Determining the accuracy of the Variational Method

In summary, the Variational Method is a technique used to obtain an upper bound on the energy of the ground state and sometimes excited states. However, without an analytic or numerical solution, it may be difficult to determine the accuracy of this upper bound. To address this, there are other methods, such as finding a lower bound for the ground state energy, which can help estimate the true ground state energy. By averaging the upper and lower bounds, the error bars can be easily determined.
  • #1
joel31
2
0
The Variational Method allows us to obtain an upper bound on energy of the ground state (and sometimes excited states).

Is there any way of determining an upper bound on the error of the energy obtained by the variational method without an analytic or numerical solution to the problem?

i.e. Is there any way of knowing how close our resulting energy is to the true ground state energy?
 
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  • #2
In addition to the variational method for finding an upper bound for the energy, there are some ways to determine a lower bound for the ground state energy, as is described here: https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.21297

Then you can set the estimate of ground state energy to be the average of the upper and lower bounds and the error bars are obvious.
 

FAQ: Determining the accuracy of the Variational Method

1. How is the accuracy of the Variational Method determined?

The accuracy of the Variational Method is determined by comparing the calculated energy values to the exact energy values obtained through other methods, such as the Schrödinger equation. If the calculated values closely match the exact values, then the method is considered accurate.

2. What factors can affect the accuracy of the Variational Method?

Several factors can affect the accuracy of the Variational Method, including the choice of trial wavefunction, the number of basis functions used, and the choice of variational parameters. Additionally, the accuracy can also be affected by the complexity of the system being studied.

3. How does the Variational Method compare to other methods in terms of accuracy?

The Variational Method is generally considered to be less accurate than other more advanced methods, such as the Hartree-Fock method or density functional theory. However, it is still a valuable tool for obtaining approximate solutions and can provide insights into the behavior of quantum systems.

4. Can the accuracy of the Variational Method be improved?

Yes, the accuracy of the Variational Method can be improved by using more sophisticated trial wavefunctions, increasing the number of basis functions, and optimizing the variational parameters. Additionally, using more advanced techniques, such as incorporating perturbation theory, can also improve the accuracy.

5. Are there any limitations to using the Variational Method to determine accuracy?

Yes, the Variational Method has several limitations, including its inability to accurately describe systems with strong electron correlation and its reliance on the choice of trial wavefunction. It also becomes increasingly difficult to apply to larger and more complex systems, making it less practical for certain applications.

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