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Contents General remarks The “classical” region Tunneling The connection formulas Literature
The WKB approximationQuantum mechanics 2 - Lecture 4
Igor Lukacevic
UJJS, Dept. of Physics, Osijek
12. studenog 2013.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
1 General remarks
2 The “classical” region
3 Tunneling
4 The connection formulas
5 Literature
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Basic idea:
1 particle Epotential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
√2m(E − V )
~
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Basic idea:
1 particle Epotential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
√2m(E − V )
~
A question
What’s the character of A and λ = 2π/k here?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Basic idea:
1 particle Epotential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
√2m(E − V )
~2 suppose V (x) not constant, but varies slowly wrt λ
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Basic idea:
1 particle Epotential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
√2m(E − V )
~2 suppose V (x) not constant, but varies slowly wrt λ
A question
What can we say about ψ, A and λ now?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Basic idea:
1 particle Epotential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
√2m(E − V )
~2 suppose V (x) not constant, but varies slowly wrt λ
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Basic idea:
1 particle Epotential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
√2m(E − V )
~2 suppose V (x) not constant, but varies slowly wrt λ
3 if E < V , the reasoning is analogous
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Basic idea:
1 particle Epotential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
√2m(E − V )
~2 suppose V (x) not constant, but varies slowly wrt λ
3 if E < V , the reasoning is analogous
A question
What if E ≈ V ?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Basic idea:
1 particle Epotential V (x) constant
if E > V ⇒ ψ(x) = Ae±ikx , k =
√2m(E − V )
~2 suppose V (x) not constant, but varies slowly wrt λ
3 if E < V , the reasoning is analogous
A question
What if E ≈ V ? Turning points
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
S.E.
− ~2
2m∆ψ + V (x)ψ = Eψ ⇐⇒ ∆ψ = −p2
~2ψ , p(x) =
√2m [E − V (x)]
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
S.E.
− ~2
2m∆ψ + V (x)ψ = Eψ ⇐⇒ ∆ψ = −p2
~2ψ , p(x) =
√2m [E − V (x)]
“Classical” region
99K E > V (x) , p real
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
S.E.
− ~2
2m∆ψ + V (x)ψ = Eψ ⇐⇒ ∆ψ = −p2
~2ψ , p(x) =
√2m [E − V (x)]
“Classical” region
99K E > V (x) , p real
99K ψ(x) = A(x)e iφ(x)
A(x) and φ(x) real
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Putting ψ(x) into S.E. gives two equations:
A′′ = A
[(φ′)2 − p2
~2
](1)(
A2φ′)′
= 0 (2)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Putting ψ(x) into S.E. gives two equations:
A′′ = A
[(φ′)2 − p2
~2
](1)(
A2φ′)′
= 0 (2)
Solve (2)
A =C√φ′, C ∈ R
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Putting ψ(x) into S.E. gives two equations:
A′′ = A
[(φ′)2 − p2
~2
](1)(
A2φ′)′
= 0 (2)
Solve (2)
A =C√φ′, C ∈ R
Solve (1)
Assumption: A varies slowly
⇒ A′′ ≈ 0
φ(x) = ±1
~
∫p(x)dx
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Solve (2)
A =C√φ′, C ∈ R
Solve (1)
Assumption: A varies slowly
⇒ A′′ ≈ 0
φ(x) = ±1
~
∫p(x)dx
Resulting wavefunction
ψ(x) ≈ C√p(x)
e±i~
∫p(x)dx
Note: general solution is a linear combination of these.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Solve (1)
A =C√φ′, C ∈ R
Solve (2)
Assumption: A varies slowly
⇒ A′′ ≈ 0
φ(x) = ±1
~
∫p(x)dx
Resulting wavefunction
ψ(x) ≈ C√p(x)
e±i~
∫p(x)dx
Note: general solution is a linear combination of these.
Probability of finding a particle at x
|ψ(x)|2 ≈ |C |2
p(x)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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) =⇒
ψ(x) ≈ 1√p(x)
[C+e
iφ(x) + C−e−iφ(x)
]=
1√p(x)
[C1 sinφ(x) + C2 cosφ(x)]
where
φ(x) =1
~
∫ x
0
p(x ′)dx ′
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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, . . .⇒∫ a
0
p(x)dx = nπ~
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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, . . .⇒∫ a
0
p(x)dx = nπ~
Take, for example, V (x) = 0⇒
En =n2π2~2
2ma2
We got an exact result...is this strange?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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, . . .⇒∫ a
0
p(x)dx = nπ~
Take, for example, V (x) = 0⇒
En =n2π2~2
2ma2
We got an exact result...is this strange? No, since A =√
2/a = const.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Now, assume E < V :
ψ(x) ≈ C√|p(x)|
e±1~
∫|p(x)|dx
where p(x) is imaginary.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Now, assume E < V :
ψ(x) ≈ C√|p(x)|
e±1~
∫|p(x)|dx
where p(x) is imaginary.
Consider the potential:
V (x) =
{some function , 0 < x < a0 , otherwise
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
x < 0
ψ(x) = Ae ikx + Be−ikx
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
x < 0 x > a
ψ(x) = Ae ikx + Be−ikx ψ(x) = Fe ikx
Transmission probability: T =|F |2
|A|2
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
x < 0 0 ≤ x ≤ a x > a
ψ(x) = Ae ikx + Be−ikx ψ(x) ≈ C√|p(x)|
e1~
∫ x0 |p(x
′)|dx′ ψ(x) = Fe ikx
+ D√|p(x)|
e−1~
∫ x0 |p(x
′)|dx′
Transmission probability: T =|F |2
|A|2
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
x < 0 0 ≤ x ≤ a x > a
ψ(x) = Ae ikx + Be−ikx ψ(x) ≈ C√|p(x)|
e1~
∫ x0 |p(x
′)|dx′ ψ(x) = Fe ikx
+ D√|p(x)|
e−1~
∫ x0 |p(x
′)|dx′
Transmission probability:
T =|F |2
|A|2High, broad barrier 1st termgoes to 0Why?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
x < 0 0 ≤ x ≤ a x > a
ψ(x) = Ae ikx + Be−ikx ψ(x) ≈ C√|p(x)|
e1~
∫ x0 |p(x
′)|dx′ ψ(x) = Fe ikx
+ D√|p(x)|
e−1~
∫ x0 |p(x
′)|dx′
Transmission probability:
T =|F |2
|A|2 ∼ e−2~
∫ a0 |p(x
′)|dx′
High, broad barrier 1st termgoes to 0Why?
T ≈ e−2γ , γ =1
~
∫ a
0
|p(x)|dx
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Gamow’s theory of alpha decay
first time that quantummechanics had beenapplied to nuclearphysics
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Gamow’s theory of alpha decay (cont.)
first time that quantummechanics had beenapplied to nuclearphysics
turning points:1 r1 7−→ nucleus radius
(6.63 fm for U238)
2 r2 7−→1
4πε0
2Ze2
r2= E
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Gamow’s theory of alpha decay (cont.)
γ =1
~
∫ r2
r1
√2m
(1
4πε0
2Ze2
r2− E
)dr =
√2mE
~
∫ r2
r1
√r2r− 1dr
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Gamow’s theory of alpha decay (cont.)
γ =1
~
∫ r2
r1
√2m
(1
4πε0
2Ze2
r2− E
)dr =
√2mE
~
∫ r2
r1
√r2r− 1dr
Substituting r = r2 sin2 u gives
γ =
√2mE
~
[r2
(π
2− sin−1
√r1r2
)−√
r1(r2 − r1)
]
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Gamow’s theory of alpha decay (cont.)
γ =1
~
∫ r2
r1
√2m
(1
4πε0
2Ze2
r2− E
)dr =
√2mE
~
∫ r2
r1
√r2r− 1dr
Substituting r = r2 sin2 u gives
γ =
√2mE
~
{r2
[π
2− sin−1
√r1r2︸ ︷︷ ︸
r1�r2−−−→√r1/r2
]
︸ ︷︷ ︸π2r2−2√r1r2
−√
r1(r2 − r1)︸ ︷︷ ︸√
r1r2−r21
r1�r2−−−→√r1r2
}
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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
γ =
√2mE
~
{r2
[π
2− sin−1
√r1r2︸ ︷︷ ︸
r1�r2−−−→√r1/r2
]
︸ ︷︷ ︸π2r2−2√r1r2
−√
r1(r2 − r1)︸ ︷︷ ︸√
r1r2−r21
r1�r2−−−→√r1r2
}
γ ≈√
2mE
~
[π2r2 − 2
√r1r2]
= K1Z√E− K2
√Zr1
where
K1 =
(e2
4πε0
)π√
2m
~= 1.980MeV1/2
K2 =
(e2
4πε0
)1/24√m
~= 1.485 fm−1/2
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 timebetween “collisions”with the nucleuspotential “wall”
v/2r1 averagefrequancy of “collisions”
e−2γ “escape”probability
(v/2r1)e−2γ “escape” probability perunit time
Lifetime:
τ =2r1v
e2γ
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 timebetween “collisions”with the nucleuspotential “wall”
v/2r1 averagefrequancy of “collisions”
e−2γ “escape”probability
(v/2r1)e−2γ “escape” probability perunit time
Lifetime:
τ =2r1v
e2γ ⇒ ln τ ∼ 1√E
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
HW
Solve Problem 8.3 from Ref. [2].
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Let us repeat:
ψ(x) ≈
1√p(x)
[Be
i~
∫ 0x p(x′)dx′ + Ce−
i~
∫ 0x p(x′)dx′
], if x < 0
1√|p(x)|
De−1~
∫ x0 |p(x
′)|dx′ , if x > 0
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Let us repeat:
ψ(x) ≈
1√p(x)
[Be
i~
∫ 0x p(x′)dx′ + Ce−
i~
∫ 0x p(x′)dx′
], if x < 0
1√|p(x)|
De−1~
∫ x0 |p(x
′)|dx′ , if x > 0
Our mission: join these two solutions at the boundary.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Let us repeat:
ψ(x) ≈
1√p(x)
[Be
i~
∫ 0x p(x′)dx′ + Ce−
i~
∫ 0x p(x′)dx′
], if x < 0
1√|p(x)|
De−1~
∫ x0 |p(x
′)|dx′ , if x > 0
Our mission: join these two solutions at the boundary.
A problem
What happens with the w.f. whenE ≈ V ?
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Let us repeat:
ψ(x) ≈
1√p(x)
[Be
i~
∫ 0x p(x′)dx′ + Ce−
i~
∫ 0x p(x′)dx′
], if x < 0
1√|p(x)|
De−1~
∫ x0 |p(x
′)|dx′ , if x > 0
Our mission: join these two solutions at the boundary.
A problem
What happens with the w.f. whenE ≈ V ?
E ≈ V ⇒ p(x)→ 0⇒ ψ →∞ !
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Let us repeat:
ψ(x) ≈
1√p(x)
[Be
i~
∫ 0x p(x′)dx′ + Ce−
i~
∫ 0x p(x′)dx′
], if x < 0
1√|p(x)|
De−1~
∫ x0 |p(x
′)|dx′ , if x > 0
Our mission: join these two solutions at the boundary.
A problem
What happens with the w.f. whenE ≈ V ?
E ≈ V ⇒ p(x)→ 0⇒ ψ →∞ !
A solution
Construct a “patching”wavefunction ψp.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Approximation: we linearize the potential
V (x) ≈ E + V ′(0)x
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Approximation: we linearize the potential
V (x) ≈ E + V ′(0)x
From S.E. we get 99K
d2ψp
dz2= zψp , z = αx , α =
[2m
~2V ′(0)
] 13
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Approximation: we linearize the potential
V (x) ≈ E + V ′(0)x
From S.E. we get 99K
d2ψp
dz2= zψp︸ ︷︷ ︸
Airy’s equation
, z = αx , α =
[2m
~2V ′(0)
] 13
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Approximation: we linearize the potential
V (x) ≈ E + V ′(0)x
From S.E. we get 99K
d2ψp
dz2= zψp︸ ︷︷ ︸
Airy’s equation
, z = αx , α =
[2m
~2V ′(0)
] 13
ψp = a Ai(αx)︸ ︷︷ ︸Airy function
+b Bi(αx)︸ ︷︷ ︸Airy function
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
a delicate double constraint has to be satisfied
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
p(x) =√
2m(E − V ) ≈ ~α32√−x
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
p(x) ≈ ~α32√−x
OLR 2 (x > 0)∫ x
0
|p(x ′)|dx ′ ≈ 2
3~(αx)
32
ψWKB ≈D√
~α3/4x1/4e−
23(αx)3/2
ψz�0p ≈ a
2√π(αx)1/4
e−23(αx)3/2
+b√
π(αx)1/4e
23(αx)3/2
⇒ a = D
√4π
α~, b = 0
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
OLR 2 (x > 0)∫ x
0
|p(x ′)|dx ′ ≈ 2
3~(αx)
32
ψWKB ≈D√
~α3/4x1/4e−
23(αx)3/2
ψz�0p ≈ a
2√π(αx)1/4
e−23(αx)3/2
+b√
π(αx)1/4e
23(αx)3/2
⇒ a = D
√4π
α~, b = 0
OLR 1 (x < 0)∫ 0
x
p(x ′)dx ′ ≈ 2
3~(−αx)
32
ψWKB ≈1√
~α3/4(−x)1/4
[Be i
23(−αx)3/2
+Ce−i 23(−αx)3/2
]ψz�0
p ≈ a√π(−αx)1/4
1
2i
[e iπ/4e i
23(−αx)3/2
−e−iπ/4e−i 23(−αx)3/2
]
⇒
a
2i√πe iπ/4 =
B√~α
− a
2i√πe−iπ/4 =
C√~α
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
The connection formulas
B = −ie iπ/4 · D , C = ie−iπ/4 · D
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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) ≈
2D√p(x)
sin
[1
~
∫ x2
x
p(x ′)dx ′ +π
4
], if x < x2
D√|p(x)|
exp
[−1
~
∫ x
x2
|p(x ′)|dx ′], if x > x2
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Potential well with one vertical wall
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Potential well with one vertical wall
Boundary condition: ψ(0) = 0, gives for ψWKB∫ x2
0
p(x)dx =
(n − 1
4
)π~ , n = 1, 2, 3, . . .
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Potential well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:
V (x) =
1
2mω2x2 , x > 0
0 otherwise
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Potential well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:
V (x) =
1
2mω2x2 , x > 0
0 otherwise
Here we have
p(x) = mω√
x22 − x2
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Potential well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:
V (x) =
1
2mω2x2 , x > 0
0 otherwise
Here we have
p(x) = mω√
x22 − x2
So ∫ x2
0
p(x)dx =πE
2ω
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Potential well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:
V (x) =
1
2mω2x2 , x > 0
0 otherwise
Here we have
p(x) = mω√
x22 − x2
So ∫ x2
0
p(x)dx =πE
2ω
Comparisson now gives:
En =
(2n − 1
2
)~ω =
(3
2,
7
2,
11
2, . . .
)~ω
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Potential well with one vertical wall (cont.)
For instance, consider the “half-harmonic oscillator”:
V (x) =
1
2mω2x2 , x > 0
0 otherwise
Here we have
p(x) = mω√
x22 − x2
So ∫ x2
0
p(x)dx =πE
2ω
Comparisson now gives:
En =
(2n − 1
2
)~ω =
(3
2,
7
2,
11
2, . . .
)~ω
Compare this result with an exact one.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Example: Potential well with no vertical walls
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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):
ψ(x) ≈
D ′√|p(x)|
exp
[−1
~
∫ x1
x
|p(x ′)|dx ′], if x < x1
2D ′√p(x)
sin
[1
~
∫ x
x1
p(x ′)dx ′ +π
4
], if x > x1
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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:
ψ(x) ≈ 2D√p(x)
sin θ2(x) , θ2(x) =1
~
∫ x2
x
p(x ′)dx ′ +π
4
ψ(x) ≈ − 2D ′√p(x)
sin θ1(x) , θ1(x) = −1
~
∫ x
x1
p(x ′)dx ′ − π
4
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 =⇒∫ x2
x1
p(x)dx =
(n − 1
2
)π~ , n = 1, 2, 3, . . .
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 =⇒∫ x2
x1
p(x)dx =
(n − 1
2
)π~ , n = 1, 2, 3, . . .
0, two vertical walls 1/4, one vertical wall
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Conclusions
WKB advantages
good for slowly changing w.f.
good for short wavelengths
best in the semi-classicalsystems (large n)
one doesn’t even have to solvethe S.E.
WKB disadvantages
bad for rapidly changing w.f.
bad for long wavelengths
inappropriate for lower states(small n)
constraint trade-off (sometimesnot possible)
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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 Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
Contents General remarks The “classical” region Tunneling The connection formulas Literature
Literature
1 R. L. Liboff, Introductory Quantum Mechanics, Addison Wesley, SanFrancisco, 2003.
2 D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., PearsonEducation, Inc., Upper Saddle River, NJ, 2005.
3 I. Supek, Teorijska fizika i struktura materije, II. dio, Skolska knjiga,Zagreb, 1989.
4 Y. Peleg, R. Pnini, E. Zaarur, Shaum’s Outline of Theory and Problems ofQuantum Mechanics, McGraw-Hill, 1998.
Igor Lukacevic UJJS, Dept. of Physics, Osijek
The WKB approximation
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