# Variational Theory: Ground vs Non-Ground States

• greisen
In summary, variational theory is a mathematical approach used to approximate the ground state of a quantum system. It involves finding the best possible approximation to the ground state by varying a trial wave function and minimizing the energy. The ground state is the lowest energy state, while non-ground states are higher energy states. Variational theory works by using a trial wave function that is varied and optimized to minimize energy. It allows for the calculation of approximate solutions to quantum systems and can estimate the ground state energy and other properties. However, it has limitations such as only providing an approximation and being dependent on the choice of trial wave function. It may also not be applicable to highly complex systems and does not provide information about excited states.
greisen
Hi,

I am trying to understand the variational principle

(E_n - E_0) >= 0

E_0 is the ground state of the system? E_n is nonground and will the nonground state than always have a higher energy than the ground state?

Yes, by definition. The ground sate is defined as the state with lowest energy (i.e., if you know the energy spectrum of a system, you label the lowest energy state as the "ground state").

PS: The equation you wrote above is not a statement of the variational principle.

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## 1. What is variational theory?

Variational theory is a mathematical approach used to approximate the ground state (lowest energy state) of a quantum system. It involves finding the best possible approximation to the ground state by varying a trial wave function and minimizing the energy.

## 2. What is the difference between ground and non-ground states?

The ground state is the lowest energy state of a quantum system, while non-ground states are higher energy states. In variational theory, the goal is to find the best approximation to the ground state, as it is the most stable and physically meaningful state.

## 3. How does variational theory work?

Variational theory works by using a trial wave function, which is a mathematical function that approximates the true wave function of a quantum system. The trial wave function is then varied and optimized in order to minimize the energy, resulting in an approximation of the ground state.

## 4. What are the advantages of using variational theory?

Variational theory allows for the calculation of approximate solutions to quantum systems that cannot be solved exactly. It also provides a way to estimate the ground state energy and other properties of a system, which can be useful in understanding and predicting the behavior of physical systems.

## 5. Are there any limitations to variational theory?

Yes, there are some limitations to variational theory. It can only provide an approximation to the ground state, and the accuracy of the approximation depends on the choice of trial wave function. Additionally, it may not be applicable to highly complex systems, and it does not provide any information about excited states.

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