Introducing State Functions and Dirac Bra-Ket Notation (Pt. 2)

 Given another general state function  f(x) =  <x |f> 

The overlap integral between the state functions  |f> and |y >  can be written:

ò <f | x ><x |y > dx =  < f | y > 

The Dirac notation then shows clearly this is just the generalization of the scalar product of two complex vectors to the infinite continuous case. If the overlap integral is zero then we have:

< f | y >   = 0

And the two state vectors are said to be orthogonal. (Perpendicular to each other).  It then follows from previous definitions:

< f | y > =  < y | f >*

The form   u _n(x)  can be used to denote the eigenvector of an operator A^  belonging to eigenvalue  a_n  and for which the general state function:  

|f >   = u _n

Then also: 

u _n(x)    =  < x | a_n >

The applicable eigenvalue equation including the operator  ( A^) can then be written:

A^ (x , ¶ / ¶ x) < x | a_n > =  a_n < x | a_n >

Which can be abbreviated to:

< A^ | a_n > =  < a_n | a_n >

The average value  a _f of repeated observations A^  on a normalized  state | y >  can be expressed:

 a _f =  ò <y | x > A^ (x , ¶ / ¶ x) <x |y > dx =   <y | A^ | y >

Suggested Problems:

1) Given eigenvalues a_n  and eigenvectors u_n of the operator  A^   find the eigenvectors if:

A^  is of the diagonal form (e.g. for the matrix)

 (a₁ …….0)

 (0……  a₂)

2)  In a quantum mechanics exam a student writes for the normalization condition:

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

Rewrite the above equation as it should be given

3) Prove that the eigenvalue a  for an observable operator A^  is real.

4) Let a₁  and  a₂  be different eigenvalues of  A^ .   Find: 

A^ | a₁ >  and A^ | a₂ >  and thence show:

<a₁| A^ |a₂ >    =   a₂ <a₁ | a₂  >

5) Show that u(x) = exp -(½x ²)   is an eigenfunction of the operator:

A^ (x , ¶ / ¶ x)  =  ( ¶ ² / ¶ x²   -  x² )

And find the corresponding eigenvalue