(#13235) [feat] Add classical and quantum Hamiltonian functions with examples

physics/hamiltonian.py

@@ -0,0 +1,158 @@
+"""
+Hamiltonian functions for classical and quantum mechanics.
+
+This module provides two educational, minimal implementations:
+
+- classical_hamiltonian(m, p, V): Computes H = T + V for a particle/system, where
+  T = p^2 / (2 m) is the kinetic energy expressed in terms of momentum p, and V is
+  the potential energy (can be a scalar or an array broadcastable to p).
+
+- quantum_hamiltonian_1d(m, hbar, V, dx): Builds the 1D Hamiltonian matrix for a
+  particle in a potential V using second-order central finite differences for the
+  kinetic energy operator: T = - (hbar^2 / 2m) d^2/dx^2 with Dirichlet boundaries.
+
+These functions are intended for learners to quickly prototype and simulate basic
+physical systems.
+
+References
+----------
+- Classical Hamiltonian mechanics: https://en.wikipedia.org/wiki/Hamiltonian_mechanics
+- Discrete 1D Schrödinger operator: https://en.wikipedia.org/wiki/Finite_difference_method
+"""
+
+from __future__ import annotations
+
+from typing import Any
+
+import numpy as np
+
+
+def classical_hamiltonian(m: float, p: Any, V: Any) -> Any:
+    """
+    Classical Hamiltonian H = T + V with T = p^2 / (2 m).
+
+    The function supports scalars or array-like inputs for momentum ``p`` and
+    potential energy ``V``; NumPy broadcasting rules apply. If inputs are scalars,
+    a float is returned; otherwise a NumPy array is returned.
+
+    Parameters
+    ----------
+    m : float
+        Mass (must be positive).
+    p : array-like or scalar
+        Canonical momentum.
+    V : array-like or scalar
+        Potential energy evaluated for the corresponding configuration.
+
+    Returns
+    -------
+    float | np.ndarray
+        The Hamiltonian value(s) H = p^2/(2m) + V.
+
+    Examples
+    --------
+    Free particle with p = 3 kg·m/s and m = 2 kg (V = 0):
+    >>> classical_hamiltonian(2.0, 3.0, 0.0)
+    2.25
+
+    Harmonic oscillator snapshot with vectorized p and V:
+    >>> m = 1.0
+    >>> p = np.array([0.0, 1.0, 2.0])
+    >>> V = np.array([0.5, 0.5, 0.5])  # e.g., 1/2 k x^2 at three positions
+    >>> classical_hamiltonian(m, p, V).tolist()
+    [0.5, 1.0, 2.5]
+    """
+    if m <= 0:
+        raise ValueError("Mass m must be positive.")
+
+    p_arr = np.asarray(p)
+    V_arr = np.asarray(V)
+
+    T = (p_arr * p_arr) / (2.0 * m)
+    H = T + V_arr
+
+    # Preserve scalar type when both inputs are scalar
+    if np.isscalar(p) and np.isscalar(V):
+        return float(H)
+    return H
+
+
+def quantum_hamiltonian_1d(m: float, hbar: float, V: Any, dx: float) -> np.ndarray:
+    """
+    Construct the 1D quantum Hamiltonian matrix using finite differences.
+
+    Discretizes the kinetic operator with second-order central differences and
+    Dirichlet boundary conditions (wavefunction assumed zero beyond endpoints):
+
+        H = - (hbar^2 / 2m) d^2/dx^2 + V
+
+    On a uniform grid with spacing ``dx`` and N sites, the Laplacian is
+    approximated by the tridiagonal matrix with main diagonal ``-2`` and
+    off-diagonals ``+1``. The resulting kinetic term has main diagonal
+    ``(hbar^2)/(m*dx^2)`` and off-diagonals ``-(hbar^2)/(2*m*dx^2)``.
+
+    Parameters
+    ----------
+    m : float
+        Particle mass (must be positive).
+    hbar : float
+        Reduced Planck constant (can be set to 1.0 in natural units).
+    V : array-like shape (N,)
+        Potential energy values on the grid. Defines the matrix size.
+    dx : float
+        Grid spacing (must be positive).
+
+    Returns
+    -------
+    np.ndarray shape (N, N)
+        The Hermitian Hamiltonian matrix.
+
+    Examples
+    --------
+    Free particle (V=0) on a small grid: main diagonal = 1/dx^2, off = -1/(2*dx^2) in units m=hbar=1.
+    >>> N, dx = 5, 0.1
+    >>> H = quantum_hamiltonian_1d(m=1.0, hbar=1.0, V=np.zeros(N), dx=dx)
+    >>> float(H[0, 0])
+    99.99999999999999
+    >>> float(H[0, 1])
+    -49.99999999999999
+
+    Add a harmonic-like site potential to the diagonal:
+    >>> x = dx * (np.arange(N) - (N-1)/2)
+    >>> V = 0.5 * x**2  # k=m=omega=1 for illustration
+    >>> H2 = quantum_hamiltonian_1d(1.0, 1.0, V, dx)
+    >>> np.allclose(np.diag(H2) - np.diag(H), V)
+    True
+    """
+    if m <= 0:
+        raise ValueError("Mass m must be positive.")
+    if dx <= 0:
+        raise ValueError("Grid spacing dx must be positive.")
+
+    V_arr = np.asarray(V, dtype=float)
+    if V_arr.ndim != 1:
+        raise ValueError("V must be a 1D array-like of potential values.")
+
+    N = V_arr.size
+    if N == 0:
+        raise ValueError("V must contain at least one grid point.")
+
+    coeff_main = (hbar * hbar) / (m * dx * dx)
+    coeff_off = -0.5 * (hbar * hbar) / (m * dx * dx)
+
+    # Build tridiagonal kinetic matrix
+    H = np.zeros((N, N), dtype=float)
+    np.fill_diagonal(H, coeff_main)
+    idx = np.arange(N - 1)
+    H[idx, idx + 1] = coeff_off
+    H[idx + 1, idx] = coeff_off
+
+    # Add the potential on the diagonal
+    H[np.arange(N), np.arange(N)] += V_arr
+    return H
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod(verbose=True)