# Using Variational Principle to solve ground state energy

• 1missing
In summary, the conversation discusses the use of two equations in solving for a ground state energy. The first approach was abandoned due to messy constants, and the second approach involved calculating a matrix and solving for its determinant. Despite following the correct method, the equation derived does not seem to have a rational solution, causing confusion for the individual. The conversation also mentions the use of a variational method in solving for the roots of the equation, and the expectation of integer solutions. However, the expert reviewing the conversation confirms that the work done was correct and there seems to be no issue with the calculations.
1missing
Homework Statement
A certain Hamiltonian can be expressed as: ##\hat H = | \varphi_1 \rangle \langle \varphi_1| + 2 | \varphi_2 \rangle \langle \varphi_2| + 3 | \varphi_3 \rangle \langle \varphi_3| ##

where, ##| \varphi_1 \rangle## , ##| \varphi_2 \rangle## , ##| \varphi_3 \rangle## are normalized but not fully orthogonal.

##\langle \varphi_1 | \varphi_2 \rangle = \langle \varphi_2 | \varphi_1 \rangle = \langle \varphi_2 | \varphi_3 \rangle = \langle \varphi_3 | \varphi_2 \rangle = \frac {1} {2}##

##\langle \varphi_1 | \varphi_3 \rangle = \langle \varphi_3 | \varphi_1 \rangle = 0##
Relevant Equations
1: ##E_\phi = \frac { \langle \phi | \hat H | \phi \rangle } { \langle \phi | \phi \rangle } ##

2: ##\sum_{j=1}^n (H_{ij} - ES_{ij} ) C_j = 0## , i = 1, 2,..., n
First I picked an arbitrary state ##|ϕ⟩=C_1|φ_1⟩+C_2|φ_2⟩+C_3|φ_3⟩## and went to use equation 1. Realizing my answer was a mess of constants and not getting me closer to a ground state energy, I abandoned that approach and went with equation two.

I proceeded to calculate the following matrix:
##\begin{pmatrix}

( \frac {3} {2} - E) & ( \frac {3} {2} - \frac {E} {2} ) & ( \frac {1} {2} ) \\

( \frac {3} {2} - \frac {E} {2} ) & ( 3 - E ) & ( \frac {5} {2} - \frac {E} {2} ) \\

( \frac {1} {2} ) & ( \frac {5} {2} - \frac {E} {2} ) & ( \frac {7} {2} - E )

\end{pmatrix}\vec C = 0##

Calculating the determinant, I end up with ##E^3 - 6E^2 + 9E - 3 = 0## which doesn't seem to have a rational solution. I must've gone through the calculation half a dozen times now, and confirmed with my professor that equation 2 was the one I should use, but I'm just not seeing how I'm supposed to find a solution.

1missing said:
I end up with ##E^3 - 6E^2 + 9E - 3 = 0## which doesn't seem to have a rational solution.
Why do you expect the solutions to be rational? Or did you mean real instead of rational? The equation does have 3 real solutions.

I don't see how any variational method is being used here.

TSny said:
Why do you expect the solutions to be rational? Or did you mean real instead of rational? The equation does have 3 real solutions.

I don't see how any variational method is being used here.
He usually models his exam questions off the homework problems he assigns. If he gave this problem on the exam there wouldn't be any way for me to solve for those roots, which made me think I messed up somewhere. If there were at least one integer solution, that term could be factored out by polynomial long division and I'd be left with a solvable quadratic for the other two.

He called the method "trial states as linear combinations of basis states". He starts with the variational principle, takes the derivative with respect to Ci, and sets it to zero. From that he derives equation 2.

OK, that sounds good. I checked the entries for your matrix and the equation coming from setting the determinant equal to zero. I get the same results. I don't see anything wrong with your work.

Einstein44

## 1. What is the Variational Principle?

The Variational Principle is a mathematical approach used in quantum mechanics to find the most accurate approximation of the ground state energy of a system. It states that the ground state energy of a system is always equal to or less than the energy of any other trial wavefunction.

## 2. How does the Variational Principle work?

The Variational Principle works by using a trial wavefunction, which is a mathematical representation of the state of a quantum system. This wavefunction is then optimized using mathematical techniques to find the lowest possible energy state of the system.

## 3. What is the significance of using the Variational Principle to solve ground state energy?

Using the Variational Principle allows us to find the most accurate approximation of the ground state energy of a quantum system. This is important because the ground state energy is the lowest possible energy state of a system, and understanding it is crucial in understanding the behavior of the system.

## 4. Can the Variational Principle be applied to any quantum system?

Yes, the Variational Principle can be applied to any quantum system, as long as a trial wavefunction can be constructed for that system. However, the accuracy of the results may vary depending on the complexity of the system and the quality of the trial wavefunction.

## 5. Are there any limitations to using the Variational Principle?

The Variational Principle is a powerful tool, but it does have its limitations. It can only provide an approximation of the ground state energy, and the accuracy of the results depends on the quality of the trial wavefunction. Additionally, it may not be able to accurately predict the behavior of systems with strong interactions or highly excited states.

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