The WKB approximation - UNIOS

Contents General remarks The “classical” region

The WKB ap

Quantum mechan

Igor Lu

UJJS, Dept. of

29. listopa

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

pproximation

nics 2 - Lecture 4

ukaˇcevi´c

Physics, Osijek

ada 2012.

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

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

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

WKB = Wentzel, Kramers, Brillou
in Holland it’s KWB
in France it’s BKW
in England it’s JWKB (for Jeffreys)

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

uin
)

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

WKB = Wentzel, Kramers, Brillou
in Holland it’s KWB
in France it’s BKW
in England it’s JWKB (for Jeffreys)

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

uin
)

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

WKB = Wentzel, Kramers, Brillou
in Holland it’s KWB
in France it’s BKW
in England it’s JWKB (for Jeffreys)

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

uin
)

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

WKB = Wentzel, Kramers, Brillou
in Holland it’s KWB
in France it’s BKW
in England it’s JWKB (for Jeffreys)

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

uin
)

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Basic idea: k=
1 particle E
potential V (x) constant

if E > V ⇒ ψ(x) = Ae±ikx ,

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

2m(E − V )
=

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Basic idea: k=
1 particle E
potential V (x) constant

if E > V ⇒ ψ(x) = Ae±ikx ,

A question
What’s the character of A and λ = 2π/

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

2m(E − V )
=
/k here?

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Basic idea:
1 particle E
potential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =

2 suppose V (x) not constant, but va

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

2m(E − V )
=
aries slowly wrt λ

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Basic idea:
1 particle E
potential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
2 suppose V (x) not constant, but va

A question
What can we say about ψ, A and λ now

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

2m(E − V )
=
aries slowly wrt λ
w?

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Basic idea:
1 particle E
potential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
2 suppose V (x) not constant, but va

A question
What can we say about ψ, A and λ now

We still have oscillating ψ, but with slow

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

2m(E − V )
=
aries slowly wrt λ
w?

wly changable A and λ.

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Basic idea:
1 particle E
potential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =

2 suppose V (x) not constant, but va
3 if E < V , the reasoning is analogou

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

2m(E − V )
=
aries slowly wrt λ

us

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Basic idea:
1 particle E
potential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =

2 suppose V (x) not constant, but va
3 if E < V , the reasoning is analogou

A question
What if E ≈ V ?

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

2m(E − V )
=
aries slowly wrt λ

us

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Basic idea:
1 particle E
potential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =

2 suppose V (x) not constant, but va
3 if E < V , the reasoning is analogou

A question
What if E ≈ V ? Turning points

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

2m(E − V )
=
aries slowly wrt λ

us

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

S.E.

2

− ∆ψ + V (x)ψ = E ψ ⇐⇒ ∆ψ
2m

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ψ = − p2 ψ , p(x) = 2m [E − V (x)]
2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

S.E.

2

− ∆ψ + V (x)ψ = E ψ ⇐⇒ ∆ψ
2m

“Classical” region
E > V (x) , p real

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ψ = − p2 ψ , p(x ) = 2m [E − V (x)]

2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

S.E.

2

− ∆ψ + V (x)ψ = E ψ ⇐⇒ ∆ψ
2m

“Classical” region
E > V (x) , p real
ψ(x ) = A(x )eiφ(x)

A(x) and φ(x) real

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ψ = − p2 ψ , p(x ) = 2m [E − V (x)]

2

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Putting ψ(x) into S.E. gives two equatio
A =A

A2φ = 0

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ons:

A (φ )2 − p2 (1)
(2)
2

0

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Putting ψ(x) into S.E. gives two equatio
A =A

A2φ = 0

Solve (2)

A= √C , C ∈R
φ

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ons:

A (φ )2 − p2 (1)
(2)
2

0

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Putting ψ(x) into S.E. gives two equatio
A =A

A2φ = 0

Solve (2) Solv
Assu
A= √C , C ∈R ⇒A
φ

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ons:

A (φ )2 − p2 (1)
(2)
2

0

ve (1)
umption: A varies slowly
A ≈0

φ(x) = ± 1 p(x)dx

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Solve (2) Solv
Assu
A= √C , C ∈R ⇒A
φ

Resulting wavefunction
ψ(x) ≈ C
p(x

Note: general solution is a linear combin

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ve (1)
umption: A varies slowly
A ≈0

φ(x) = ± 1 p(x)dx

e ± i p(x)dx
x)
nation of these.

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Solve (1) Solv
Assu
A= √C , C ∈R ⇒A
φ

Resulting wavefunction
ψ(x) ≈ C
p(x

Note: general solution is a linear combin

Probability of finding a particle at x
|ψ(x )|2

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

ve (2)
umption: A varies slowly
A ≈0

φ(x) = ± 1 p(x)dx

e ± i p(x)dx
x)
nation of these.

≈ |C |2
p(x )

UJJS, Dept. of Physics, Osijek

Contents General remarks The “classical” region

Example: potential well with two vertica

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

Igor Lukaˇcevi´c
The WKB approximation

Tunneling The connection formulas Literature

al walls

UJJS, Dept. of Physics, Osijek