Numerical solution to SE - variational method, many electrons

In summary, the conversation discusses the use of the Variational method (VM) to find the electron wavefunction of a system. The problem at hand involves a simple system, an infinitely deep square quantum well, and the question is whether a specific procedure can accurately determine the wavefunction of a second electron. The proposed procedure involves finding the ground state and its wavefunction for the system without any electrons, then adding in one electron and re-calculating the ground state and wavefunction. It is acknowledged that this procedure only takes into account the influence of the first electron on the second, and not vice versa. It is suggested to look into the Hartree-Fock method, which is a commonly used approach in quantum chemistry.
  • #1
Nemanja989
79
2
Hi everyone,

I am trying to find electron wavefunction of a system I am working in. Numerical method I choose is the Variational method (VM). This method is convenient to find the ground state of the system. More details are available here.

Problem I have can be explained on a very simple system like an infinitely deep square quantum well, [itex] U_0 [/itex] .

Let's say we have two electrons, and we are thinking how the wavefunction of the electron in the higher state looks like. We would also like to take into account electric potential these two electrons have on each other.

Now my question is, would the following procedure make sense:

1. we first find the ground state and its wavefunction with the VM of the infinite well with no electrons, [itex] E_1 [/itex] and [itex] \psi_1[/itex] .
2. then we "fill" that state with an electron.
3. we now have a new "structure" which consists of a quantum well plus the electron. We construct a new potential [itex] U_{new}=U_0-e|\psi_1|^2 [/itex].
4. then we find the ground state and its wavefunction, [itex] E_2 [/itex] and [itex] \psi_2[/itex], which corresponds to the potential [itex] U_{new}[/itex].

What do you think, does [itex] \psi_2[/itex] corresponds to the real value of the wavefunction of the second electron?

I understand that in this procedure it is only taken into account influence of the first electron to the second, and there is no influence of the second electron to the first. This is a clear drawback of this procedure, but I cannot estimate if this would be a big problem or not.

Is there anyone who has experience with this kind of problems? I assume this is a routine problem in quantum chemistry.
 
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  • #2
You should look up "Hartree-Fock" method. There are many programs, also freely available to solve quantum chemical problems, so there is no need to program it yourself.
 

FAQ: Numerical solution to SE - variational method, many electrons

What is the variational method in quantum mechanics?

The variational method is a technique used in quantum mechanics to approximate solutions to the Schrödinger equation. It involves finding a trial wavefunction and minimizing its energy with respect to any free parameters, such as the electron-electron repulsion term.

How does the variational method work?

The variational method works by using a trial wavefunction, which is a proposed solution to the Schrödinger equation. This wavefunction is then used to calculate the energy of the system. The energy is then minimized by varying the parameters of the trial wavefunction until the lowest possible energy is obtained, which is an approximation to the true energy of the system.

What is the significance of the variational method in quantum mechanics?

The variational method is significant because it allows for the calculation of approximate solutions to the Schrödinger equation for many-electron systems, which cannot be solved exactly. It also provides a way to improve the accuracy of the solution by using more sophisticated trial wavefunctions.

What are the limitations of the variational method?

One limitation of the variational method is that it is an approximation technique and therefore may not provide an exact solution to the Schrödinger equation. The accuracy of the solution also depends on the choice of the trial wavefunction, which may not always be known or easily determined. Additionally, the variational method may be computationally intensive, especially for systems with a large number of electrons.

How is the variational method used in practical applications?

The variational method is used in practical applications, such as computational chemistry, to calculate the electronic structure and properties of molecules and materials. It is also used in the field of condensed matter physics to study the behavior of many-electron systems, such as superconductors and semiconductors. Additionally, the variational method is used in quantum chemistry software packages to predict the behavior of chemical systems.

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