Physics 227 Lecture 7 Appendix A 1 Autumn 2007
Lecture 7 – Appendix A: Dirac (Bra Ket) Notation
In Lectures 6 and 7 we have developed some facility with vector and matrix notation,
including an effort to be able to think in terms of general vectors and linear operators.
In concrete notation, i.e., with an explicit choice of basis vectors, these objects are
represented by N-tuples of numbers (1-D arrays) and matrices (2-D arrays). In this
Appendix we will attempt to connect the previous discussion to a notation due to Paul
Dirac that is standard in discussions of Quantum Mechanics and in some pure
mathematics. In QM we are typically interested in describing the state of a (quantum
mechanical) system and the function that provides this description is called the “state
vector”. The (fairly) standard Dirac notation for such a vector is in terms of the
“ket”,
2
.
r
(A.1)
In similar notation the Hermitian conjugate vector is represented by a “bra”,
1
,
r
(A.2)
and the familiar scalar product become the “bra-ket” (bracket) expression
1 2
.
r r
(A.3)
If we “operate” on the state
with a linear operator
, we have a (in principle) new
state vector
(A.4)
and a “matrix element” of the operator given by
.
M
(A.5)
If the state vector describes an electron and we want to find the “wave function” of
the elector as a function of its location, we can do so by “projecting” onto a state
vector corresponding to a localized position,
x
. In the Dirac notation this looks like
,
x x
(A.6)
