ≈ α 2 −x

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

p(x) ≈

OLR 2 (x > 0)

x ≈2 3

|p(x )|dx (αx ) 2
03

ψWKB ≈ √ D e− 2 (αx )3/2
3

α3/4x 1/4

ψpz 0≈ √ a e− 2 (αx )3/2
3

2 π(αx )1/4

+√ b e2 (αx )3/2
3
π(αx )1/4

⇒a=D 4π b=0
,
α

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

3√
α 2 −x

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

OLR 2 (x > 0) O
ψ
x ≈2 3

|p(x )|dx (αx ) 2
03

ψWKB ≈ √ D e− 2 (αx )3/2
3

α3/4x 1/4

ψpz 0≈ √ a e− 2 (αx )3/2
3

2 π(αx )1/4

+√ b e2 (αx )3/2
3
π(αx )1/4

⇒ a = D 4π , b = 0
α

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

OLR 1 (x < 0)

0 ≈2 3

p(x )dx (−αx) 2
x3

ψWKB ≈ √ 1 Be i 2 (−αx )3/2
α3/4(−x )1/4 3

+Ce −i 2 (−αx )3/2
3

ψpz 0≈√ a 1 e i π/4 e i 2 (−αx )3/2
3

π(−αx )1/4 2i

−e −i π/4 e −i 2 (−αx )3/2
3

2i √a eiπ/4 = √B
− π α
⇒ √a e−iπ/4 = √C

2i π α

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

The connection formulas
B = −ieiπ/4 · D ,

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

C = ie−iπ/4 · D

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

The connection formulas
B = −ieiπ/4 · D ,

WKB w.f. 2D 1 x2
sin
 p(x )
x

 D exp − 1
 |p(x )|


ψ(x) ≈







Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

C = ie−iπ/4 · D

π , if x < x2
p(x )dx +
4

x

|p(x )|dx , if x > x2

x2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potentail well with one vertica

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al wall

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potentail well with one vertica

Boundary condition: ψ(0) = 0, gives for

x2 n− 1

p(x)dx =
04

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al wall

r ψWKB
π , n = 1, 2, 3, . . .

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potentail well with one vertica

For instance, consider the “half-harmoni

 1 mω2


 2
V (x) =


0

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al wall (cont.)
ic oscillator”:
2x2 , x > 0

otherwise

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potentail well with one vertica

For instance, consider the “half-harmoni

 1 mω2


 2
V (x) =


0

Here we have

p(x) = mω

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al wall (cont.)
ic oscillator”:
2x2 , x > 0

otherwise

ω x22 − x 2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potentail well with one vertica

For instance, consider the “half-harmoni

 1 mω2


 2
V (x) =


0

Here we have

p(x) = mω

So
x2
p(x )d

0

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al wall (cont.)
ic oscillator”:
2x2 , x > 0

otherwise

ω x22 − x 2
πE

dx =
2ω

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potentail well with one vertica

For instance, consider the “half-harmoni

 1 mω2


 2
V (x) =


0

Here we have

p(x) = mω

So x2
Comparisson now gives:
p(x )d

0

En = 2n − 1 ω=
2

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al wall (cont.)
ic oscillator”:
2x2 , x > 0

otherwise

ω x22 − x 2
dx = πE

2ω

= 3 , 7 , 11 , . . . ω
22 2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potentail well with one vertica

For instance, consider the “half-harmoni

 1 mω2


 2
V (x) =


0

Here we have

p(x) = mω

So x2
Comparisson now gives:
p(x )d

0

En = 2n − 1 ω=
2

Compare this result with an exact one.

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al wall (cont.)
ic oscillator”:
2x2 , x > 0

otherwise

ω x22 − x 2
dx = πE

2ω

= 3 , 7 , 11 , . . . ω
22 2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potential well with no vertical

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

l walls

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potential well with no vertical

we have seen the connection formu
for downward slopes (analogous):

 D exp −
|p(x )|

 2D sin 1
 p(x )



ψ(x) ≈







x

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

l walls

ulas for upward potential slopes

−1 x1 , if x < x1

|p(x )|dx

x

xπ if x > x1
p(x )dx + ,

x1 4

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: Potential well with no vertical

we want the w.f. in the “well”, i.e.
ψ(x) ≈ 2D sin θ2(x) ,
p(x )

ψ(x) ≈ − 2D sin θ1(x) ,
p(x )

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

l walls

where x1 < x < x2:

1 x2 π
θ2(x) = p(x )dx +
x4

θ1(x) = − 1 x −π

p(x )dx
x1 4

UJJS, Dept. of Physics, Osijek