advanced_physics_computational_methods.py

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# Advanced Physics Computational Methods and Theoretical Frameworks
# Comprehensive collection of modern physics calculations and simulations

import numpy as np
import matplotlib.pyplot as plt
from scipy import constants, integrate, optimize, linalg
from scipy.special import spherical_jn, sph_harm, hermite, laguerre
import sympy as sp
from sympy import symbols, diff, integrate as sp_integrate, solve, Matrix
import pandas as pd

# Quantum Mechanics Computational Framework
class QuantumMechanics:
    def __init__(self):
        self.hbar = constants.hbar
        self.m_e = constants.m_e
        self.e = constants.e
        self.c = constants.c
        
    def time_independent_schrodinger_1d(self, V, x_range, n_points=1000, mass=None):
        if mass is None:
            mass = self.m_e
            
        x = np.linspace(x_range[0], x_range[1], n_points)
        dx = x[1] - x[0]
        
        # Kinetic energy operator (second derivative)
        kinetic = -self.hbar**2 / (2 * mass) * np.diag(np.ones(n_points-1), 1) \
                  - 2 * self.hbar**2 / (2 * mass) * np.diag(np.ones(n_points)) \
                  + -self.hbar**2 / (2 * mass) * np.diag(np.ones(n_points-1), -1)
        kinetic /= dx**2
        
        # Potential energy operator
        potential = np.diag(V(x))
        
        # Hamiltonian matrix
        H = kinetic + potential
        
        # Solve eigenvalue problem
        eigenvalues, eigenvectors = linalg.eigh(H)
        
        return x, eigenvalues, eigenvectors
    
    def hydrogen_atom_wavefunction(self, n, l, m, r, theta, phi):
        # Radial part
        a0 = 4 * np.pi * constants.epsilon_0 * self.hbar**2 / (self.m_e * self.e**2)
        rho = 2 * r / (n * a0)
        
        # Associated Laguerre polynomial
        L = laguerre(n - l - 1)
        
        # Normalization constant
        N_nl = np.sqrt((2 / (n * a0))**3 * np.math.factorial(n - l - 1) / 
                      (2 * n * np.math.factorial(n + l)))
        
        # Radial wavefunction
        R_nl = N_nl * np.exp(-rho / 2) * rho**l * L(rho)
        
        # Angular part (spherical harmonics)
        Y_lm = sph_harm(m, l, phi, theta)
        
        # Complete wavefunction
        psi = R_nl * Y_lm
        
        return psi
    
    def quantum_harmonic_oscillator(self, n, x, omega, mass=None):
        if mass is None:
            mass = self.m_e
            
        # Characteristic length
        x0 = np.sqrt(self.hbar / (mass * omega))
        
        # Dimensionless coordinate
        xi = x / x0
        
        # Hermite polynomial
        H_n = hermite(n)
        
        # Normalization
        N = 1 / np.sqrt(2**n * np.math.factorial(n)) * (mass * omega / (np.pi * self.hbar))**(1/4)
        
        # Wavefunction
        psi = N * np.exp(-xi**2 / 2) * H_n(xi)
        
        return psi
    
    def quantum_tunneling_probability(self, E, V0, width, mass=None):
        if mass is None:
            mass = self.m_e
            
        if E > V0:
            return 1.0  # Classical case
        
        kappa = np.sqrt(2 * mass * (V0 - E)) / self.hbar
        
        # Transmission probability
        T = 1 / (1 + (V0**2 * np.sinh(kappa * width)**2) / (4 * E * (V0 - E)))
        
        return T

# Statistical Mechanics and Thermodynamics
class StatisticalMechanics:
    def __init__(self):
        self.k_B = constants.k
        self.N_A = constants.N_A
        self.R = constants.R
        
    def maxwell_boltzmann_distribution(self, v, T, mass):
        normalization = 4 * np.pi * (mass / (2 * np.pi * self.k_B * T))**(3/2)
        exponential = np.exp(-mass * v**2 / (2 * self.k_B * T))
        
        return normalization * v**2 * exponential
    
    def fermi_dirac_distribution(self, E, mu, T):
        return 1 / (np.exp((E - mu) / (self.k_B * T)) + 1)
    
    def bose_einstein_distribution(self, E, mu, T):
        return 1 / (np.exp((E - mu) / (self.k_B * T)) - 1)
    
    def partition_function_harmonic_oscillator(self, T, omega):
        beta = 1 / (self.k_B * T)
        hbar_omega = constants.hbar * omega
        
        return 1 / (2 * np.sinh(beta * hbar_omega / 2))
    
    def heat_capacity_einstein_model(self, T, einstein_temperature):
        x = einstein_temperature / T
        return 3 * self.R * (x / np.sinh(x/2))**2
    
    def heat_capacity_debye_model(self, T, debye_temperature):
        def debye_function(x):
            integrand = lambda t: (t**4 * np.exp(t)) / (np.exp(t) - 1)**2
            result, _ = integrate.quad(integrand, 0, x)
            return 9 * result / x**3
        
        x = debye_temperature / T
        return 3 * self.R * debye_function(x)

# Electromagnetic Theory and Optics
class Electromagnetism:
    def __init__(self):
        self.c = constants.c
        self.mu_0 = constants.mu_0
        self.epsilon_0 = constants.epsilon_0
        self.e = constants.e
        
    def electric_field_point_charge(self, q, r_source, r_field):
        r_vec = r_field - r_source
        r_mag = np.linalg.norm(r_vec)
        
        E = (1 / (4 * np.pi * self.epsilon_0)) * q * r_vec / r_mag**3
        
        return E
    
    def magnetic_field_current_loop(self, I, R, z):
        # Magnetic field along axis of circular current loop
        mu_0 = self.mu_0
        
        B_z = (mu_0 * I * R**2) / (2 * (R**2 + z**2)**(3/2))
        
        return B_z
    
    def electromagnetic_wave_propagation(self, k, omega, r, t, E0, direction):
        phase = np.dot(k * direction, r) - omega * t
        E = E0 * np.cos(phase)
        B = np.cross(direction, E) / self.c
        
        return E, B
    
    def fresnel_coefficients(self, n1, n2, theta_i):
        # Angle of refraction
        theta_t = np.arcsin(n1 * np.sin(theta_i) / n2)
        
        # s-polarized reflection coefficient
        r_s = (n1 * np.cos(theta_i) - n2 * np.cos(theta_t)) / \
              (n1 * np.cos(theta_i) + n2 * np.cos(theta_t))
        
        # p-polarized reflection coefficient
        r_p = (n2 * np.cos(theta_i) - n1 * np.cos(theta_t)) / \
              (n2 * np.cos(theta_i) + n1 * np.cos(theta_t))
        
        # Transmission coefficients
        t_s = 2 * n1 * np.cos(theta_i) / (n1 * np.cos(theta_i) + n2 * np.cos(theta_t))
        t_p = 2 * n1 * np.cos(theta_i) / (n2 * np.cos(theta_i) + n1 * np.cos(theta_t))
        
        return r_s, r_p, t_s, t_p
    
    def fabry_perot_transmission(self, wavelength, thickness, n, R):
        # Phase change
        delta = 4 * np.pi * n * thickness / wavelength
        
        # Finesse
        F = 4 * R / (1 - R)**2
        
        # Transmission
        T = 1 / (1 + F * np.sin(delta / 2)**2)
        
        return T

# Relativity and Spacetime Physics
class Relativity:
    def __init__(self):
        self.c = constants.c
        
    def lorentz_factor(self, v):
        beta = v / self.c
        return 1 / np.sqrt(1 - beta**2)
    
    def lorentz_transformation(self, x, t, v):
        gamma = self.lorentz_factor(v)
        beta = v / self.c
        
        x_prime = gamma * (x - beta * self.c * t)
        t_prime = gamma * (t - beta * x / self.c)
        
        return x_prime, t_prime
    
    def relativistic_energy_momentum(self, m0, v):
        gamma = self.lorentz_factor(v)
        
        E = gamma * m0 * self.c**2
        p = gamma * m0 * v
        
        return E, p
    
    def schwarzschild_radius(self, M):
        G = constants.G
        return 2 * G * M / self.c**2
    
    def time_dilation(self, v, proper_time):
        gamma = self.lorentz_factor(v)
        return gamma * proper_time
    
    def length_contraction(self, v, proper_length):
        gamma = self.lorentz_factor(v)
        return proper_length / gamma

# Solid State Physics
class SolidStatePhysics:
    def __init__(self):
        self.k_B = constants.k
        self.hbar = constants.hbar
        self.m_e = constants.m_e
        
    def free_electron_density_of_states(self, E, V):
        # 3D free electron gas
        return (3 * V / (2 * np.pi**2)) * (2 * self.m_e / self.hbar**2)**(3/2) * np.sqrt(E)
    
    def fermi_energy(self, n):
        # Electron density to Fermi energy
        return (self.hbar**2 / (2 * self.m_e)) * (3 * np.pi**2 * n)**(2/3)
    
    def bloch_wave(self, k, r, u_k):
        # Bloch theorem wavefunction
        return u_k * np.exp(1j * np.dot(k, r))
    
    def band_structure_1d(self, k, V0, a):
        # Nearly free electron model
        cos_ka = np.cos(k * a)
        E_free = (self.hbar * k)**2 / (2 * self.m_e)
        
        # Energy gaps at Bragg condition
        if np.abs(cos_ka) > 1:
            E_gap = V0
        else:
            E_gap = 0
            
        return E_free + E_gap
    
    def phonon_dispersion_1d_monatomic(self, k, M, K, a):
        # 1D monatomic chain
        omega = np.sqrt(4 * K / M) * np.abs(np.sin(k * a / 2))
        return omega
    
    def phonon_dispersion_1d_diatomic(self, k, M1, M2, K, a):
        # 1D diatomic chain
        mu = M1 * M2 / (M1 + M2)  # Reduced mass
        
        omega_plus = np.sqrt(K * (M1 + M2) / (M1 * M2)) * np.sqrt(1 + np.sqrt(1 - (4 * mu * np.sin(k * a / 2)**2) / (M1 + M2)))
        omega_minus = np.sqrt(K * (M1 + M2) / (M1 * M2)) * np.sqrt(1 - np.sqrt(1 - (4 * mu * np.sin(k * a / 2)**2) / (M1 + M2)))
        
        return omega_plus, omega_minus

# Nuclear and Particle Physics
class NuclearPhysics:
    def __init__(self):
        self.m_p = constants.m_p
        self.m_n = constants.m_n
        self.c = constants.c
        self.hbar = constants.hbar
        
    def binding_energy(self, A, Z):
        # Semi-empirical mass formula (SEMF)
        a_v = 15.75  # Volume term (MeV)
        a_s = 17.8   # Surface term (MeV)
        a_c = 0.711  # Coulomb term (MeV)
        a_a = 23.7   # Asymmetry term (MeV)
        a_p = 11.18  # Pairing term (MeV)
        
        N = A - Z
        
        # Pairing term
        if A % 2 == 0:  # Even A
            if Z % 2 == 0:  # Even Z, even N
                delta = a_p / np.sqrt(A)
            else:  # Odd Z, odd N
                delta = -a_p / np.sqrt(A)
        else:  # Odd A
            delta = 0
        
        BE = (a_v * A - 
              a_s * A**(2/3) - 
              a_c * Z**2 / A**(1/3) - 
              a_a * (N - Z)**2 / A + 
              delta)
        
        return BE
    
    def radioactive_decay(self, N0, t, half_life):
        decay_constant = np.log(2) / half_life
        return N0 * np.exp(-decay_constant * t)
    
    def cross_section_rutherford(self, Z1, Z2, E, theta):
        # Rutherford scattering cross section
        alpha = constants.fine_structure
        
        prefactor = (Z1 * Z2 * alpha * self.hbar * self.c / (4 * E))**2
        angular_factor = 1 / np.sin(theta / 2)**4
        
        return prefactor * angular_factor
    
    def neutron_absorption_cross_section(self, v):
        # 1/v law for thermal neutrons
        sigma_0 = 1e-24  # Reference cross section (cm^2)
        v_0 = 2200  # Reference velocity (m/s)
        
        return sigma_0 * v_0 / v

# Plasma Physics
class PlasmaPhysics:
    def __init__(self):
        self.k_B = constants.k
        self.e = constants.e
        self.m_e = constants.m_e
        self.m_p = constants.m_p
        self.epsilon_0 = constants.epsilon_0
        
    def plasma_frequency(self, n_e):
        # Electron plasma frequency
        return np.sqrt(n_e * self.e**2 / (self.m_e * self.epsilon_0))
    
    def debye_length(self, T_e, n_e):
        # Electron Debye screening length
        return np.sqrt(self.epsilon_0 * self.k_B * T_e / (n_e * self.e**2))
    
    def cyclotron_frequency(self, B, mass, charge):
        # Charged particle cyclotron frequency
        return charge * B / mass
    
    def magnetohydrodynamic_wave_speed(self, B, rho, mu_0):
        # Alfvén wave speed
        return B / np.sqrt(mu_0 * rho)
    
    def fusion_reaction_rate(self, T, sigma_v):
        # Reaction rate for fusion processes
        n_fuel = 1e20  # Typical fuel density (m^-3)
        return 0.25 * n_fuel**2 * sigma_v

# Computational Physics Methods
class ComputationalPhysics:
    def __init__(self):
        pass
    
    def finite_difference_laplacian_2d(self, u, dx, dy):
        # 2D Laplacian using finite differences
        u_xx = (np.roll(u, 1, axis=1) - 2*u + np.roll(u, -1, axis=1)) / dx**2
        u_yy = (np.roll(u, 1, axis=0) - 2*u + np.roll(u, -1, axis=0)) / dy**2
        
        return u_xx + u_yy
    
    def runge_kutta_4th_order(self, f, y0, t_span, h):
        # 4th order Runge-Kutta integration
        t = np.arange(t_span[0], t_span[1] + h, h)
        y = np.zeros((len(t), len(y0)))
        y[0] = y0
        
        for i in range(len(t) - 1):
            k1 = h * f(t[i], y[i])
            k2 = h * f(t[i] + h/2, y[i] + k1/2)
            k3 = h * f(t[i] + h/2, y[i] + k2/2)
            k4 = h * f(t[i] + h, y[i] + k3)
            
            y[i+1] = y[i] + (k1 + 2*k2 + 2*k3 + k4) / 6
        
        return t, y
    
    def monte_carlo_integration(self, func, bounds, n_samples):
        # Monte Carlo integration
        ndim = len(bounds)
        
        # Generate random points
        points = np.random.uniform(
            low=[b[0] for b in bounds],
            high=[b[1] for b in bounds],
            size=(n_samples, ndim)
        )
        
        # Evaluate function at random points
        values = np.array([func(point) for point in points])
        
        # Calculate volume of integration domain
        volume = np.prod([b[1] - b[0] for b in bounds])
        
        # Monte Carlo estimate
        integral = volume * np.mean(values)
        error = volume * np.std(values) / np.sqrt(n_samples)
        
        return integral, error
    
    def molecular_dynamics_verlet(self, positions, velocities, forces_func, dt, n_steps):
        # Velocity Verlet integration for molecular dynamics
        trajectory = np.zeros((n_steps, *positions.shape))
        
        for step in range(n_steps):
            forces = forces_func(positions)
            
            # Update positions
            positions += velocities * dt + 0.5 * forces * dt**2
            
            # Update velocities
            new_forces = forces_func(positions)
            velocities += 0.5 * (forces + new_forces) * dt
            
            trajectory[step] = positions.copy()
        
        return trajectory

# Advanced Mathematical Physics
class MathematicalPhysics:
    def __init__(self):
        pass
    
    def green_function_1d_wave(self, x, t, x_source, t_source, c):
        # Green's function for 1D wave equation
        if t <= t_source:
            return 0
        
        tau = t - t_source
        distance = np.abs(x - x_source)
        
        if distance <= c * tau:
            return 1 / (2 * c)
        else:
            return 0
    
    def fourier_transform_gaussian(self, x, sigma):
        # Analytical Fourier transform of Gaussian
        k = sp.symbols('k')
        gaussian = sp.exp(-x**2 / (2 * sigma**2))
        ft = sp.sqrt(2 * sp.pi) * sigma * sp.exp(-sigma**2 * k**2 / 2)
        
        return ft
    
    def legendre_polynomial_expansion(self, f, x_range, n_terms):
        # Expand function in Legendre polynomials
        from scipy.special import legendre
        
        coefficients = []
        x = np.linspace(x_range[0], x_range[1], 1000)
        
        for n in range(n_terms):
            P_n = legendre(n)
            coefficient = (2*n + 1) / 2 * integrate.trapz(f(x) * P_n(x), x)
            coefficients.append(coefficient)
        
        return coefficients
    
    def solve_laplace_equation_2d(self, boundary_conditions, grid_size):
        # Solve 2D Laplace equation using finite differences
        nx, ny = grid_size
        u = np.zeros((ny, nx))
        
        # Apply boundary conditions
        u[0, :] = boundary_conditions['top']
        u[-1, :] = boundary_conditions['bottom']
        u[:, 0] = boundary_conditions['left']
        u[:, -1] = boundary_conditions['right']
        
        # Iterative solution (Gauss-Seidel)
        for iteration in range(1000):
            u_old = u.copy()
            
            for i in range(1, ny-1):
                for j in range(1, nx-1):
                    u[i, j] = 0.25 * (u[i+1, j] + u[i-1, j] + u[i, j+1] + u[i, j-1])
            
            # Check convergence
            if np.max(np.abs(u - u_old)) < 1e-6:
                break
        
        return u

# Example Usage and Demonstrations
def physics_simulation_examples():
    # Initialize physics modules
    qm = QuantumMechanics()
    sm = StatisticalMechanics()
    em = Electromagnetism()
    rel = Relativity()
    solid = SolidStatePhysics()
    nuclear = NuclearPhysics()
    plasma = PlasmaPhysics()
    comp = ComputationalPhysics()
    math_phys = MathematicalPhysics()
    
    print("Advanced Physics Computational Framework")
    print("Modules: Quantum Mechanics, Statistical Mechanics, Electromagnetism,")
    print("Relativity, Solid State, Nuclear, Plasma, Computational, Mathematical")
    
    # Example: Quantum harmonic oscillator
    x = np.linspace(-5, 5, 1000)
    omega = 1.0
    for n in [0, 1, 2, 3]:
        psi = qm.quantum_harmonic_oscillator(n, x, omega)
        plt.plot(x, np.abs(psi)**2, label=f'n={n}')
    
    plt.xlabel('Position')
    plt.ylabel('Probability Density')
    plt.title('Quantum Harmonic Oscillator Wavefunctions')
    plt.legend()
    plt.show()
    
    return {
        'quantum_mechanics': qm,
        'statistical_mechanics': sm,
        'electromagnetism': em,
        'relativity': rel,
        'solid_state': solid,
        'nuclear': nuclear,
        'plasma': plasma,
        'computational': comp,
        'mathematical': math_phys
    }

if __name__ == "__main__":
    physics_modules = physics_simulation_examples()

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