Imaginary time propagation to find eigenfunctions

In summary, the conversation discusses the use of imaginary time propagation to find ground state and excited states eigen functions. The results obtained are different from the analytical solution and it is mentioned that multiple orthogonal functions need to be propagated to find excited states. The question is raised about the requirement for the initial guess used for propagation and it is mentioned that the starting point for finding excited states must be orthogonal to the ground state. It is also stated that starting from a random guess requires it to be first made orthogonal to the ground state.
  • #1
semc
368
5
Hi, I have been trying to use imaginary time propagation to get the ground state and excited states eigen function but the results I got is different from the analytical solution. I know that to get excited states, I should propagate 2 or more orthogonal functions depending on the number of excited states that you want. I just would like to check whether there is any requirement on the initial guess that you use to propagate. If I choose a random set of numbers to propagate, will it always contain my ground state?
 
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  • #2
semc said:
If I choose a random set of numbers to propagate, will it always contain my ground state?
Most probably. After having found the ground state, any starting point to find the first excited state must be orthogonal to the ground state. If you start from a random guess, you need to first make it orthogonal to the ground state.
 
  • #3
Hey semc.

What is your mathematical system you are solving for?
 

Related to Imaginary time propagation to find eigenfunctions

1. What is imaginary time propagation?

Imaginary time propagation is a mathematical technique used in quantum mechanics to find the eigenfunctions of a system. It involves transforming the time variable in the Schrödinger equation from real time to imaginary time, which simplifies the equation and allows for easier calculation of the system's eigenfunctions.

2. How does imaginary time propagation work?

Imaginary time propagation works by transforming the time variable in the Schrödinger equation to an imaginary value, typically denoted as τ. This transforms the equation from a time-dependent problem to a steady-state problem, making it easier to solve for the system's eigenfunctions. The resulting eigenfunctions can then be transformed back to real time to obtain the system's wavefunction.

3. When is imaginary time propagation used?

Imaginary time propagation is often used in quantum mechanics when solving for the ground state of a system. This is because the lowest energy state of a system is equivalent to the steady-state solution in imaginary time. It is also useful for systems that exhibit complex dynamics, as it can help to simplify the calculation of the system's eigenfunctions.

4. What are the advantages of using imaginary time propagation?

There are several advantages of using imaginary time propagation. Firstly, it simplifies the Schrödinger equation, making it easier to solve for the system's eigenfunctions. It also allows for the calculation of the ground state energy of a system, which is often difficult to obtain through other methods. Additionally, it can be used to calculate the eigenfunctions of systems with complex dynamics, which may be difficult to solve for using other techniques.

5. Are there any limitations to using imaginary time propagation?

While imaginary time propagation can be a useful tool in solving for eigenfunctions, it does have some limitations. It is only applicable to systems that exhibit time-independent behavior, and it may not accurately capture the behavior of systems with rapidly-changing dynamics. Additionally, it can only be used to find the lowest energy state of a system, and other methods may be needed to find higher energy states.

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