Braket notation

 

In quantum mechanics spectroscopy it is frequently necessary to solve integrals of the type

where the Ym and Yn are wavefunctions, A (hat) is an operator, and the volume element dt represents the integral over all space. The significance of this expression is that it gives the average of the physical quantity corresponding to operator A for the wavefunctions in question. There are several equivalent ways of writing this integral. It is important to introduce these early in the process of teaching spectroscopy since they are frequently used and students often get confused by these. We can rewrite the above integral using the following different abbreviations

 

Keep in mind that all of these expression mean integration over all space. These expressions are referred to as matrix elements. This is because often there are many different wavefunctions for different electronic, vibrational, or rotational states of a molecule and the operator may represent the transitions between them. This is frequency the case in spectroscopy. For example, if the operator A is the dipole operator then the above matrix element represent transition dipoles between states m and n.

In the same manner the overlap of two states can be expressed as

Another way to view the above terms is that they represent the projection of one state onto another.

The shorthand notation with angle brackets is due to Dirac. This is known as braket notation. Note one important aspect of the shorthand notation is that the bra ám| implies that the complex conjugate of Ym* is represented. The ket |nñ represents Yn. Note that since

We have the identity

 

There is no particular significance to the choice of symbol for the wavefunction. We will use variously Y, y, f, c, or the ket |nñ to represent a wavefunction.

We require that average properties and probabilities are real. In other words we require that Y*Y be real. Wavefunctions that satisfy this condition are called Hermitian. Detailed discussion of the properties of Hermitian operators are given in Levine's "Quantum Chemistry" Chapter 7.