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.

 

[denian]

Determining the Hamiltonian matrix is a crucial step in understanding the behavior of a system, such as the one described in your question. The Hamiltonian matrix is a mathematical representation of the total energy of a system, and it can be used to calculate the energy levels and the corresponding eigenstates of the system.

To determine the Hamiltonian matrix for the given system, you will need to consider the contributions of both protons and the electron to the total energy. The Hamiltonian matrix can be written as a sum of the kinetic energy operator and the potential energy operator.

In this case, the kinetic energy operator will include the kinetic energy of the electron, which is given by the expression p^2/2m, where p is the electron's momentum and m is its mass. The potential energy operator will include the Coulombic interaction between the two protons and the electron, given by the expression -e^2/r, where e is the elementary charge and r is the distance between the particles.

To determine the Hamiltonian matrix, you will need to express the kinetic energy operator and the potential energy operator in terms of the basis states of the system. In this case, the basis states are the stationary states of the electron around the nucleus. Once you have the Hamiltonian matrix in terms of the basis states, you can solve for the eigenvalues and eigenvectors, which will give you the energy levels and the corresponding stationary states of the system.

I hope this helps to give you a starting point for determining the Hamiltonian matrix for this system. It may also be helpful to consult a textbook or consult with a colleague for further guidance. Good luck!