Examples of changing the coordinate system

From the separability theorem (Section 1.6.3) it follows that if an operator (e.g. the Hamiltonian) depending on N coordinates can be written as a sum of operators that only depend on one coordinate, the corresponding N coordinate wave function can be written as a product of one-coordinate functions, and the total energy as a sum of energies. 

Instead of solving one equation with N variables, the problem is transformed into solving N equations with only one variable. When the differential operator is transformed into a matrix representation, the separation is equivalent to finding a coordinate system where the representation is diagonal. 

Consider a matrix A expressed in a coordinate system xi, X2, X3. X v - The coordinate axes are the x, vectors, and these may be simple Cartesian axes, or one-variable functions, or many-variable functions. The matrix A is typically defined by an operator working on the coordinates. Some examples are  

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