One of the most fundamental concepts in quantum theory lies in the idea that "xp" does not equal "px". We have seen elsewhere how pure numbers cannot behave in this fashion, but operators can. This is why we needed to alter our understanding of these physical properties from being represented by numbers to being represented by operators.
From our experience elsewhere, we also know that matrices have the same non-commutativity as operators and they too can be the mathematical formalism of quantum theory. In fact, Heisenberg's original approach to quantum theory, which is called Matrix Mechanics, was to find matrices x and p, such that xp - px = i
h - the commutation relation we seen elsewhere.
Matrices are represented as rectangular arrays of numbers arranged and indexed by the row and column number. For our present purposes, we remind you about matrix multiplication.
Each number in the matrix array is called a matrix element and is signified by Crc (for example, C13, being the element in row 1 and column 3). Because matrix multiplication is in general not commutative AB - BA is not zero.
We recall that in quantum theory we continually run into integrals of this form (in Dirac notation). Such an integral is sometimes further abbreviated as Wnm.
The correlation between this notation and that of natrices is obvious, and in fact such a integral is often called a matrix element.
In quantum theory we often encounter products of these integrals - these matrix elements - which take the form
Clearly, if we regard these as matrix elements, we can write
The sum is simply the matrix element of the product of the operators A and B and leads us to the conclusion that
This is known as the completeness relation and is widely used in quantum theory, general inthe reverse process of what we have just done, that is to break up a product of operators into a sum of the two operators separately.
Consider the Schrödinger equation, which is an eigenvalue equation using the Hamiltonian operator. Consider furthermore applying it to an arbitrary wavefunction which is a linear combination of eigenstates. We have
Now left multiply be some bra <m|.
Now suppose that we set out to find the collection of states such that the Hamiltonian matrix is a diagonal matrix, that is that all Hmn = 0 unless m=n. Then we have
Solving Schrödinger Equation is equivalent to diagonalizing the Hamiltonian Matrix! This is a direct link between Schrödinger's Wave Mechanics and Heisenberg's Matrix Mechanics. Indeed, it has been reported that when Heisenberg was looking for ways to diagonalize his matrices, Hilbert suggested that he look for the corresponding differential equation instead. If he had, he would have invented wave mechanics too!