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 .

## Leave a Reply