The Klein Paradox. Short history Scattering from potential step Bosons and fermions Resolution with pair production In- and out-states Conclusion

Transcription

1 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

2 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, : A. Nikishov 1981: Hansen-Ravndal 1998: B. Holstein

3 Scattering on potential step e!ipx e iqx e ipx ev x Schrödinger equation: ( 1 2m d 2 dx 2 + ev ) ψ(x) = Eψ(x)

4 Solutions x < 0: ψ(x) = 1 p eipx + R 1 p e ipx incident reflected with p = 2mE x > 0: ψ(x) = T 1 q eiqx transmitted where q = 2m(E ev ) When ev > E, then q imaginary and we have total reflection

5 Reflection probability: R 2 = 1 κ 1 + κ 2 with κ = q/p Transmission probability: T 2 = 4κ 1 + κ 2 satisfying unitarity R 2 + T 2 = 1 For strong potential ev > E, thenκ becomes imaginary and R 2 = 1 with T 2 = 0 which describes total reflection.

6 Klein-Gordon equation: [(E ev ) 2 d2 dx 2 + m2] φ(x) = 0 Again plane-wave solutions with p = E 2 m 2 and q = (E ev ) 2 m 2 Reflection probability: R 2 = 1 κ 1 + κ Transmission probability: T 2 = 4κ 1 + κ 2 2 with κ = q/p again satisfying unitarity R 2 + T 2 = 1 Notice that q real for ev > E + m > 2m

7 Classical motion from Lagrangian which here becomes L = m 2 g µνẋ µ ẋ ν + ea µ ẋ µ L = m 2 (ṫ2 ẋ 2) + ev ṫ Conserved energy E = L ṫ = mṫ + ev or for motion from A to B: ṫ A = ṫ B + ev m

8 t B B t A x A x E > ev + m ev + m > E > ev - m transmission total reflection

9 Pair production ev + m t E ev ev! m e! e + x A B E < ev - m x

10 Dirac equation: iγ x dψ dx + mψ = γ 0(E ev )ψ Again plane-wave solutions with p = E 2 m 2 and q = (E ev ) 2 m 2 Reflection probability: R 2 = 1 + κ 1 κ 2 Transmission probability: But now κ = q p satisfying unitarity T 2 = 4κ 1 κ 2 E + m E ev + m > 0 R 2 T 2 = 1

11 Klein s Paradox: When ev > E + m > 2m 1) Transmission into classically forbidden region! 2) R 2 > 1 i.e. more particles reflected than incident!! 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.

12 Rel. prob. for production of one pair: ω = T 2 Rel. prob. for reflection of one particle: σ = R 2 Prob. for no pair production: C 0 Prob. for producing one pair: C 0 ω Prob. for producing two pairs: C 0 ω 2 etc. Bosonic unitarity: 1 = C 0 [1 + ω + ω 2 + ] = C 0 1 ω i.e. C 0 = 1 ω = 1 T 2 = R 2 Fermionic unitarity: 1 = C 0 (1 + ω) i.e. C 0 = ω = T 2 = 1 R 2

13 Average number produced bosonic pairs: n = C 0 [ω + 2ω 2 + 3ω 3 + ] = C 0ω (1 ω) 2 2 = T (q) R(q) = T ( q) 2 F. Hund, 1940 Average number produced fermionic pairs: n = C 0 ω = ω ω = T (q) R(q) = T ( q) 2 Probability for elastic scattering of fermions: S el = C 0 σ = 1 R 2 R 2 = 1 Consistent with Pauli principle

14 In- and out-states Particle in-state: Particle out-state: p 1 (x) e ipx p 2 (x) e ipx for x < 0 Antiparticle in-state: Antiparticle out-state: n 1 (x) e iqx n 2 (x) e iqx for x > 0 Forming complete sets of states so that [ a1k p 1k (x) + b 1k n 1k(x) ] ψ(x) = k = k [ a2k p 2k (x) + b 2k n 2k(x) ]

15 Orthonormality (p 1k, p 1k ) = (p 2k, p 2k ) = δ kk (p 1k, n 1k ) = (p 2k, n 2k ) = 0 and (n 1k, n 1k ) = (n 2k, n 2k ) = ɛδ kk with ɛ = ±1 for fermions/bosons Thus a 1 = Aa 2 + Bb 2 and ɛb 1 = Ba 2 + Ãb 2 a 2 = A a 1 + B b 1 ɛb 2 = B a 1 + Ã b 1 where Bogoliubov coefficients A = (p 1, p 2 ) B = (p 1, n 2 ) Ã = (n 1, n 2 ) B = (n 1, p 2 )

16 Quantization [a 1, a 1 ] ɛ = [a 2, a 2 ] ɛ = 1 [b 1, b 1 ] ɛ = [b 2, b 2 ] ɛ = 1 which gives A 2 + ɛ B 2 = 1 Since R = -1/A and T = B/A then R 2 ɛ T 2 = 1 In-vacuum 0 1 is empty: a = b = 0 Same for out-vacuum: a = b = 0 General in- and out-states related by S-matrix operator: Ψ 1 = S Ψ 2

17 Incoming 1-particle state p 1 = a 1p 0 1 = S p 2 = Sa 2p 0 2 i.e. a 1 = Sa 2 S S-matrix elements: S fi = f i Example 1: Pair production i = 0 1 f = a 2 b = S pair = 0 2 b 2 a 2 0 1

18 Now a = 0 = (Aa 2 + Bb 2 ) 0 1 i.e. a = B A b Thus S pair = B A 0 2 b 2 b = B A eiw where vacuum-to-vacuum amplitude e iw = Prob. to create one pair: where ω = B A and C 0 = e iw 2 = e 2ImW is prob. that vacuum remains a vacuum. 2 P pair = S pair 2 = C 0 ω

19 Unitarity S S = 1 i.e. 0 1 S S 0 1 = 1 = p 2 n after inserting complete set of in-states. Thus for bosons 1 = C 0 [1 + ω + ω 2 + ] = C 0 1 ω as before. Example 2: Scattering i = p 1 = a f = p 2 = a S scatt = 0 2 a 2 a where a 1 = A a 2 + B b 2

20 S scatt = A B 0 2 a 2 b = A e iw + ɛb ( B ) = 1 A A eiw Probability for elastic scattering: For fermions C 0 = P scatt = S scatt 2 1 = C 0 2 A 1 2 = A 2 R Thus P scatt = 1 and therefore only elastic scattering. But for bosons incident particle can induce pair production in same mode.

21 Scattering amp. including production of n pairs S (n) scatt = 1 n!(n + 1)! 0 2 (a 2 b 2 ) n a 2 a = n + 1e iw [ A + B ( B A ) ] ( B A ) n and probability: S (n) scatt 2 = C 0 σ(n + 1)ω n Unitarity: n=0 S (n) scatt 2 = C 0σ (1 ω) 2 = 1

22 Sauter potential: ev V (x) = V 2 tanh ax 2 e! e + Klein step potential when a. x T 2 = cosh(2π/a)(p + q) cosh(2π/a)(p q) cosh(2π/a)ev cosh(2π/a)(p + q)

23 In limit a 0 and V with av 4E obtain linear potential corresponding to constant electric field E. with Tunneling: Pair production prob. ω = T 2 = n 1 n n = e πm2 /ee n = e 2 x2 x 1 dxk(x,m T ) WKB with m T = (k 2 T + m 2 ) 1/2 and k = ( m 2 T (ε eex) 2) 1/2 Vacuum-to-vacuum persistence prob: C 0 = e 2ImW = e 2 d 4 ximl eff

24 Now for fermions 2 C 0 = k T [1 n(k T )] d 4 ximl eff = k T log[1 n(k T )] or ImL eff = 1 8π ( ee π ) 2 n=1 1 2 n 2 e nπm /ee Schwinger, 1951

25 Conclusion 1) Hawking radiation, ) 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)

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