Metodi approssimati Flashcards
What is the reason that many-electron systems are not exactly solvable in quantum mechanics?
The fundamental reason is the electron-electron interaction.
What is the goal of using approximate solutions to the Schrödinger equation?
The goal is to find solutions that are not too far from reality and can be compared to experimental data.
What is the variational theorem?
The variational theorem states that for any trial function ϕ and the ground state energy E0 of a system with Hamiltonian H, the expectation value of H is greater than or equal to E0.
What is the integral I in the variational method?
The integral I is the variational integral, which is the expectation value of the difference between the Hamiltonian and the ground state energy.
What is the process of finding the best variational function?
The process involves choosing a function with parameters and then minimizing the variational integral with respect to these parameters.
What is the importance of choosing a good trial function in the variational method?
Choosing a good trial function is important because if it is too different from the true wave function, the minimization process will not be effective.
What is the difficulty in approximating the wave function compared to approximating the energy?
Approximating the wave function is more difficult than approximating the energy.
What is the linear variational method?
It is a method used to determine the best approximation to the energy of a system by expressing the variational function as a linear combination of an appropriate set of functions.
What are the integrals used in the linear variational method?
The overlap integral (Sij) and the Hamiltonian integral (<fi|H|fj>).
What is the expression for the integral variazionale (W)?
W = <ϕ|H|ϕ>/<ϕ|ϕ> = (∑cjckHjk)/(∑cjckSjk).
What is the condition for finding the minimum of the integral variazionale?
∂W/∂ci = 0.
What is the condition for finding the minimum of the integral variazionale?
∂W/∂ci = 0.
What is the equation obtained by applying the condition for finding the minimum of the integral variazionale?
∑(Hik - SikW)ck = 0.
What is the condition for finding thenon trivial solutions of the system of equations obtained by applying the condition for finding the minimum of the integral variazionale?
determinante |Hik - SikW| = 0.
What is the role of the matrix S in the linear variational method?
It is used to transform the set of functions fi into a new set of functions in which the matrix S is diagonal and unitary.
