Contents General remarks The “classical” region Tunneling The connection formulas Literature The WKB approximation Quantum mechanics 2 - Lecture 4 Igor Lukaˇ cevi´ c UJJS, Dept. of Physics, Osijek 12. studenog 2013. Igor Lukaˇ cevi´ c UJJS, Dept. of Physics, Osijek The WKB approximation
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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