Download The WKB approximation - Quantum mechanics 2

Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
The WKB approximation
Quantum mechanics 2 - Lecture 4
Igor Lukačević
UJJS, Dept. of Physics, Osijek
29. listopada 2012.
Igor Lukačević
The WKB approximation
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 Lukačević
The WKB approximation
Tunneling
The connection formulas
Literature
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Contents
1
General remarks
2
The “classical” region
3
Tunneling
4
The connection formulas
5
Literature
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
WKB = Wentzel, Kramers, Brillouin
in Holland it’s KWB
in France it’s BKW
in England it’s JWKB (for Jeffreys)
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
WKB = Wentzel, Kramers, Brillouin
in Holland it’s KWB
in France it’s BKW
in England it’s JWKB (for Jeffreys)
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
WKB = Wentzel, Kramers, Brillouin
in Holland it’s KWB
in France it’s BKW
in England it’s JWKB (for Jeffreys)
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
WKB = Wentzel, Kramers, Brillouin
in Holland it’s KWB
in France it’s BKW
in England it’s JWKB (for Jeffreys)
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Basic idea:
1
particle
E
potential V (x) constant
if E > V ⇒ ψ(x) = Ae
Igor Lukačević
The WKB approximation
±ikx
p
,
k=
2m(E − V )
~
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Basic idea:
1
particle
E
potential V (x) constant
if E > V ⇒ ψ(x) = Ae ±ikx ,
p
k=
2m(E − V )
~
A question
What’s the character of A and λ = 2π/k here?
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Basic idea:
1
particle
E
potential V (x) constant
2
p
2m(E − V )
~
suppose V (x) not constant, but varies slowly wrt λ
if E > V ⇒ ψ(x) = Ae ±ikx ,
Igor Lukačević
The WKB approximation
k=
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Basic idea:
1
particle
E
potential V (x) constant
2
p
2m(E − V )
~
suppose V (x) not constant, but varies slowly wrt λ
if E > V ⇒ ψ(x) = Ae ±ikx ,
k=
A question
What can we say about ψ, A and λ now?
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Basic idea:
1
particle
E
potential V (x) constant
2
p
2m(E − V )
~
suppose V (x) not constant, but varies slowly wrt λ
if E > V ⇒ ψ(x) = Ae ±ikx ,
k=
A question
What can we say about ψ, A and λ now?
We still have oscillating ψ, but with slowly changable A and λ.
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Basic idea:
1
particle
E
potential V (x) constant
p
2
2m(E − V )
~
suppose V (x) not constant, but varies slowly wrt λ
3
if E < V , the reasoning is analogous
if E > V ⇒ ψ(x) = Ae
Igor Lukačević
The WKB approximation
±ikx
,
k=
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Basic idea:
1
particle
E
potential V (x) constant
p
2
2m(E − V )
~
suppose V (x) not constant, but varies slowly wrt λ
3
if E < V , the reasoning is analogous
if E > V ⇒ ψ(x) = Ae
±ikx
,
k=
A question
What if E ≈ V ?
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Basic idea:
1
particle
E
potential V (x) constant
p
2
2m(E − V )
~
suppose V (x) not constant, but varies slowly wrt λ
3
if E < V , the reasoning is analogous
if E > V ⇒ ψ(x) = Ae
±ikx
,
k=
A question
What if E ≈ V ? Turning points
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Contents
1
General remarks
2
The “classical” region
3
Tunneling
4
The connection formulas
5
Literature
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
S.E.
−
p
~2
p2
∆ψ + V (x)ψ = E ψ ⇐⇒ ∆ψ = − 2 ψ , p(x) = 2m [E − V (x)]
2m
~
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
S.E.
−
p
~2
p2
∆ψ + V (x)ψ = E ψ ⇐⇒ ∆ψ = − 2 ψ , p(x) = 2m [E − V (x)]
2m
~
“Classical” region
99K E > V (x) , p real
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
S.E.
−
p
~2
p2
∆ψ + V (x)ψ = E ψ ⇐⇒ ∆ψ = − 2 ψ , p(x) = 2m [E − V (x)]
2m
~
“Classical” region
99K E > V (x) , p real
99K ψ(x) = A(x)e iφ(x)
A(x) and φ(x) real
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
Putting ψ(x) into S.E. gives two equations:
p2
A00 = A (φ0 )2 − 2
~
0
2 0
A φ
= 0
Igor Lukačević
The WKB approximation
The connection formulas
Literature
(1)
(2)
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
Putting ψ(x) into S.E. gives two equations:
p2
A00 = A (φ0 )2 − 2
~
0
A2 φ0
= 0
The connection formulas
Literature
(1)
(2)
Solve (2)
C
A= √ 0, C ∈R
φ
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Putting ψ(x) into S.E. gives two equations:
p2
A00 = A (φ0 )2 − 2
~
0
2 0
A φ
= 0
Solve (2)
C
A= √ 0, C ∈R
φ
The WKB approximation
(1)
(2)
Solve (1)
Assumption: A varies slowly
⇒ A00 ≈ 0
φ(x) = ±
Igor Lukačević
Literature
1
~
Z
p(x)dx
UJJS, Dept. of Physics, Osijek
Contents
General remarks
Solve (2)
C
A= √ 0, C ∈R
φ
The “classical” region
Tunneling
The connection formulas
Literature
Solve (1)
Assumption: A varies slowly
⇒ A00 ≈ 0
φ(x) = ±
1
~
Z
p(x)dx
Resulting wavefunction
i R
C
ψ(x) ≈ p
e ± ~ p(x)dx
p(x)
Note: general solution is a linear combination of these.
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Solve (2)
Solve (1)
Assumption: A varies slowly
C
A= √ 0, C ∈R
φ
⇒ A00 ≈ 0
φ(x) = ±
1
~
Z
p(x)dx
Resulting wavefunction
i R
C
ψ(x) ≈ p
e ± ~ p(x)dx
p(x)
Note: general solution is a linear combination of these.
Probability of finding a particle at x
|ψ(x)|2 ≈
Igor Lukačević
The WKB approximation
|C |2
p(x)
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: potential well with two vertical walls
V (x) =
Igor Lukačević
The WKB approximation
some function , 0 < x < a
∞,
otherwise
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: potential well with two vertical walls
V (x) =
some function , 0 < x < a
∞,
otherwise
Again, assume E > V (x) =⇒
h
i
1
1
ψ(x) ≈ p
C+ e iφ(x) + C− e −iφ(x) = p
[C1 sin φ(x) + C2 cos φ(x)]
p(x)
p(x)
where
φ(x) =
Igor Lukačević
The WKB approximation
1
~
Z
x
p(x 0 )dx 0
0
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example (cont.)
Boundary conditions: ψ(0) = 0, ψ(a) = 0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒
Z a
p(x)dx = nπ~
0
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example (cont.)
Boundary conditions: ψ(0) = 0, ψ(a) = 0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒
Z a
p(x)dx = nπ~
0
Take, for example, V (x) = 0 ⇒
En =
n 2 π 2 ~2
2ma2
We got an exact result...is this strange?
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example (cont.)
Boundary conditions: ψ(0) = 0, ψ(a) = 0 ⇒ φ(a) = nπ , n = 1, 2, 3, . . . ⇒
Z a
p(x)dx = nπ~
0
Take, for example, V (x) = 0 ⇒
En =
n 2 π 2 ~2
2ma2
We got an exact result...is this strange? No, since A =
Igor Lukačević
The WKB approximation
p
2/a = const.
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Contents
1
General remarks
2
The “classical” region
3
Tunneling
4
The connection formulas
5
Literature
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Now, assume E < V :
1 R
C
e ± ~ |p(x)|dx
ψ(x) ≈ p
|p(x)|
where p(x) is imaginary.
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Now, assume E < V :
1 R
C
e ± ~ |p(x)|dx
ψ(x) ≈ p
|p(x)|
where p(x) is imaginary.
Consider the potential:
some function , 0 < x < a
V (x) =
0,
otherwise
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
x <0
ψ(x) = Ae ikx + Be −ikx
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
x <0
x >a
ψ(x) = Ae ikx + Be −ikx
ψ(x) = Fe ikx
Transmission probability: T =
Igor Lukačević
The WKB approximation
Literature
|F |2
|A|2
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
x <0
0≤x ≤a
ψ(x) = Ae ikx + Be −ikx
ψ(x) ≈ √ C
+√ D
|p(x)|
Transmission probability: T =
Igor Lukačević
The WKB approximation
e
The connection formulas
0
0
1 Rx
e ~ 0 |p(x )|dx
|p(x)|
1 R x |p(x 0 )|dx 0
−~
0
Literature
x >a
ψ(x) = Fe ikx
|F |2
|A|2
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
x <0
0≤x ≤a
ψ(x) = Ae ikx + Be −ikx
ψ(x) ≈ √ C
+√ D
|p(x)|
e
The connection formulas
Literature
x >a
0
0
0 |p(x )|dx
|p(x)|
R
x
0
0
1
−~
0 |p(x )|dx
1
e~
Rx
ψ(x) = Fe ikx
Transmission probability:
|F |2
T =
|A|2
High, broad barrier
1st term
goes to 0
Why?
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
x <0
0≤x ≤a
ψ(x) = Ae ikx + Be −ikx
ψ(x) ≈ √ C
+√ D
|p(x)|
e
The connection formulas
Literature
x >a
0
0
1 Rx
e ~ 0 |p(x )|dx
|p(x)|
R
x
0
0
1
−~
0 |p(x )|dx
ψ(x) = Fe ikx
Transmission probability:
0
0
2 Ra
|F |2
T =
∼ e − ~ 0 |p(x )|dx
2
|A|
High, broad barrier
1st term
goes to 0
Why?
T ≈ e −2γ ,
Igor Lukačević
The WKB approximation
γ=
1
~
Z
a
|p(x)|dx
0
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Gamow’s theory of alpha decay
first time that quantum
mechanics had been
applied to nuclear
physics
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Gamow’s theory of alpha decay (cont.)
first time that quantum
mechanics had been
applied to nuclear
physics
turning points:
1
r1 7−→ nucleus radius
(6.63 fm for U238 )
2
r2 −
7 →
1 2Ze 2
=E
4π0 r2
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Gamow’s theory of alpha decay (cont.)
√
Z s
Z r
1 r2
2mE r2 r2
1 2Ze 2
γ=
2m
− E dr =
− 1dr
~ r1
4π0 r2
~
r
r1
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Gamow’s theory of alpha decay (cont.)
√
Z s
Z r
1 r2
1 2Ze 2
2mE r2 r2
γ=
2m
− E dr =
− 1dr
~ r1
4π0 r2
~
r
r1
Substituting r = r2 sin2 u gives
√
r p
2mE
π
r1
γ=
− sin−1
− r1 (r2 − r1 )
r2
~
2
r2
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Gamow’s theory of alpha decay (cont.)
√
Z s
Z r
1 r2
1 2Ze 2
2mE r2 r2
γ=
2m
− E dr =
− 1dr
~ r1
4π0 r2
~
r
r1
Substituting r = r2 sin2 u gives
( )
√
r
p
2mE
π
r1
−1
−
γ=
r2
− sin
r1 (r2 − r1 )
~
2
r2
|
{z
}
| {z }
√
r1 r2 √
r1 r2 −r12 −−−→ r1 r2
r1 r2 √
−−−→ r1 /r2
|
{z
}
π r −2√r r
1 2
2 2
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Gamow’s theory of alpha decay (cont.)
Substituting r = r2 sin2 u gives
( )
√
r
p
2mE
r1
π
−1
γ=
r2
− sin
−
r1 (r2 − r1 )
~
2
r2
{z
}
|
| {z }
√
r1 r2 √
2
r1 r2 −r1 −−−→ r1 r2
r1 r2 √
−−
−→ r1 /r2
{z
}
|
π r −2√r r
1 2
2 2
√
γ≈
i
√
√
2mE h π
Z
r2 − 2 r1 r2 = K1 √ − K2 Zr1
~
2
E
where
√
π 2m
= 1.980 MeV1/2
~
2 1/2 √
e
4 m
= 1.485 fm−1/2
4π0
~
Igor Lukačević
The WKB approximation
K1
=
K2
=
e2
4π0
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Gamow’s theory of alpha decay (cont.)
v
average velocity
2r1 /v
average time
between “collisions”
with the nucleus
potential “wall”
v /2r1
average
frequancy of “collisions”
“escape”
e −2γ
probability
(v /2r1 )e −2γ
“escape” probability per
unit time
Lifetime:
τ =
Igor Lukačević
The WKB approximation
2r1 2γ
e
v
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Gamow’s theory of alpha decay (cont.)
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 =
1
2r1 2γ
e ⇒ ln τ ∼ √
v
E
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
HW
Solve Problem 8.3 from Ref. [2].
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Contents
1
General remarks
2
The “classical” region
3
Tunneling
4
The connection formulas
5
Literature
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
Igor Lukačević
The WKB approximation
The “classical” region
Tunneling
The connection formulas
Literature
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Let us repeat:
ψ(x) ≈

 √1
p(x)
 √1
p(x)
Igor Lukačević
The WKB approximation
h
i
Be ~
R0
x
p(x 0 )dx 0
i
+ Ce − ~
0
0
1 Rx
De − ~ 0 |p(x )|dx
,
R0
x
p(x 0 )dx 0
i
,
if x < 0
if x > 0
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Let us repeat:
ψ(x) ≈

 √1

h
i
Be ~
R0
x
p(x 0 )dx 0
i
+ Ce − ~
p(x)
0
0
1 Rx
√ 1 De − ~ 0 |p(x )|dx
p(x)
R0
x
p(x 0 )dx 0
,
i
,
if x < 0
if x > 0
Our mission: join these two solutions at the boundary.
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Let us repeat:
ψ(x) ≈

 √1
p(x)
 √1
p(x)
h
i
Be ~
R0
x
p(x 0 )dx 0
i
+ Ce − ~
0
0
1 Rx
De − ~ 0 |p(x )|dx
R0
x
p(x 0 )dx 0
,
i
,
if x < 0
if x > 0
Our mission: join these two solutions at the boundary.
A problem
What happens with the w.f. when
E ≈ V?
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Let us repeat:
ψ(x) ≈

 √1

h
i
Be ~
R0
x
p(x 0 )dx 0
i
+ Ce − ~
p(x)
0
0
1 Rx
√ 1 De − ~ 0 |p(x )|dx
p(x)
R0
x
p(x 0 )dx 0
,
i
,
if x < 0
if x > 0
Our mission: join these two solutions at the boundary.
A problem
What happens with the w.f. when
E ≈ V?
E ≈ V ⇒ p(x) → 0 ⇒ ψ → ∞ !
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Let us repeat:
ψ(x) ≈

 √1

h
i
Be ~
R0
x
p(x 0 )dx 0
i
+ Ce − ~
p(x)
0
0
1 Rx
√ 1 De − ~ 0 |p(x )|dx
p(x)
R0
x
p(x 0 )dx 0
,
i
,
if x < 0
if x > 0
Our mission: join these two solutions at the boundary.
A problem
What happens with the w.f. when
E ≈ V?
A solution
Construct a “patching”
wavefunction ψp .
E ≈ V ⇒ p(x) → 0 ⇒ ψ → ∞ !
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Approximation: we linearize the potential
V (x) ≈ E + V 0 (0)x
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Approximation: we linearize the potential
V (x) ≈ E + V 0 (0)x
From S.E. we get 99K
d2 ψp
= zψp ,
dz 2
Igor Lukačević
The WKB approximation
z = αx ,
α=
1
3
2m 0
V
(0)
~2
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Approximation: we linearize the potential
V (x) ≈ E + V 0 (0)x
From S.E. we get 99K
d2 ψp
= zψp ,
2
|dz {z
}
z = αx ,
α=
1
3
2m 0
V
(0)
~2
Airy’s equation
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Approximation: we linearize the potential
V (x) ≈ E + V 0 (0)x
From S.E. we get 99K
d2 ψp
= zψp ,
2
|dz {z
}
1
3
2m 0
V (0)
α=
~2
z = αx ,
Airy’s equation
ψp = a
Ai(αx)
| {z }
Airy function
Igor Lukačević
The WKB approximation
+b
Bi(αx)
| {z }
Airy function
UJJS, Dept. of Physics, Osijek
Contents
General remarks
Igor Lukačević
The WKB approximation
The “classical” region
Tunneling
The connection formulas
Literature
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
a delicate double constraint has to be satisfied
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
a delicate double constraint has to be satisfied
we need WKB w.f. and ψp for both overlap regions (OLR)
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
p(x) =
Igor Lukačević
The WKB approximation
Tunneling
The connection formulas
Literature
p
3√
2m(E − V ) ≈ ~α 2 −x
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
3
p(x) ≈ ~α 2
√
The connection formulas
Literature
−x
OLR 2 (x > 0)
Z x
3
2
|p(x 0 )|dx 0 ≈ ~(αx) 2
3
0
ψWKB ≈ √
D
~α3/4 x 1/4
2
e − 3 (αx)
3/2
3/2
2
a
√
e − 3 (αx)
2 π(αx)1/4
3/2
2
b
+√
e 3 (αx)
π(αx)1/4
r
4π
⇒a=D
, b=0
α~
ψpz0 ≈
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
OLR 2 (x > 0)
Z x
3
2
|p(x 0 )|dx 0 ≈ ~(αx) 2
3
0
ψWKB ≈ √
D
~α3/4 x 1/4
2
e − 3 (αx)
Tunneling
0
Z
p(x 0 )dx 0 ≈
x
3/2
3/2
2
a
√
e − 3 (αx)
2 π(αx)1/4
3/2
2
b
+√
e 3 (αx)
1/4
π(αx)
r
4π
⇒a=D
, b=0
α~
The WKB approximation
Literature
OLR 1 (x < 0)
3
2
~(−αx) 2
3
h
3/2
2
1
Be i 3 (−αx)
3/4
1/4
~α (−x)
i
3/2
2
+Ce −i 3 (−αx)
a
1 h iπ/4 i 23 (−αx)3/2
e
e
≈ √
1/4
π(−αx) 2i
i
3/2
2
−e −iπ/4 e −i 3 (−αx)
ψWKB ≈ √
ψpz0 ≈
Igor Lukačević
The connection formulas
ψpz0
a
B
√ e iπ/4
= √
2i
π
~α
⇒
a
C
−iπ/4
− √ e
= √
2i π
~α
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
The connection formulas
B = −ie iπ/4 · D ,
Igor Lukačević
The WKB approximation
C = ie −iπ/4 · D
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
The connection formulas
B = −ie iπ/4 · D ,
C = ie −iπ/4 · D
WKB w.f.
ψ(x) ≈
Igor Lukačević
The WKB approximation
Z x2

2D
π
1


p
,
sin
p(x 0 )dx 0 +


~ x
4
p(x)

if x < x2
Z


D
1 x

0
0

p
exp
−
|p(x
)|dx
,

~ x2
|p(x)|
if x > x2
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potentail well with one vertical wall
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potentail well with one vertical wall
Boundary condition: ψ(0) = 0, gives for ψWKB
Z x2
1
p(x)dx = n −
π~ , n = 1, 2, 3, . . .
4
0
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potentail well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:

1

 mω 2 x 2 , x > 0
2
V (x) =


0
otherwise
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potentail well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:

1

 mω 2 x 2 , x > 0
2
V (x) =


0
otherwise
Here we have
p(x) = mω
Igor Lukačević
The WKB approximation
q
x22 − x 2
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potentail well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:

1

 mω 2 x 2 , x > 0
2
V (x) =


0
otherwise
Here we have
p(x) = mω
So
q
x22 − x 2
x2
Z
p(x)dx =
0
Igor Lukačević
The WKB approximation
πE
2ω
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potentail well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:

1

 mω 2 x 2 , x > 0
2
V (x) =


0
otherwise
Here we have
p(x) = mω
So
q
x22 − x 2
x2
Z
p(x)dx =
0
πE
2ω
Comparisson now gives:
En =
Igor Lukačević
The WKB approximation
1
2n −
2
~ω =
3 7 11
, , ,...
2 2 2
~ω
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potentail well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:

1

 mω 2 x 2 , x > 0
2
V (x) =


0
otherwise
Here we have
p(x) = mω
So
q
x22 − x 2
x2
Z
p(x)dx =
0
πE
2ω
Comparisson now gives:
En =
2n −
1
2
~ω =
3 7 11
, , ,...
2 2 2
~ω
Compare this result with an exact one.
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potential well with no vertical walls
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potential well with no vertical walls
we have seen the connection formulas for upward potential slopes
for downward slopes (analogous):
Z

1 x1
D0
0
0


p
|p(x
)|dx
exp
−
,


~ x
|p(x)|

ψ(x) ≈
Z x


2D 0
1
π

0
0

sin
p(x )dx +
,
 p
~ x1
4
p(x)
Igor Lukačević
The WKB approximation
if x < x1
if x > x1
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potential well with no vertical walls
we want the w.f. in the “well”, i.e. where x1 < x < x2 :
Z
2D
π
1 x2
ψ(x) ≈ p
sin θ2 (x) , θ2 (x) =
p(x 0 )dx 0 +
~ x
4
p(x)
Z x
0
2D
1
π
p(x 0 )dx 0 −
ψ(x) ≈ − p
sin θ1 (x) , θ1 (x) = −
~ x1
4
p(x)
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potential well with no vertical walls
sin θ1 = sin θ2 =⇒ θ2 = θ1 + nπ =⇒
Z x2
1
p(x)dx = n −
π~ , n = 1, 2, 3, . . .
2
x1
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Example: Potential well with no vertical walls
sin θ1 = sin θ2 =⇒ θ2 = θ1 + nπ =⇒
Z x2
1
p(x)dx = n −
π~ , n = 1, 2, 3, . . .
2
x1
0, two vertical walls
1/4, one vertical wall
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Conclusions
WKB advantages
WKB disadvantages
good for slowly changing w.f.
bad for rapidly changing w.f.
good for short wavelengths
bad for long wavelengths
best in the semi-classical
systems (large n)
inappropriate for lower states
(small n)
one doesn’t even have to solve
the S.E.
constraint trade-off (sometimes
not possible)
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Contents
1
General remarks
2
The “classical” region
3
Tunneling
4
The connection formulas
5
Literature
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek
Contents
General remarks
The “classical” region
Tunneling
The connection formulas
Literature
Literature
1
R. L. Liboff, Introductory Quantum Mechanics, Addison Wesley, San
Francisco, 2003.
2
D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., Pearson
Education, Inc., Upper Saddle River, NJ, 2005.
3
I. Supek, Teorijska fizika i struktura materije, II. dio, Školska knjiga,
Zagreb, 1989.
4
Y. Peleg, R. Pnini, E. Zaarur, Shaum’s Outline of Theory and Problems of
Quantum Mechanics, McGraw-Hill, 1998.
Igor Lukačević
The WKB approximation
UJJS, Dept. of Physics, Osijek