diff --git a/physics/hamiltonian.py b/physics/hamiltonian.py
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+++ b/physics/hamiltonian.py
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+"""
+The Hamiltonian is a central concept in both classical and quantum mechanics.
+It represents the total energy of a system and is used to describe how that
+system evolves over time.
+
+Classical mechanics:
+    H = T + V
+    where T is kinetic energy and V is potential energy.
+
+Quantum mechanics (1D, finite-difference form):
+    The time-independent Schrodinger equation, H|psi> = E|psi>, can be solved
+    numerically by discretizing space into points and approximating the second
+    derivative with the finite-difference method. This turns the continuous
+    Hamiltonian operator into a matrix:
+
+        H[i][i]   = hbar^2 / (m * dx^2) + V(x_i)
+        H[i][i+1] = H[i][i-1] = -hbar^2 / (2 * m * dx^2)
+
+References:
+    - https://en.wikipedia.org/wiki/Hamiltonian_mechanics
+    - https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)
+    - https://en.wikipedia.org/wiki/Finite_difference_method
+"""
+
+
+def classical_hamiltonian(
+    mass: float, velocity: float, potential_energy: float
+) -> float:
+    """
+    Compute the classical Hamiltonian H = T + V for a particle,
+    where T = 0.5 * m * v^2 is the kinetic energy.
+
+    >>> classical_hamiltonian(2, 3, 5)
+    14.0
+    >>> classical_hamiltonian(1, 0, 10)
+    10.0
+    >>> classical_hamiltonian(2, -4, 0)
+    16.0
+    >>> classical_hamiltonian(-1, 2, 5)
+    Traceback (most recent call last):
+        ...
+    ValueError: mass must be positive
+    """
+    if mass <= 0:
+        raise ValueError("mass must be positive")
+
+    kinetic_energy = 0.5 * mass * velocity**2
+    return kinetic_energy + potential_energy
+
+
+def quantum_hamiltonian(
+    num_points: int,
+    potential: list[float],
+    mass: float = 1.0,
+    hbar: float = 1.0,
+    dx: float = 1.0,
+) -> list[list[float]]:
+    """
+    Construct the Hamiltonian matrix for a particle in a 1D potential
+    using finite-difference discretization of the time-independent
+    Schrodinger equation.
+
+    >>> quantum_hamiltonian(3, [0.0, 0.0, 0.0])
+    [[1.0, -0.5, 0.0], [-0.5, 1.0, -0.5], [0.0, -0.5, 1.0]]
+    >>> quantum_hamiltonian(2, [1.0, 2.0], mass=2.0, hbar=1.0, dx=1.0)
+    [[1.5, -0.25], [-0.25, 2.5]]
+    >>> quantum_hamiltonian(3, [0.0, 0.0])
+    Traceback (most recent call last):
+        ...
+    ValueError: potential must have length num_points
+    >>> quantum_hamiltonian(0, [])
+    Traceback (most recent call last):
+        ...
+    ValueError: num_points must be positive
+    """
+    if num_points <= 0:
+        raise ValueError("num_points must be positive")
+    if len(potential) != num_points:
+        raise ValueError("potential must have length num_points")
+
+    diag_term = hbar**2 / (mass * dx**2)
+    off_diag_term = -(hbar**2) / (2 * mass * dx**2)
+
+    hamiltonian = [[0.0] * num_points for _ in range(num_points)]
+    for i in range(num_points):
+        hamiltonian[i][i] = diag_term + potential[i]
+        if i > 0:
+            hamiltonian[i][i - 1] = off_diag_term
+        if i < num_points - 1:
+            hamiltonian[i][i + 1] = off_diag_term
+
+    return hamiltonian
+
+
+if __name__ == "__main__":
+    from doctest import testmod
+
+    testmod()