Required math: calculus, vectors
Required physics: none
References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Section 3.6; Problem 3.22.
As a simple example of the bra-ket notation and the matrix representation of operators, suppose we have a three-dimensional vector space spanned by an orthonormal basis . We begin with a couple of vectors given by
The corresponding bras are found by taking the complex conjugate:
Remember that these bras are not vectors in their own right; rather, they are operators which acquire meaning only when they are applied to vectors to yield a complex number.
We can form the two inner products of these two vectors:
If we define an operator we can write it in terms of the basis above:
From here, we can obtain its matrix elements in this basis by using the orthonomal property of the basis. For example
Doing similar calculations for the other elements, we get
The matrix is not hermitian, as it is not equal to its conjugate transpose.