Variational Method and Energy Upper Bound

Using the variational method, find an upper bound on the energy of the ground state for a quantum system. If the trial function deviates by an order $\epsilon$, how does this affect the upper bound?

In quantum mechanics, one of the fundamental challenges is to calculate the energy levels of a quantum system. The variational method is a powerful tool for estimating the ground state energy, particularly for systems where exact solutions are difficult to obtain. At the heart of this method is the concept of a trial wavefunction, which is an educated guess that approximates the true ground state wavefunction. The goal is to minimize the expectation value of the Hamiltonian with respect to this trial wavefunction to find an upper bound for the ground state energy.

The variational principle states that for any trial wavefunction, the expectation value of the Hamiltonian provides an upper bound to the true ground state energy. This means that by systematically improving the trial wavefunction, you can get a tighter bound on the energy. The choice of trial function is crucial; often, functions that closely resemble the expected behavior of the system's true wavefunction under known conditions yield the best results. The closer the trial function matches the true wavefunction, the lower the expectation value, and thus, the tighter the upper bound.

In practical terms, if the trial function deviates by a small amount, say by an order epsilon, this will proportionally affect the calculated upper bound. Specifically, the variational method's sensitivity means that small deviations can lead to noticeable changes in the calculated energy. Understanding this dependency helps in assessing the accuracy of your approximation and guides you in choosing a better trial function for improved results. This iterative process of refining the trial function is vital for effectively employing the variational method in quantum mechanics.