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## Background and physical system
Semiclassical trajectory models provide an interpretable account of continuum
electron motion, but their predictive value depends on how the outgoing
wavepacket is represented as trajectory input. Investigate whether a
quantum-informed trajectory description can give a quantitatively credible
account of strong-field ionization while retaining a semiclassical description
of continuum dynamics.
Work in atomic units with the one-dimensional single-electron potential
$$
V(x)=-\frac{1}{\sqrt{x^2+a^2}},\qquad a=1,
$$
and initialize the electron in its field-free ground state. The laser field is
defined by
$$
A(t)=(-1)^{n+1}\frac{F_0}{\omega}
\sin^2\!\left(\frac{\omega t}{2n}\right)
\sin(\omega t+\phi),
\qquad 0\leq t\leq t_f=\frac{2\pi n}{\omega},
$$
with $A(t)=0$ outside the pulse and $F(t)=-dA/dt$. Use
$$
n=4,\qquad F_0=0.0757,\qquad \omega=0.057,\qquad \phi=0.
$$
## Research problem
Assess whether a semiclassical two-step model with quantum input gives a
defensible account of the final photoelectron momentum distribution for this
target and pulse when judged against both a conventional one-dimensional
semiclassical two-step (SCTS) calculation and an independently converged direct
TDSE result. Determine which information carried by the quantum input is
material to the agreement or disagreement you establish, and delimit the
physical and numerical scope supported by your calculated evidence.
How you derive the model, organize the comparisons and construct evidence for
the final judgment is for you to decide. Base the conclusion on your own
calculated final photoelectron signal; qualitative resemblance by itself does
not establish model validity.
## Model identity and scope
In the quantum-input model, the trajectory input is the **complex**
position--momentum amplitude obtained from a Gaussian-window Gabor
representation of the newly outgoing wavefunction removed by an absorbing
boundary. Classical trajectories evolve in the stated laser and soft-core
fields. Their asymptotic momentum contributions are combined coherently, with
the task-specific semiclassical trajectory phase and the finite postpulse
potential contribution retained. These conditions define the commissioned
model; deriving a consistent computable realization remains part of the task.
All discretization, TDSE extraction, absorbing-boundary, Gabor-window,
sampling, propagation, asymptotic-matching, momentum-binning, normalization
and display choices not fixed above are yours. State choices that materially
affect your claims and provide evidence sufficiently reliable and
reconstructible to support them. A declared common scale may be used;
absolute yield is not a grading target unless independently established by
your calculation.
Do not add multielectron, three-dimensional, relativistic, focal-volume,
macroscopic-propagation or detector-response physics to the required
production model. Clearly separated extensions are permitted but cannot
replace the stated commission.
## Submission and resources
Submit one self-contained scientific report and an auditable offline
calculation package containing the task-specific derivation, code, numerical
data underlying all material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run
and environment information. No fixed file tree, filename scheme, panel count
or complete exploration history is required.
Resource limits are supplied exclusively by the accompanying
`resource_limits.json`. Its fixed CPU allocation, candidate-active
wall time, network boundary are binding. No separate aggregate CPU-hour limit applies. Greater computational expense
receives no additional credit.
## Reference and integrity policy
This examination is conducted without network access or live literature
retrieval. You may rely on your existing general scientific knowledge and, if any are
explicitly supplied with the examination, on those offline materials or
software documentation. Record every such source actually consulted. For formulas,
methods, constants, or conventions used from general knowledge rather than a
consulted source, state enough of the method basis, definition, value, units,
and convention to make the calculation auditable; do not invent a citation.
You are not required to identify or prove a publication date or to retrieve
literature during the examination.
The hidden target article and its identifying metadata, figures or captions,
supplementary material, author code or data, target-specific reproduction
code, solution or scoring material, online solutions, and worked
reproductions of this case must not be used. All submitted scientific figures
and numerical data must be generated by your own implementation of the stated
problem.
See [submission requirements](../../submission_requirements.md) for the common submission entry points.
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