A diagonal matrix is a type of square matrix in which all the elements outside of the main diagonal (which runs from the top left corner to the bottom right corner) are zero. This means that all the nonzero elements are located on the main diagonal.
In quantum mechanics, a matrix in "position representation" represents a physical observable in terms of position coordinates. This means that the matrix elements correspond to the probabilities of a particle being in a given position.
The operator r, also known as the position operator, is used to represent the position of a particle in quantum mechanics. It is a diagonal matrix in position representation, with its diagonal elements corresponding to the possible position values of the particle.
A diagonal matrix represents the position operator in position representation because the elements of the diagonal matrix correspond to the eigenvalues of the position operator. This means that the diagonal elements represent the possible position values of a particle in a given state.
The position representation is important in quantum mechanics because it allows us to describe the position of a particle and calculate the probabilities of finding the particle in a specific position. It also helps us understand the relationship between the position operator and other physical observables in quantum mechanics.