Numerical solution of the time-dependent Schr

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• Numerical solution of the time-dependent
  Schrödinger equation (TDSE)

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Hydrogen atom in a laser field
Time-Dependent Schrödinger Equation (TDSE)
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Dipole approximation
?????(Differential equation)
?????(Difference equation)
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3??(3D)
?3??(3D)
??
???????(Dipole approximation)?? m 0 ????
?2??(2D)!
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Peaceman-Rachford ?(??????)
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atomic Hamiltonian
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( l ??)
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Interaction
l ?
???(couple)???
???(selection rule)???
???
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???(Selection rule)
??(???????)???????
???
Selection rule
??????????
s
p
d
f
????(allowed transition)
????(forbidden transition)
1s???m 0????????????????m?????
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Peaceman-Rachford ?(????)
temporal evolution
j ?? l ????3???(tridiagonal)
l ?? j ????3???(tridiagonal)
Alternating directions implicit scheme

• t ????2????(2nd order in t)
• ????????(unitary)

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???????????
??
2p
10.2 eV
1s
??(mm)???????(eV)1.24
1.24 eV
1 eV
??????
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2p
1s
8
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???
10.2 eV
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??
1 ????????
1 ?????
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1????(??)
Single-photon ionization (theory)
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????(1s)???????
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1????(????????)
Single-photon ionization (simulation)
Energy of the ejected electrion
Evolution of ionization in time
??
?????????????
??
Ip13.6 eV
????
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Photon energy dependence of single-photon
ionization
Simulation
theory
?????????
????????????????????? ????????????
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1??????????
Intensity-dependence of single-photon ionization
2?
??(linear)??? ??????(linear optical effect)
Ionization
Intensity
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?????(3????)
Multi-photon ionization (3-photon ionization)
????40fs
???????
3
Ionization
Intensity
???(nonlinear)??? ???????(nonlinear optical
effect)
??? n ??????
n
Ionization
Intensity
??????? ? ????????????????
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?????(????????????)
Electron energy spectrum
???????????
??
Ip
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