加速求解一维薛定谔方程的C++代码

以下是一个简单的C++代码示例，用于求解一维薛定谔方程的时间演化。

#include <iostream>
#include <cmath>
#include <complex>
#include <vector>

const double hbar = 1.0;  // Planck constant / (2 * pi)
const double mass = 1.0;  // particle mass
const int num_points = 1000;  // number of spatial grid points
const double dx = 0.01;  // spatial step size
const double dt = 0.001;  // time step size
const double x0 = -5.0;  // initial position
const double sigma = 0.5;  // initial width of wave packet
const double k0 = 10.0;  // initial wave number

typedef std::complex<double> complex;

// Function to compute the potential energy at a given position x
double potential(double x) {
    // Example potential function: harmonic oscillator
    return 0.5 * mass * std::pow(x, 2);
}

// Function to compute the initial wave function at a given position x
complex initial_wavefunction(double x) {
    return std::exp(-std::pow(x - x0, 2) / (2 * std::pow(sigma, 2))) *
           std::exp(complex(0.0, k0 * x / hbar));
}

int main() {
    // Initialize the wave function
    std::vector<complex> wavefunction(num_points);
    for (int i = 0; i < num_points; ++i) {
        double x = x0 + i * dx;
        wavefunction[i] = initial_wavefunction(x);
    }

    // Time evolution loop
    for (int t = 0; t < 1000; ++t) {
        // Compute the potential energy at each point
        std::vector<double> potential_energy(num_points);
        for (int i = 0; i < num_points; ++i) {
            double x = x0 + i * dx;
            potential_energy[i] = potential(x);
        }

        // Update the wave function using the time-dependent Schrödinger equation
        std::vector<complex> new_wavefunction(num_points);
        for (int i = 1; i < num_points - 1; ++i) {
            complex psi = wavefunction[i];
            complex psi_plus = wavefunction[i + 1];
            complex psi_minus = wavefunction[i - 1];
            double v = potential_energy[i];

            new_wavefunction[i] = psi + complex(0.0, dt / (hbar * hbar)) *
                                              (psi_plus - 2.0 * psi + psi_minus) +
                                 complex(0.0, dt / hbar) * v * psi;
        }

        // Update the wave function at the boundaries
        new_wavefunction[0] = new_wavefunction[1];
        new_wavefunction[num_points - 1] = new_wavefunction[num_points - 2];

        // Normalize the wave function
        double norm = 0.0;
        for (int i = 0; i < num_points; ++i) {
            norm += std::norm(new_wavefunction[i]);
        }
        norm = std::sqrt(norm);
        for (int i = 0; i < num_points; ++i) {
            new_wavefunction[i] /= norm;
        }

        // Update the wave function for the next time step
        wavefunction = new_wavefunction;
    }

    // Print the final wave function
    for (int i = 0; i < num_points; ++i) {
        double x = x0 + i * dx;
        std::cout << x << "\t" << std::real(wavefunction[i]) << std::endl;
    }

    return 0;
}

在上述代码中，我们首先定义了一些常数和类型别名，然后实现了计算势能和初始波函数的函数。主要的求解过程在主函数main()中完成。我们首先初始化波函数，并在时间演化循环中计算势能和更新波函数。最后，我们将最终的波函数打印出来。

这个代码示例使用有限差分方法来近似求解一维薛定谔方程的时间演化。在循环中，我们使用一阶差分近似来计算波函数的二阶导数，并使用时间