Introducing State Functions and Dirac Bra-Ket Notation For Quantum Mechanics. (Part 1)

The necessity for using operators to represent the operations of measurement on quantum systems was noted in previous posts, e.g.

Introduction to Quantum Mechanical Operators (Pt.1)

For example, the Hamiltonian for the hydrogen atom Schrodinger equation can also be written in concise form using two operators ( H _op   and  E_op  ) as:

H _op y  =  E_op  y

Where: 

E_op  =  i h  (¶ /¶t)

And:

H _op    = p^r ²/ 2m +  ℓ^²/ 2mr² + V(r)

which contains two further operators:

p^r    = - i ħ (1/r  ¶² /¶ r² )

And:

  ℓ^²  =

- ħ²/ sin² q [ sin q ¶ / ¶ q (sin q ¶ / ¶ q)  + ¶² / ¶ j² ]

This paves the way for introducing a new notation, due to Paul M. Dirac, for describing quantum systems and operations.  In this regard, it is useful to use analogies to describing electron spin. Thus, Dirac's notation is designed to exploit this analogy to the fullest.  

In Dirac's notation we write the general state function (not an eigenfunction), of any given observable :

y (x)  = <x |y >

And its complex conjugate:

y* (x)  = < y | x >

By analogy the spin vector has two components designated by:

  <+ ½ |y >       and:    <- ½ |y >

To refresh memories I cite my earlier post on electron spin:

 An Introduction to Quantum Mechanics (2) 

Wherein the spin quantization is shown:

This has associated spin vector S _z values  of + ½.

The expression <x |y >   can be thought of in the same way, i.e. as the x-component of a state vector |y >.  Only now we have an infinity of components specified by the continuous variable x which run together to form the state function: 

y (x)  = <x |y >

The normalizing condition can then be written:

ò < y | x ><x |y > dx =     ò |< y | x >|²   dx =    1

with the integration over the full physical range of the integration variable.  In terms of state vectors this can be abbreviated:

< y | y > =   1