physics.hamiltonian

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:

Functions

classical_hamiltonian(→ float)

Compute the classical Hamiltonian H = T + V for a particle,

quantum_hamiltonian(→ list[list[float]])

Construct the Hamiltonian matrix for a particle in a 1D potential

Module Contents

physics.hamiltonian.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
physics.hamiltonian.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