Up to now, I’ve used mathematical notation for state vectors, inner products and operators. However, physicists use something called “Dirac” notation (“bras” and “kets”) which we will now discuss.
Recall: our vectors are integrable functions where converges.
Our inner product is:
Here is the Dirac notation version of this:
A “ket” can be thought of as the vector . Of course, there is an easy vector space isomorphism (Hilbert space isomorphism really) between the vector space of state vectors and kets given by . The kets are denoted by .
Similarly there are the “bra” vectors which are “dual” to the “kets”; these are denoted by and the vector space isomorphism is given by . I chose this isomorphism because in the bra vector space, . Then there is a vector space isomorphism between the bras and the kets given by .
Now is the inner product; that is
By convention: if is a linear operator, and Now if is a Hermitian operator (the ones that correspond to observables are), then there is no ambiguity in writing .
This leads to the following: let be an operator corresponding to an observable with eigenvectors and eigenvalues . Let be a state vector.
Then and if is a random variable corresponding to the observed value of , then and the expectation .