Formulate Quantum Problem

Physical system → well-posed QM problem: ID DOFs → build H + Hilbert space → BCs → pick approx method → validate vs known limits.

Use When

• Set up QM problem for analytic/numerical solution
• QChem calc (MOs, electronic structure)
• Physical scenario → Dirac/Schrödinger
• Choose perturbation / variational / DFT / exact diag
• Theoretical model for spectroscopic/scattering comparison

In

• Required: system desc (atom, molecule, solid, field)
• Required: target observable (spectrum, rates, ground state)
• Optional: experimental constraints
• Optional: accuracy / compute budget
• Optional: formalism (wave mech, matrix mech, 2nd quant, path int)

Do

Step 1: ID system + DOFs

1. Particles: list (electrons, nuclei, photons, phonons) + quantum nums (spin, charge, mass)
2. Symmetries: spatial (sph/cyl/trans/crystal), internal (spin/gauge), discrete (P, T)
3. Energy scales: which DOFs active vs frozen/adiabatic
4. Reduction: Born-Oppenheimer if nuclear/electronic timescales separate; collective coords for many-body

## System Characterization
- **Particles**: [list with quantum numbers]
- **Active degrees of freedom**: [coordinates, spins, fields]
- **Frozen degrees of freedom**: [and justification for freezing]
- **Symmetry group**: [continuous and discrete]
- **Energy scale hierarchy**: [e.g., electronic >> vibrational >> rotational]

→ Complete inventory: particles, QNs, symmetries, active vs frozen justified.

If err: hierarchy unclear → keep all DOFs, flag for scale analysis. Premature truncation → wrong physics.

Step 2: Build H + Hilbert space

1. Hilbert space: finite-dim → basis (|↑>, |↓>). Infinite → function space (L²(R³) for 3D single particle).
2. Kinetic: each particle. Position rep: T = -ℏ²/(2m) nabla².
3. Potential: all interactions (Coulomb, harmonic, spin-orbit, external). Explicit form + coupling.
4. Composite H: H = T + V, group by type. Multi-particle: exchange/correlation or note via approx.
5. Operator algebra: H Hermitian? Constants of motion ([H,O]=0) → block-diagonalize.

## Hamiltonian Structure
- **Hilbert space**: [definition and basis]
- **H = T + V decomposition**:
  - T = [kinetic terms]
  - V = [potential terms, grouped by type]
- **Constants of motion**: [operators commuting with H]
- **Symmetry-adapted basis**: [if block diagonalization is possible]

→ Complete Hermitian H w/ all terms, Hilbert space defined, constants of motion ID'd.

If err: not Hermitian → missing conjugate / gauge phase. Ambiguous Hilbert space (relativistic) → specify formalism.

Step 3: BCs + initial conditions

1. BCs: bound → normalizability (psi→0 at ∞). Scattering → incoming wave. Periodic → Bloch / Born-von Karman.
2. Domain: spatial. Box walls. H atom: radial + angular. Lattice + topology.
3. Initial state (time-dep): t=0 expansion in eigenbasis or wave packet w/ center + width.
4. Constraints: indistinguishable → sym (bosons) / antisym (fermions). Gauge → gauge-fixing.

## Boundary and Initial Conditions
- **Spatial domain**: [definition]
- **Boundary type**: [Dirichlet / Neumann / periodic / scattering]
- **Normalization**: [condition]
- **Particle statistics**: [bosonic / fermionic / distinguishable]
- **Initial state** (if time-dependent): [specification]

→ BCs physically motivated, consistent w/ H domain, unique solution (or scattering matrix).

If err: over/under-determined → check self-adjointness on domain. Non-self-adjoint → handle deficiency indices.

Step 4: Pick approx method

1. Exact solvable: matches known model (HO, H atom, Ising)? → exact + perturbation corrections.

2. Perturbation (weak coupling):

   • H = H0 + lambda V, H0 solvable
   • lambda V small vs H0 level spacing
   • Degeneracy? → degenerate perturbation
   • Fits: weak interaction, few-body, analytic needed

3. Variational (ground state):

   • Trial wf w/ params
   • Satisfies BCs + symmetry
   • Fits: ground state energy primary, many-body

4. DFT (many-electron):

   • XC functional (LDA, GGA, hybrid)
   • Basis (plane waves, Gaussian, NAOs)
   • Fits: many-electron, ground state density + energy

5. Numerical exact (small/benchmark):

   • Exact diag for small Hilbert
   • QMC for ground state sampling
   • DMRG for 1D/quasi-1D
   • Fits: high accuracy, small system

## Approximation Method Selection
- **Method chosen**: [name]
- **Justification**: [why this method fits the problem structure]
- **Expected accuracy**: [order of perturbation, variational bound quality, DFT functional accuracy]
- **Computational cost**: [scaling with system size]
- **Alternatives considered**: [and why they were rejected]

→ Justified method + expected accuracy + compute cost + alternatives.

If err: no single method fits → formulate for 2 + compare. Disagreement reveals difficulty.

Step 5: Validate vs limits

1. Classical (ℏ→0 or large QNs): H reduces to classical mech.
2. Non-interacting: couplings → 0 → product of single-particle states.
3. Symmetry: respects all symmetries. H transforms correctly under group.
4. Dimensional: every H term = energy. Length/energy/time scales reasonable.
5. Known exact: special cases (H atom Z=1, HO quadratic) → reproduced.

## Validation Checks
| Check | Expected Result | Status |
|-------|----------------|--------|
| Classical limit (hbar -> 0) | [classical Hamiltonian] | [Pass/Fail] |
| Non-interacting limit | [product states] | [Pass/Fail] |
| Symmetry transformation | [correct representation] | [Pass/Fail] |
| Dimensional analysis | [all terms in energy units] | [Pass/Fail] |
| Known exact case | [reproduced result] | [Pass/Fail] |

→ All pass. Self-consistent, ready to solve.

If err: fail → err in H construction or BCs. Trace to specific term/condition, fix before solving.

Check

• Particles + QNs listed
• Hilbert space + basis
• H Hermitian + correct units
• Constants of motion used
• BCs physically + mathematically sufficient
• Statistics (bos/ferm) enforced
• Method justified + accuracy stated
• Classical, non-interacting, symmetry limits checked
• Known exact reproduced
• Reproducible

Traps

• Premature DOF drop: freeze w/o energy scale arg → wrong physics. Justify every reduction.
• Non-Hermitian H: missing conjugate in spin-orbit / complex V. Verify H=H† explicitly.
• Wrong scattering BCs: bound-state BCs for scattering → discards continuum. Match to question.
• Degeneracy in perturbation: non-deg perturbation on deg level → divergent. Check first.
• Single-method reliance: variational → upper bound but misses excited; perturbation diverges at strong coupling. Cross-validate.
• Unit inconsistency: mixing natural (ℏ=1) + SI. Pick consistent system, state explicitly.

→

• derive-theoretical-result — analytic from formulated problem
• survey-theoretical-literature — prior work on similar QM systems