Variational Method: Best Estimate of Ground State Energy

[hhhmortal]

Homework Statement

In the variational method the ground state energy of a quantum system has been calculated as (ћω_o)f(λ), where f(λ) is a function of an arbitrary parameter λ. If f(λ)= λ² + λ, then what is the best estimate of the ground state energy.

The Attempt at a Solution

Is this simply (ћω_o)f(λ² + λ). As the parameters of the ground state wave function have been found which give the lowest expectation value for the ground state energy.

 

[Jhero]

Thank you for your question. In the variational method, the ground state energy is calculated by minimizing the expectation value of the energy with respect to a trial wavefunction that depends on an arbitrary parameter λ. In this case, f(λ) represents this trial wavefunction and can be written as f(λ) = λ² + λ.

To find the best estimate of the ground state energy, we need to minimize the expectation value of the energy with respect to λ. This can be done by taking the derivative of the expectation value with respect to λ and setting it equal to zero. Solving for λ will give us the optimal value for λ that minimizes the energy.

Once we have the optimal value for λ, we can plug it back into the expression for f(λ) and then multiply by the constant (ћω_o) to obtain the best estimate of the ground state energy.

I hope this helps clarify the process for finding the best estimate of the ground state energy in the variational method. Please let me know if you have any further questions or if I can be of any assistance.