RK4 to solve Schrödinger equation

Question

Let $m = 1$, $\hbar = 1$, the Schrödinger equation becomes: \begin{equation} \frac{\partial \psi}{\partial t}(x,t) = \frac{i}{2} \left( \nabla^2 - V(x)\right) \psi(x,t) \end{equation} given $x_k = x_0 + kh_x$, define $f_k(t) = \psi(x_k,t)$, then: \begin{equation} \frac{\partial f_k}{\partial t}(t) = \frac{\partial \psi}{\partial t}(x_k,t) = F_k(t,f_k(t)) \end{equation} where (aproximating the laplacian): \begin{align*} F_k(t,f_k)&=\frac{i}{2} \left( \frac{\psi(x_k+h_x,t)-2\psi(x_k,t)+\psi(x_k-h_x,t)}{h_x^2}- V(x_k)f_k \right) \\ F_k(t,f_k)&=\frac{i}{2} \left( \frac{f_{k+1}(t)-2f_k(t)+f_{k-1}(t)}{h_x^2}- V(x_k)f_k \right) \end{align*} let $t_n = t_0 + nh_t$, define: \begin{equation} f_k^n \approx f_k(t_n) \end{equation} we want to find \begin{equation} f_{k}^{n+1} \end{equation} given that we already know \begin{equation} f_k^0 = \psi(x_k,0) \end{equation} Using RK4: \begin{align*} k_1 &= F_k(t_n,f_k^n) \\ k_2 &= F_k(t_n+\frac{h_t}{2},f_k^n +\frac{h_t}{2}k_1) \\ k_3 &= F_k(t_n+\frac{h_t}{2},f_k^n +\frac{h_t}{2}k_2) \\ k_4 &= F_k(t_n+h_t,f_k^n + h_tk_3) \end{align*} then: \begin{equation} f_k^{n+1} = f_k^n + \frac{h_t}{6}(k_1 + 2k_2 + 2k_3 + k_4) \end{equation} usign derivative: \begin{align*} f_{k+1}(t_n + \frac{h_t}{2}) &= f_{k+1}(t_n) + \frac{\partial f_{k+1}}{\partial t}(t_n) \frac{h_t}{2} \\ f_{k+1}(t_n + \frac{h_t}{2}) &= f_{k+1}(t_n) + F_{k+1}(t_n, f_{k+1}(t_n))\frac{h_t}{2} \end{align*} introducing the notation: \begin{align*} F_{k}^n &= F_k(t_n, f_k^n) \\ f_{k+1}(t_n + \frac{h_t}{2}) &= f_{k+1}^n + F_{k+1}^n \frac{h_t}{2} \end{align*} likewise: \begin{align*} f_{k}(t_n + \frac{h_t}{2}) &= f_{k}^n + F_{k}^n\frac{h_t}{2}\\ f_{k-1}(t_n + \frac{h_t}{2}) &= f_{k-1}^n + F_{k-1}^n\frac{h_t}{2} \end{align*} replacing: \begin{align*} k_1 = F_k^n &=\frac{i}{2}(\nabla^2 - V)f_k^n \\ k_2 = F_k(t_n+\frac{h_t}{2},f_k^n +\frac{h_t}{2}k_1) &= \frac{i}{2} (\nabla^2 - V) \left( f_k^n +\frac{h_t}{2}k_1 \right) \\ k_3 = F_k(t_n+\frac{h_t}{2},f_k^n +\frac{h_t}{2}k_2) &= \frac{i}{2} \left( \nabla^2 \left( f_k^n +\frac{h_t}{2}k_1 \right) - V \left( f_k^n +\frac{h_t}{2}k_2 \right) \right) \\ k_4 = F_k(t_n+h_t,f_k^n +k_3) &= \frac{i}{2} \left( \nabla^2 \left( f_k^n + h_tk_1 \right) - V \left( f_k^n + h_tk_3 \right) \right) \end{align*} I am not sure if I arrive at the correct coefficients they don't look like this: