Hamiltonian

Hamiltonian(operator1, operator2, ...)

\[\hat{H} = \sum_i \hat{o}_i\]

The Hamiltonian is the input for each solver. This is an example for the non-relativistic Hamiltonian of hydrogen atom in atomic units:

\[\hat{H} = - \frac{1}{2} \nabla^2 - \frac{1}{r}\]

hamiltonian = Hamiltonian(
  NonRelativisticKinetic(ℏ =1 , m = 1),
  CoulombPotential(coefficient = -1),
)

NonRelativisticKinetic(ℏ=1, m=1)

\[-\frac{\hbar^2}{2m} \nabla^2\]

RestEnergy(c=1, m=1)

\[m c^2\]

Use c = 137.035999177 (from 2022 CODATA) in the atomic units.

RelativisticCorrection(c=1, m=1, n=2) The p^{2n} term of the Taylor expansion:

\[\begin{aligned} \sqrt{p^2 c^2 + m^2 c^4} =& m \times c^2 \\ &+ 1 / 2 / m \times p^2 (n=1) \\ &- 1 / 8 / m^3 / c^2 \times p^4 (n=2) \\ &+ 1 / 16 / m^5 / c^4 \times p^6 (n=3) \\ &- 5 / 128 / m^7 / c^6 \times p^8 (n=4) \\ &+ \cdots \end{aligned}\]

Use c = 137.035999177 (from 2022 CODATA) in the atomic units.

RelativisticKinetic(c=1, m=1)

\[\sqrt{p^2 c^2 + m^2 c^4} - m c^2\]

Use c = 137.035999177 (from 2022 CODATA) in the atomic units.

ConstantPotential(constant=1)

\[+ \mathrm{const.}\]

LinearPotential(coefficient=1)

\[+ \mathrm{coeff.} \times r \]

CoulombPotential(coefficient=1)

\[+ \mathrm{coeff.} \times \frac{1}{r}\]

PowerLawPotential(coefficient=1, exponent=1)

\[+ \mathrm{coeff.} \times r^\mathrm{expon.}\]

GaussianPotential(coefficient=1, exponent=1)

\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r^2)\]

ExponentialPotential(coefficient=1, exponent=1)

\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r)\]

YukawaPotential(coefficient=1, exponent=1)

\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r) / r\]

DeltaPotential(coefficient=1)

\[+ \mathrm{coeff.} \times δ(r)\]

FunctionPotential(f)

\[+ f(r)\]

UniformGridPotential(R, V)