Variational Method vs Inverse Rayleigh Method with Shifting

[HasuChObe]

I must fundamentally misunderstand what the variational method is. According to my textbook, it's used to find the minimum eigen energy of an operator (in particular, the time-independent Schrödinger equation). This appears to be synonymous to finding the eigenvalues of the matrix representation of the operator after you operate on some finite basis and project the result back to the same basis. My book doesn't go into detail about how to choose or vary the guesses for the coefficients of the of the basis expansion of the wave function. In the back of my mind, I can't help but think why you wouldn't just do the inverse rayleigh method with shifting, which is a pretty standard way to finding specific eigenvalues. Hopefully what I just wrote isn't too vague. If you need clarification on what I'm talking about, don't hesitate to ask.

Edit: I think I kind of see the difference when the expansion functions used aren't the actual eigenfunctions of the operator. I have to think carefully about what the cross terms mean though.. I'm guessing the minimum eigenvalue of the operator matrix when there are cross terms present is not the same as the minimum eigenvalue of the operator in general.

 

[azntoon]

The variational method is a way of approximating the ground state energy of a system. It involves starting with an ansatz for the wavefunction, and then using the variational principle to adjust the parameters of the ansatz until the energy is minimized. This can be done either analytically or numerically, depending on the system. The idea is that the optimal wavefunction will be a linear combination of a set of basis functions, and the variational method seeks to find the coefficients that minimize the energy. This approach is particularly useful for systems with many degrees of freedom, since it allows for an efficient search of the parameter space. Additionally, it can be used to calculate excited states by starting with an ansatz that includes more than one basis function.