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If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?
A density matrix is a mathematical representation used in quantum mechanics to describe the probability of finding a quantum system in a particular state. It is essentially a matrix of numbers that contains information about the probabilities of different states of the system.
A density matrix is said to be on a diagonal when all of its off-diagonal elements are equal to zero. This means that the system is in a pure state, where the probability of being in one specific state is 1 and the probabilities of all other states are 0.
A Hamiltonian is a mathematical operator that represents the total energy of a quantum system. It takes into account the kinetic and potential energies of the particles in the system and is used to describe the time evolution of the system.
The density matrix on a diagonal can be obtained by diagonalizing the Hamiltonian. This means finding the eigenvalues and eigenvectors of the Hamiltonian, which represent the energy levels and corresponding states of the system. The diagonal elements of the density matrix are then equal to the probabilities of the system being in these energy levels.
In some cases, the Hamiltonian of a system can be divided into smaller blocks that do not interact with each other. In these cases, the density matrix on a diagonal by blocks is used to describe the probabilities of finding the system in these separate blocks. This allows for a more efficient and accurate description of the system's behavior.