quantum-mechanics

# Quantum Mechanics ## What I Do I provide comprehensive quantum mechanics tools including wave function manipulation, operator algebra, Schrödinger equation solving, quantum measurement, superposition, entanglement, and perturbation theory for physics and quantum computing applications. ## When to Use Me - Quantum state evolution - Atomic and molecular systems - Quantum computing operations - Spectroscopy calculations - Quantum measurement theory - Perturbation analysis ## Core Concepts - **Wave Functions**: Probability amplitudes, normalization - **Operators**: Position, momentum, Hamiltonian - **Schrödinger Equation**: Time-dependent and time-independent - **Superposition**: Linear combinations of states - **Entanglement**: Non-local quantum correlations - **Uncertainty Principle**: Position-momentum uncertainty - **Angular Momentum**: Spin, orbital angular momentum - **Perturbation Theory**: Non-degenerate and degenerate ## Code Examples ### Quantum States and Basis ```python import numpy as np from scipy.linalg import expm ket_0 = np.array([1, 0]) ket_1 = np.array([0, 1]) plus = (ket_0 + ket_1) / np.sqrt(2) minus = (ket_0 - ket_1) / np.sqrt(2) print(f"|+⟩ = {plus}") print(f"|-⟩ = {minus}") def normalize(state): return state / np.linalg.norm(state) def inner_product(psi, phi): return np.vdot(psi, phi) print(f"⟨0|1⟩ = {inner_product(ket_0, ket_1)}") ``` ### Quantum Operators ```python sigma_x = np.array([[0, 1], [1, 0]]) sigma_y = np.array([[0, -1j], [1j, 0]]) sigma_z = np.array([[1, 0], [0, -1]]) identity = np.eye(2) def commutator(A, B): return A @ B - B @ A print(f"[σx, σy] = {commutator(sigma_x, sigma_y)}") def expectation(operator, state): return np.real(np.vdot(state, operator @ state)) psi = plus print(f"⟨σz⟩ in |+⟩: {expectation(sigma_z, psi)}") ``` ### Time Evolution ```python def time_evolution(psi0, H, t): U = expm(-1j * H * t) return U @ psi0 H = np.array([[1, 0], [0, -1]]) t = np.pi / 2 psi_t = time_evolution(ket_0, H, t) print(f"State at t=π/2: {psi_t}") def adiabatic_evolution(psi0, H_initial, H_final, T, steps=1000): psi = psi0.copy() dt = T / steps for i in range(steps): s = i / steps H = (1 - s) * H_initial + s * H_final U = expm(-1j * H * dt) psi = U @ psi return psi ``` ### Hydrogen Atom Wave Functions ```python from scipy.special import spherical_jn, eval_hermite from numpy.polynomial.hermite import hermval def hydrogen_wavefunction(n, l, m, r, theta, phi, a0=1): R = 0 if n == 1 and l == 0: R = 2 * np.exp(-r / a0) / a0**1.5 elif n == 2 and l == 0: R = (1 / np.sqrt(8)) * (2 - r / a0) * np.exp(-r / (2 * a0)) / a0**1.5 elif n == 2 and l == 1: R = (1 / np.sqrt(24)) * (r / a0) * np.exp(-r / (2 * a0)) / a0**1.5 Y = spherical_jn(l, m) return R * Y r = np.linspace(0, 5, 100) psi_1s = hydrogen_wavefunction(1, 0, 0, r, np.pi/4, 0) print(f"1s wavefunction at r=1: {psi_1s[50]:.4f}") ``` ### Spin Systems ```python def spin_eigenvalues(S): return np.arange(S, -S - 1, -1) def spin_raising(S, m): return np.sqrt(S * (S + 1) - m * (m + 1)) S = 1/2 m_values = spin_eigenvalues(S) print(f"Spin-1/2 eigenvalues: {m_values}") def pauli_vector(theta, phi): return np.array([ np.sin(theta) * np.cos(phi), np.sin(theta) * np.sin(phi), np.cos(theta) ]) spin_direction = pauli_vector(np.pi/3, np.pi/4) print(f"Spin direction: {spin_direction}") ``` ## Best Practices 1. **Normalization**: Always normalize states 2. **Complex Numbers**: Handle complex arithmetic carefully 3. **Hermitian Operators**: Ensure observables use Hermitian matrices 4. **Units**: Use consistent units (atomic units often simplest) 5. **Basis Choice**: Choose appropriate basis for problem ## Common Patterns ```python # Density matrix def density_matrix(state): return np.outer(state, np.conj(state)) def partial_trace(rho, subsystem, dims): traced = np.trace(rho, axis1=subsystem, axis2=subsystem + len(dims)) return traced # Quantum entropy def von_neumann_entropy(rho): eigenvalues = np.linalg.eigvalsh(rho) return -sum(e * np.log(e + 1e-15) for e in eigenvalues if e > 0) ``` ## Core Competencies 1. Wave function manipulation 2. Operator algebra and expectations 3. Schrödinger equation solving 4. Quantum measurement theory 5. Perturbation analysis