时间依赖的1D Schroedinger方程使用Numpy和SciPy solve_ivp

Question

我试图用有限差分方法求解一维时间依赖的Schroedinger方程，下面是方程看起来如何以及如何进行离散化

假设我有N个空间点（x_i从0变为N-1），并假设我的时间跨度是K个时间点。

我努力获得K×N矩阵。每行（j）将是时间t_j的函数

我怀疑我的问题是我以错误的方式定义了耦合方程的系统。

我的边界条件是psi = 0，或者在框的两侧有一些常数，所以我将X跨度两侧的颂歌设为零

我的代码：

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

#Defining the length and the resolution of our x vector
length = 2*np.pi
delta_x = .01

# create a vector of X values, and the number of X values
def create_x_vector(length, delta_x):
    x = np.arange(-length, length, delta_x)
    N = len(x)
    return x, N

# create initial condition vector
def create_initial_cond(x,x0,Gausswidth):
    psi0 = np.exp((-(x-x0)**2)/Gausswidth)
    return psi0

#create the system of ODEs
def ode_system(psi,t,delta_x,N):
    psi_t = np.zeros(N)
    psi_t[0]=0
    psi_t[N-1]=0
    for i in range(1,N-1):
        psi_t[i] = (psi[i+1]-2*psi[i]+psi[i-1])/(delta_x)**2
    return psi_t

#Create the actual time, x and initial condition vectors using the functions
t = np.linspace(0,15,5000)
x,N= create_x_vector(length,delta_x)
psi0 = create_initial_cond(x,0,1)

psi = np.zeros(N)
psi= solve_ivp(ode_system(psi,t,delta_x,N),[0,15],psi0,method='Radau',max_step=0.1)

运行后我得到一个错误：



runfile('D:/Studies/Project/Simulation Test/Test2.py', wdir='D:/Studies/Project/Simulation Test')
Traceback (most recent call last):

  File "<ipython-input-16-bff0a1fd9937>", line 1, in <module>
    runfile('D:/Studies/Project/Simulation Test/Test2.py', wdir='D:/Studies/Project/Simulation Test')

  File "C:UsersPashaAnaconda3libsite-packagesspyder_kernelscustomizespydercustomize.py", line 704, in runfile
    execfile(filename, namespace)

  File "C:UsersPashaAnaconda3libsite-packagesspyder_kernelscustomizespydercustomize.py", line 108, in execfile
    exec(compile(f.read(), filename, 'exec'), namespace)

  File "D:/Studies/Project/Simulation Test/Test2.py", line 35, in <module>
    psi= solve_ivp(ode_system(psi,t,delta_x,N),[0,15],psi0,method='Radau',max_step=0.1)

  File "C:UsersPashaAnaconda3libsite-packagesscipyintegrate\_ivpivp.py", line 454, in solve_ivp
    solver = method(fun, t0, y0, tf, vectorized=vectorized, **options)

  File "C:UsersPashaAnaconda3libsite-packagesscipyintegrate\_ivp
adau.py", line 288, in __init__
    self.f = self.fun(self.t, self.y)

  File "C:UsersPashaAnaconda3libsite-packagesscipyintegrate\_ivpase.py", line 139, in fun
    return self.fun_single(t, y)

  File "C:UsersPashaAnaconda3libsite-packagesscipyintegrate\_ivpase.py", line 21, in fun_wrapped
    return np.asarray(fun(t, y), dtype=dtype)

TypeError: 'numpy.ndarray' object is not callable

更一般地说，如何在不手动定义每个和其中一个的情况下使python解决N ode？

我想要一个名为xdot的大向量，这个向量中的每个单元格都是X [i]的函数，我似乎没有做到这一点？或者我的方法完全错了？

我也觉得ivp_solve的“Vectorized”参数可以连接，但我不理解SciPy文档中的解释。