Fightfish said:The time dependent Schrödinger equation is a partial differential equation, not an ordinary differential equation. One of the more commonly used finite difference schemes for numerically evolving the dynamics of a wavepacket is the Crank-Nicolson method.
Not the time-dependent Schrödinger equation.Frank Castle said:Yes, I appreciate that, but if one is solving it for the one-dimensional case it can be treated as an ODE.
DrClaude said:Not the time-dependent Schrödinger equation.
You can use ODE methods with the TDSE if you express it in terms of a basis set, since you then get a set of coupled ODEs for the basis coefficients.
Fightfish said:Technically you can still use RK4 (or related schemes) for the temporal evolution (while discretising the spatial Laplacian part of the Schrodinger equation via finite differences). For the same time step size, this will be faster than implicit methods such as Crank-Nicolson, but it is less stable and in general requires smaller step sizes in order to achieve sufficient accuracy, which is why the Crank-Nicolson method is still largely preferred in most cases.
Unless the number of discrete states of the system is small, grid methods are usually more efficient (fewer grid points are needed than basis functions for a similar problem).Frank Castle said:In general, are there cases where it is better to use a finite difference (interpolation) method than using an ODE solver such as ode45?
mfig said:Why not use PDEPE?
Finite difference numerical integration is a method used to approximate the solution of differential equations by discretizing the continuous problem into a set of algebraic equations. It involves dividing the continuous domain into a finite number of discrete points and approximating the derivatives at each point using finite difference equations.
Finite difference numerical integration differs from other numerical integration methods in that it uses a discrete approach to approximate the solution of differential equations, rather than an analytical approach. It is also a relatively simple and easy-to-implement method, making it a popular choice for solving a wide range of scientific problems.
Ode45 is a specific algorithm used for solving ordinary differential equations (ODEs) numerically. It is a Runge-Kutta method that uses a combination of fourth and fifth order accurate formulas to approximate the solution of the ODE. Ode45 is particularly useful for solving stiff ODEs, which are equations that exhibit very rapid changes in behavior.
The main advantage of using finite difference numerical integration is its simplicity and ease of implementation. It is also a very flexible method that can be applied to a wide range of problems. Additionally, it allows for the use of varying time steps, making it useful for solving problems with changing dynamics.
One limitation of finite difference numerical integration is that it can only approximate the solution of a differential equation, rather than providing an exact solution. Additionally, it may not be the most accurate method for problems with rapidly changing dynamics or steep gradients. It also requires a sufficient number of discrete points to accurately approximate the solution, which can be computationally expensive for large and complex problems.