Solving 3D TDSE with Runge-Kutta Method

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In summary, the conversation discusses the use of the Runge-Kutta method in 3-D to solve the 3-D TDSE and whether there are any differences compared to the 1-D method. The question also arises about implementation of the spatial part and the importance of boundary conditions. It is concluded that there should be no difference in using the Runge-Kutta method for the time step in 1-D or 3-D, and the wavefunction can be propagated forward in time using finite differences for partial derivatives with respect to spatial coordinates.
  • #1
thatboi
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Hey all,
For the Runge-Kutta method in 3-D (specifically to solve the 3-D TDSE), I was wondering if there were any subtleties I should expect, or if I could just simply use the 1-d method and add on the respective contributions from the other 2 dimensions.
Thanks.
 
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  • #2
I guess you mean applying a Runge-Kutta method for the time step. How do you plan to implement the spatial part?

In principle, there should be no difference between 1D and 3D for the time part.
 
  • #3
DrClaude said:
I guess you mean applying a Runge-Kutta method for the time step. How do you plan to implement the spatial part?

In principle, there should be no difference between 1D and 3D for the time part.
Since I just need to propagate the wavefunction forward in time I figured I could just discretize the space and use finite differences for any partial derivatives with respect to spacial coordinates.
 
  • #4
Sure, but what about boundary conditions?
 

1. What is the 3D TDSE?

The 3D TDSE (Time-Dependent Schrödinger Equation) is a mathematical equation that describes the evolution of a quantum system over time. It is used to model the behavior of particles in three-dimensional space and is a fundamental equation in quantum mechanics.

2. What is the Runge-Kutta Method?

The Runge-Kutta Method is a numerical method used to solve ordinary differential equations, such as the 3D TDSE. It is a popular method due to its accuracy and ability to handle complex systems. It involves breaking down the equation into smaller steps and using iterative calculations to approximate the solution.

3. Why is the Runge-Kutta Method used to solve the 3D TDSE?

The 3D TDSE is a complex equation that cannot be solved analytically. Therefore, numerical methods like the Runge-Kutta Method are used to approximate the solution. The Runge-Kutta Method is particularly useful for solving time-dependent equations, making it a suitable choice for the 3D TDSE.

4. What are the advantages of using the Runge-Kutta Method for solving the 3D TDSE?

The Runge-Kutta Method offers several advantages for solving the 3D TDSE. It is a highly accurate method, meaning that the solutions are close to the exact solution. It is also a stable method, meaning that small changes in the initial conditions do not greatly affect the results. Additionally, the Runge-Kutta Method can handle systems with multiple variables and complex boundary conditions.

5. Are there any limitations to using the Runge-Kutta Method for solving the 3D TDSE?

Like any numerical method, the Runge-Kutta Method has its limitations. It can become computationally expensive for large systems or when high accuracy is required. It also requires a significant amount of memory and processing power. Additionally, the Runge-Kutta Method may not be suitable for systems with highly oscillatory behavior or those with discontinuous solutions.

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