How to dertermine the Hamiltonian matrix

[hxwgter]

Hi, guys,

I do not know how to determine the Hamiltonian matrix of the following question with the basis of two stationary state. Pls give me some hints about it.

Consider first a single Hydrogen atom, made up of a proton at some location A in space, and an electron. We assume that the electron is in a stationary state with an energy centered around the position of the nucleus. We will investigate the effect of introducing a second proton, at location B on the stationary states and the energy of the system.

 

[azntoon]

The Hamiltonian matrix for this system can be determined by using the basis of two stationary states. The first state is the original single Hydrogen atom, and the second state is the two Hydrogen atom system. The Hamiltonian is then written as a matrix with each element representing the energy of each state. For example, if the energy of the single Hydrogen atom is E1 and the energy of the two Hydrogen atom system is E2, then the matrix will have elements E1 and E2. To determine the Hamiltonian matrix in this way, you need to calculate the energies of each state. This can be done by solving the Schrödinger equation for each state, or by using the variational principle.