Dirac Notation:
  1. State Function:
      2. Each state function is denoted by a ket (reason for Diracís choice of this name will soon be clear); written as | > individual kets are distinguished by labels placed inside the ket symbol, as |A>, |1>, |a,b> or |y >
       
  2. Observable Quantites:
      3. If we denote an observable quantity by Q , we will denote the corresponding quantum mechanical operator by Q (i.e. the same symbol, but bold-faced).

      A quantum mechanical operator operates on kets and transforms them into other kets, as
Q |A> = |B>
       
      Q is defined if its effect on all allowable kets is known.
      In general, quantum mechanical operators need not commute; i.e. Q1Q 2 |A> need not equal Q2Q1 |A>.

      The commutator of two operators (itself an operator)
[Q1,Q2] ºQ 1Q2 - Q 2Q1
       
      is a measure of whether or not two operators commute, and plays a very important role in quantum mechanics
       
  1. Eigenvalues and Eigenkets (Eigenvectors)
      2.  
      If Q |1> = a1 |1>
        -- i.e. the operator acts on |1> and returns |1> multiplied by a complex number a1 --
      |1> is called an eigenket of the operator Q and a1 is called the associated eigenvalue

      A ket is often labelled by its eigenvalues, as  Q |a1> = a1 |a1>

      The completeness postulate then says that an arbitrary ket |y > may be expanded in terms of the eigenkets of Q , as

      |y > = c1 |a1> + c2 |a2> + c3 |a3> + ........

      where the ci are called the expansion coefficients of |y > , and-- as noted before, |c1|2 = c1*c1 (with proper normalization) gives the probability that if a measurement of Q is made, the result will be a1
       

  2. Dual (Bra) Space and Scalar Products
      3.  
      To each ket |A>, there corresponds a dual or adjoint quantity called by Dirac a bra; it is not a ket-- rather it exists in a totally different space. The generalized scalar product is defined in analogy with the ordinary scalar product that you are familiar with as a combination of a bra and a ket to form a bracket (Diracís pun-- the reason for his nomenclature)-- thegeneralized scalar product is simply a complex number associated with a pair of kets.

      This notion is familiar for the case of a real vector space, where a real number (the scalar product denoted (x,y) or x· y) was associated with every pair of vectors x and y and was required to have certain properties (more on this below)

      You are also probably familiar with the generalization of this to complex vector spaces, where a single complex number (x,y) was associated with each pair of complex vectors

      The scalar product (as part of its definition) was required to have the following properties:

       
      (x,y) = (y,x)*

      (x,x) > 0 unless x=0

      (ax,y) = a* (x,y)

      (x,by) = b (x,y)
       

             according to the mathematicians, this defines the scalar product-- although we physicists are more familiar with evaluating it in a representation(more about this later) as
        (x,y) º S xi*yi
                    For the generalized scalar product (bracket), we will assume-- following the mathematicians-- that it has the same abstract properties as the more familiar scalar product defined above; here too, we will later find out how to evaluate the scalar product in a representation.
       
  3. A Bit More About Operators (some definitions and a theorem):
      4.  
      The Adjoint Operator: To each operator Q , we define a corresponding adjoint operator Q + so that the bra that corresponds to |Q f> is <fQ+| note that here Q + acts to the left on the bra <f|

      Hermitian (Self Adjoint) Operator:

      An operator Q is called Hermitian, or self-adjoint, if it is its ownadjoint operator; i.e. if

       
      <gQ |f> = <g|Q f>
                    Every quantum mechanical operator representing an observable must be Hermitian. The reason for this is the following famous theorem: The eigenvalues of a Hermitian Operator must be real.
       
       
       
  4. Linear Operators
      5.  
      In addition, the operators representing observables must be linear; this means that
        Q [ a |1> + b |2> ] = a Q |1> + b Q |2>
         
  5. Orthogonality
      6.  
      Definition: we say that |1> and |2> are orthogonal if <1|2> = 0

      Another famous theorem: eigenkets of the same Hermitian Operator having different eigenvalues are automaticallyorthogonal)

      (in general, if two kets have the same eignevalue, while they are not automatically orthogonal, they may be made orthogonal, provided they are linearly independent)
       

  6. Normalized Kets
      7.  
      a ket |k> is called normalizedif <k|k> = 1;

      two different kets are called orthonormalif

        <k|l> = dkl
         
  7. Proof of the Two Famous Theorems Regarding Hermitian Operators:
      8.  
      Let q be a Hermitian Operator and let |1> and |2> be two eigenfunctions of q with respective eigenvalues a1 and a2 q |1> = a1 |1>

      q |2> = a2 |2>

             Thus it follows that <2|q |1> = a1 <2|1> and

      <1|q |2> = a2 <1|2> or

      <2|q |1> = a2*<2|1>

             and, therefore (equating the two expressions for <2|q |1>), (a1 - a2*) <2|1> = 0 By setting |2> = |1> the first theorem follows, and

      by assuming a1 ? a2 the second theorem results.
       

      Finally, we show that if q |i> = ai |i>,

      <i|q+ = ai <i| one may expand <j|q+ = S cja<a | and compute <j|q+|i> = S cja<a |i> = cji = ai <j|i> = ai d ji establishing the desired result.
       
       
       
  8. Expansion of State Functions Again... Closure
      9.  
      the completeness postulate guarantees that if Q represents an observable, and Q |k> = k |k> then an arbitrary state function |y > may be expanded as |y > = S ck |k> if the eigenkets are orthonormal, then <l|y > = S ck <l|k> = S ck dlk = cl i.e. the expansion coefficient cl is the scalar product of |l> and the state function |y >

      This results in an important result called closure:

        since |y > = S ck |k> and ck = <k|y > we have |y > = S <k|y > |k> = S |k><k|y > and we can recognize the operator S |k><k| as the identity operator, i.e. I = S |k><k|
         
  9. Probability of Obtaining a Given Result
      10.  
      We recall that if Q is measured for a particle described by the state function |y >, the only results possible are the eigenvalues of Q , k; the probability of observing a particular k is given by |ck|2 = |<k|y >|2 assuming that |y > is normalized so that <y |y > = 1. This principle of expressing |y > as a superposition of eigenstates, and the statistical interpretation thereof is regarded by Dirac as perhaps the fundamental principle of Quantum Mechanics.
       
  10. Projection Operators
      11.  
      It is convenient to define yet one more operator, the projection operator, Pk-- which operates on |y > and "projects out" or "selects" only that part of |y > which has the eigenvalue k; thus
        Pk º |k> <k|


      so that Pk |y > = ck |k>

      It is clear that S Pk = I
       

  11. Expectation Values:
      12.  
      For a system described by the State Function |y >, if the observable Q is measured many times, the result will be on the average <y | Q |y >.... the so-called expectation value of Q ; this is exactly what one would expect, since expanding |y > in terms of the eigenstates of Q gives (using the orthonormality of the eigenstates):
         
        S ck*ck ak
         
      the weighted average obtained by assuming that ck*ck gives the probability of measuring ak
       
  12. Generalized Uncertainty Principle:
      14. Whenever the operators representing two observable A and B do not commute, it is not possible to exactly specify both of the observables; rather, there is an Uncertainty Principle which limits the accuracy of the simultaneous measurements, D A and D B according to the condition
        D A2 D B2 => < i [A , B] >2 / 4
                    see Townsend, sec. 3.5 (p.78) for the proof of this very important theorem
       
  14. A Corrolary :
      16. When [A,B] = 0 (i.e. A and B commute) and there is no required uncertainty, one can prove that one can find kets which are simultaneously eigenkets of both A and B:
        i.e. if [A,B] = 0 then there exists a ket |a,b> such that A|a,b> = a |a,b> and B|a,b> = b |a,b>