The Klein Paradox
Finn Ravndal, Dept of Physics, UiO
• Short history
• Scattering from potential step
• Bosons and fermions
• Resolution with pair production
• In- and out-states
• Conclusion
Gausdal, 4/1 - 2011
Short history:
1926: Klein-Gordon equation
1928: Dirac equation
1929: Klein-Nishina formula
1929: Potential step paradox
1932: F. Sauter realistic potential
1940: F. Hund second quantization
Oskar Klein, 1894 - 1977
1970: A. Nikishov
1981: Hansen-Ravndal
1998: B. Holstein
Scattering on potential step
e
e
!ipx
e
ipx
iqx
eV
x
Schrödinger equation:
!
2
1 d
−
+
eV
2m dx2
"
ψ(x) = Eψ(x)
Solutions
x < 0:
ψ(x) =
!
1 ipx
e +R
p
incident
with
x > 0:
p=
√
!
1 −ipx
e
p
reflected
2mE
ψ(x) = T
!
1 iqx
e
q
transmitted
where
!
q = 2m(E − eV )
When eV > E, then q
imaginary and we have
total reflection
Reflection probability:
!
!2
!1 − κ!
2
!
|R| = !!
1 + κ!
Transmission probability:
4κ
|T | =
|1 + κ|2
satisfying unitarity
with
κ = q/p
2
2
2
|R| + |T | = 1
For strong potential eV > E, then κ becomes imaginary
and
2
|R| = 1
total reflection.
2
with |T | = 0 which describes
2
d
Klein-Gordon equation: (E − eV ) − 2 + m2 φ(x) = 0
dx
!
Again plane-wave solutions with p = E 2 − m2
!
and q = (E − eV )2 − m2
!
2
Reflection probability:
!
!2
!1 − κ!
2
!
|R| = !!
1 + κ!
Transmission probability:
4κ
|T | =
|1 + κ|2
again satisfying unitarity
|R|2 + |T |2 = 1
2
Notice that q real for eV > E + m > 2m
"
with
κ = q/p
Classical motion from Lagrangian
m
L = gµν ẋµ ẋν + eAµ ẋµ
2
which here becomes
"
m! 2
2
L=
ṫ − ẋ + eV ṫ
2
Conserved energy
∂L
= mṫ + eV
E=
∂ ṫ
or for motion from A to B:
eV
ṫA = ṫB +
m
t
t
B
B
A
A
x
x
E > eV + m
transmission
eV + m > E > eV - m
total reflection
Pair production
eV + m
E
eV
t
eV ! m
e
!
e
+
x
A
B
E < eV - m
x
dψ
+ mψ = γ0 (E − eV )ψ
iγx
Dirac equation:
dx
!
Again plane-wave solutions with p = E 2 − m2
!
and q = (E − eV )2 − m2
Reflection probability:
Transmission probability:
But now
!
!2
!
!
1
+
κ
!
|R|2 = !!
1 − κ!
4κ
|T | =
|1 − κ|2
2
q
E+m
κ=−
>0
p E − eV + m
satisfying ‘unitarity’
|R|2 − |T |2 = 1
Klein’s Paradox: When eV > E + m > 2m
1) Transmission into classically forbidden region !
2
|R|
> 1 i.e. more particles reflected than incident !!
2)
Underlying physics the same for Klein-Gordon equation,
but paradox not so visible.
Both the Klein-Gordon and the Dirac equation are no
1-particle wave-equations, but relativistic field equations.
Rel. prob. for production of one pair:
2
ω = |T |
2
σ
=
|R|
Rel. prob. for reflection of one particle:
Prob. for no pair production:
C0
Prob. for producing one pair:
C0 ω
etc.
Prob. for producing two pairs: C0 ω 2
C0
Bosonic unitarity: 1 = C0 [1 + ω + ω + · · ·] =
1−ω
2
2
i.e. C0 = 1 − ω = 1 − |T | = |R|
2
Fermionic unitarity: 1 = C0 (1 + ω)
i.e.
1
1
1
=
=
C0 =
2
1+ω
1 + |T |
|R|2
Average number produced bosonic pairs:
C0 ω
n̄ = C0 [ω + 2ω + 3ω + · · ·] =
2
(1
−
ω)
!
!2
! T (q) !
! = |T (−q)|2
= !!
R(q) !
2
3
F. Hund, 1940
Average number produced fermionic pairs:
!
!2
! T (q) !
ω
! = |T (−q)|2
n̄ = C0 ω =
= !!
1+ω
R(q) !
Probability for elastic scattering of fermions:
1
2
|R|
=1
Sel = C0 σ =
2
|R|
Consistent with Pauli
principle
In- and out-states
ipx
Particle in-state:
p1 (x) ! e
Particle out-state:
p2 (x) ! e
Antiparticle in-state:
for x < 0
−ipx
n1 (x) ! eiqx
Antiparticle out-state: n2 (x) ! e
for x > 0
−iqx
Forming complete sets of states so that
!"
#
†
ψ(x) =
a1k p1k (x) + b1k n1k (x)
k
!
#
"
†
=
a2k p2k (x) + b2k n2k (x)
k
Orthonormality
(p1k , p1k! ) = (p2k , p2k! ) = δkk!
(p1k , n1k! ) = (p2k , n2k! ) = 0
and
with
Thus
(n1k , n1k! ) = (n2k , n2k! ) = !δkk!
! = ±1
a1 =
a2 =
for fermions/bosons
Aa2 + Bb†2
A∗ a1 + B̃ ∗ b†1
and
where Bogoliubov coefficients
A = (p1 , p2 )
B = (p1 , n2 )
à = (n1 , n2 )
B̃ = (n1 , p2 )
†
!b1
†
!b2
†
= B̃a2 + Ãb2
∗
∗ †
= B a1 + Ã b1
Quantization
which gives
†
[a1 , a1 ]!
†
[b1 , b1 ]!
=
=
†
[a2 , a2 ]!
†
[b2 , b2 ]!
=1
=1
|A|2 + !|B|2 = 1
2
2
|R|
−
!|T
|
=1
Since R = -1/A and T = B/A then
In-vacuum |01! is empty: a1 |01! = b1 |01! = 0
Same for out-vacuum: a2 |02! = b2 |02! = 0
General in- and out-states related by S-matrix
operator:
|Ψ1! = S|Ψ2!
Incoming 1-particle state |p1! =
=
i.e.
a1 = Sa2 S
†
a1p |01! =
†
Sa2p |02!
†
S-matrix elements: Sf i = !f |i"
Example 1: Pair production
|i! = |01 !
=⇒
|f ! =
† †
a2 b2 |02 !
Spair = !02 |b2 a2 |01 "
S|p2!
Now
i.e.
Thus
a1 |01! = 0 = (Aa2 +
†
Bb2 )|01!
B †
a2 |01 ! = − b2 |01!
A
B
B iW
†
Spair = − "02 |b2 b2 |01 # = − e
A
A
iW
where vacuum-to-vacuum amplitude e
= !01 |02 "
Prob. to create one pair: Ppair = |Spair |2 = C0 ω
! !2
!B !
where ω = !! !!
A
and C0 = |eiW |2 = e−2ImW
remains a vacuum.
is prob. that vacuum
Unitarity
S†S = 1
†
2
2
!0
|S
S|0
"
=
1
=
|!0
|0
"|
+
|!0
|p
n
"|
+ ···
i.e. 1
1
1 2
1 2 2
after inserting complete set of in-states. Thus for bosons
C0
1 = C0 [1 + ω + ω + · · ·] =
1−ω
2
as before.
Example 2: Scattering
|i! = |p1 ! =
=⇒
where
†
a1
a†1 |01 !
Sscatt =
=
∗ †
A a2
|f ! = |p2 ! =
†
!02 |a2 a1 |01 "
+ B ∗ b2
†
a2 |02 !
Sscatt = A∗ !02 |01 " + B ∗ !02 |a2 b2 |01 "
∗ iW
=A e
+ !B
∗
!B "
A
1 iW
= e
A
Probability for elastic scattering:
For fermions
! 1 !2
! !
2
Pscatt = |Sscatt | = C0 ! !
A
! 1 !2
! !
C0 = ! ! = |A|2
R
Thus Pscatt = 1 and therefore only elastic scattering.
But for bosons incident particle can induce pair
production in same mode.
Scattering amp. including production of n pairs
(n)
Sscatt
=
=
and probability:
Unitarity:
!
1
!02 |(a2 b2 )n a2 a†1 |01 "
n!(n + 1)!
n + 1eiW
!
" B #$ " B #n
−
A∗ + B ∗ −
A
A
(n) 2
|Sscatt |
= C0 σ(n + 1)ω n
√
∞
!
n=0
(n) 2
|Sscatt |
C0 σ
=1
=
2
(1 − ω)
Sauter potential:
eV
e!
V
ax
V (x) = tanh
2
2
e+
x
Klein step potential when
a → ∞.
cosh(2π/a)(p + q) − cosh(2π/a)(p − q)
|T | =
cosh(2π/a)eV − cosh(2π/a)(p + q)
2
In limit a → 0 and V → ∞ with aV → 4E
obtain linear potential corresponding to constant
electric field E.
=⇒ Pair production prob. ω = |T |2 = n̄
1 − n̄
with
−πm2 /eE
n̄ = e
Tunneling:
with mT =
−2
n̄ = e
(kT2
! x2
2 1/2
+m )
x1
dxk(x,mT )
and k =
!
m2T
Vacuum-to-vacuum persistence prob:
! 4
C0 = e−2ImW = e−2 d xImLef f
WKB
"
2 1/2
− (ε − eEx)
Now for fermions
!
[1 − n̄(kT )]
C0 =
kT
=⇒
!
"
2 d4 xImLef f = −
log[1 − n̄(kT )]
kT
or
ImLef f
1
=
8π
!
eE
π
"2 #
∞
1 −nπm2 /eE
e
2
n
n=1
Schwinger, 1951
Conclusion
1) Hawking radiation, 1974
2) Geim graphene, 2010
Literature
O. Klein, Z. Phys. 53, 157 (1929)
F. Sauter, Z. Phys. 73, 547 (1931)
F. Hund, Z. Phys. 117, 1 (1940)
J. Schwinger, Phys. Rev. 82, 664 (1951)
J.D. Bjorken and S.D. Drell, Rel. Quant. Mech. (1964)
A. I. Nikishov, Nucl. Phys. B21, 346 (1970)
A. Hansen and F. Ravndal, Phys. Scr. 23, 1030 (1981)
B. Holstein, Am. J. Phys. 66, 507 (1998)
N. Dombey and A. Calogeracos, Phys. Rep. 315, 41 (1999)