Variational method (quantum mechanics)

The variational method, also known as the Rayleigh-Ritz method, is a powerful technique in quantum mechanics for approximating the ground state energy (the lowest energy eigenvalue) and ground state wavefunction of a quantum system. It is particularly useful when solving the time-independent Schrödinger equation analytically is not possible.

The fundamental principle of the variational method is based on the variational principle, which states that the expectation value of the Hamiltonian operator, H, calculated with any arbitrary, well-behaved trial wavefunction, ψ, will always be greater than or equal to the true ground state energy, E₀. Mathematically, this is expressed as:

Here, $\backslash langle\; \backslash psi\; |\; H\; |\; \backslash psi\; \backslash rangle$ represents the expectation value of the Hamiltonian, and $\backslash langle\; \backslash psi\; |\; \backslash psi\; \backslash rangle$ is the normalization integral.

The method involves choosing a trial wavefunction, ψ, that contains one or more adjustable parameters. The expectation value of the energy is then calculated as a function of these parameters. The variational principle guarantees that the minimum value of this energy expectation value will be the best approximation to the true ground state energy achievable with the chosen trial wavefunction. The values of the adjustable parameters are then varied to minimize the energy expectation value.

The resulting energy is an upper bound to the true ground state energy. Furthermore, the trial wavefunction corresponding to the minimum energy provides an approximation to the true ground state wavefunction. While the energy is guaranteed to be an upper bound, the wavefunction is not guaranteed to be a good approximation. Often, the better the energy, the better the wavefunction approximation will be, but this is not always the case.

The accuracy of the variational method depends crucially on the choice of the trial wavefunction. A trial wavefunction that closely resembles the true ground state wavefunction will yield a more accurate energy estimate. However, even a relatively simple trial wavefunction can often provide a reasonably good approximation to the ground state energy.

The variational method can also be extended to approximate excited state energies and wavefunctions. However, for excited states, the variational principle only guarantees that the calculated energy will be an upper bound if the trial wavefunction is orthogonal to all lower energy eigenstates. In practice, enforcing this orthogonality can be challenging.

In summary, the variational method provides a systematic approach for approximating the ground state energy and wavefunction of quantum mechanical systems, particularly those for which analytical solutions are not available. The method relies on choosing a trial wavefunction with adjustable parameters and minimizing the energy expectation value with respect to these parameters. This minimum energy is an upper bound to the true ground state energy, and the corresponding trial wavefunction provides an approximation to the true ground state wavefunction.