Abstract

Potential scattering of free-electron wave packets is considered in the framework of non-stationary quantum-mechanical theory. The general expression for the average angle of scattering is obtained. The traditional quantum-mechanical plane-wave approximation and classical results are shown to be incorporated in the results derived.

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References

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  1. R. Newton, Scattering theory of waves and particles (McGraw-Hill, New York 1968).
  2. P. H. Bucksbaum, D. W. Schumacher, and M. Bashkansky, "High Intensity Kapitza-Dirac Effect", Phys. Rev. Lett. 61, 1182 (1988).
    [CrossRef] [PubMed]
  3. L. D. Landau and E. M. Lifshitz, Mechanics (Permagon Press, Oxford-New York 1976).
  4. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, (Permagon Press, Oxford-New York 1977).
  5. M. V. Fedorov, S. P. Goreslavsky, and V. S. Letokhov, Ponderomotive forces and stimulated Compton scattering on free electrons in a laser field, Phys. Rev. E, 55, 1015 (1997)
    [CrossRef]
  6. M. V. Fedorov and J. Peatross, Strong-field photoionization and emission of light in the wave-packet-spreading regime, Phys. Rev. A 52, 504 (1995).
    [CrossRef] [PubMed]
  7. I. S. Gradstein and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York 1980).

Other

R. Newton, Scattering theory of waves and particles (McGraw-Hill, New York 1968).

P. H. Bucksbaum, D. W. Schumacher, and M. Bashkansky, "High Intensity Kapitza-Dirac Effect", Phys. Rev. Lett. 61, 1182 (1988).
[CrossRef] [PubMed]

L. D. Landau and E. M. Lifshitz, Mechanics (Permagon Press, Oxford-New York 1976).

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, (Permagon Press, Oxford-New York 1977).

M. V. Fedorov, S. P. Goreslavsky, and V. S. Letokhov, Ponderomotive forces and stimulated Compton scattering on free electrons in a laser field, Phys. Rev. E, 55, 1015 (1997)
[CrossRef]

M. V. Fedorov and J. Peatross, Strong-field photoionization and emission of light in the wave-packet-spreading regime, Phys. Rev. A 52, 504 (1995).
[CrossRef] [PubMed]

I. S. Gradstein and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York 1980).

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Figures (1)

Fig 1.
Fig 1.

The first-order angle of scattering θ¯(1)r 0) of the Gaussian wave packet scattered by a pure Coulomb potential.

Equations (26)

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Ψ ( r , t ) t = ( ħ 2 2 m 2 + U ( r ) ) Ψ ( r , t )
Ψ ( r , t ) = 1 ( 2 πħ ) 3 / 2 d p C p ( t ) exp [ i ħ ( r E p t ) ]
C ˙ p ( t ) = d p′ C p′ ( t ) U ˜ ( p′ p ) exp [ i ħ ( E p E p′ ) t ]
C p ( t ) t = 0 C p ( 0 ) = d r exp ( i ħ p · r ) Ψ ( 0 ) ( r ) .
U ˜ ( q ) = 1 ( 2 πħ ) 3 d r U ( r ) exp ( i q · r ħ ) .
C p ( t ) = C p ( 0 ) + C p ( 1 ) ( t ) + C p ( 2 ) ( t ) +
C p ( 1 ) ( t ) = i ħ 0 t dt d p′ C p′ ( 0 ) U ˜ ( p′ p ) exp [ i ħ ( E p E p′ ) t′ ]
C p ( 0 ) = ( Δ r 0 ) 3 / 2 π 3 / 4 ħ 3 / 2 exp [ ( Δ r 0 ) 2 2 ħ 2 ( p p 0 ) 2 i ħ r 0 ] ,
Δ r ( t ) = ( Δ r 0 ) 2 + ( ħ t m Δ r 0 ) 2
p x ¯ ( ρ ) = p x C p ( t ) 2 d p ( 2 πħ ) 3 .
θ ¯ ( ρ ) = sin 1 ( p x ¯ ( ρ ) p 0 ) p x ¯ ( ρ ) p 0 .
θ ¯ ( 1 ) = 1 p 0 p x ( C p ( 0 ) C p ( 1 ) * + C p ( 1 ) C p ( 0 ) * ) d p ( 2 πħ ) 3 =
= 1 π 3 / 2 p 0 0 t dt′ 1 [ Δ r ( t′ ) ] 3 d r U x exp { ( r r 0 v 0 t ) 2 [ Δ r ( t′ ) ] 2 } .
θ ¯ ( 1 ) = 1 p 0 0 t dt′ [ x U eff ( r , t ) ] r = r 0 + v 0 t .
U eff ( r , t ) = d r U ( r + r ) Ψ c . m . ( 0 ) ( r , t ) 2 ,
Ψ c . m . ( 0 ) ( r , t ) 2 = 1 π 3 / 2 [ Δ r ( t ) ] 3 exp [ r 2 [ Δ r ( t ) ] 2 ] .
U eff ( r , t ) = α r Erf ( r Δ r ( t ) ) ,
U eff ( 0 , t ) = 2 α π Δ r ( t ) .
θ ¯ ( 1 ) = θ cl ( 1 ) [ 1 exp ( ρ 2 Δ r 0 2 ) ] ,
θ ( 1 ) Δ r 0 >> ρ ρ 2 Δ r 0 2 θ cl ( 1 ) .
θ ( 1 ) Δ r 0 > > ρ , d = 8 ρ m v 0 2 Δ r 0 4 0 r 2 dr U ( r ) .
θ ¯ ( 1 ) Δ r 0 > > ρ , d = 8 αρ d 2 m v 0 2 Δ r 0 4 = ( 2 ρ d Δ r 0 2 ) 2 θ cl ( 1 ) .
t spr > > t int = md p 0 .
( ƛ dB d ) 1 / 2 < < Δ r 0 < < d ,
( ƛ dB ρ ) 1 / 2 < < Δ r 0 < < ρ ,
10 2 cm < < Δ r 0 < < 10 5 cm .

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