Short review
      Short review
        Linear Operators
        Linear Operators
   •  V,W: Vector spaces.
      V,W: Vector spaces.
   •  A linear operator A from V to W is a linear function A:VW.  An 
                                                                      
      A linear operator A from V to W is a linear function A:V          W.  An 
      operator on V is an operator from V to itself.
      operator on V is an operator from V to itself.
   •  Given bases for V and W, we can represent linear operators as 
      Given bases for V and W, we can represent linear operators as 
      matrices.
      matrices.
                                                                        †
   •                                                                     †
      An operator A on V is Hermitian iff it is self-adjoint (A=A ).  
      An operator A on V is Hermitian iff it is self-adjoint (A=A ).  
   •  Its diagonal elements are real.
      Its diagonal elements are real.
  Eigenvalues & Eigenvectors
  Eigenvalues & Eigenvectors
  • v is called an eigenvector of linear operator A iff A just 
   v is called an eigenvector of linear operator A iff A just 
   multiplies v by a scalar x, i.e. Av=xv 
   multiplies v by a scalar x, i.e. Av=xv 
   – “eigen” (German) = “characteristic”.
    “eigen” (German) = “characteristic”.
  • x, the eigenvalue corresponding to eigenvector v, is just 
   x, the eigenvalue corresponding to eigenvector v, is just 
   the scalar that A multiplies v by.
   the scalar that A multiplies v by.
  • the eigenvalue x is degenerate if it is shared by 2 
   the eigenvalue x is degenerate if it is shared by 2 
   eigenvectors that are not scalar multiples of each other.
   eigenvectors that are not scalar multiples of each other.
   – (Two different eigenvectors have the same eigenvalue)
    (Two different eigenvectors have the same eigenvalue)
  • Any Hermitian operator has all real-valued eigenvectors, 
   Any Hermitian operator has all real-valued eigenvectors, 
   which are orthogonal (for distinct eigenvalues).
   which are orthogonal (for distinct eigenvalues).
       Exam Problems
       Exam Problems
 • Find eigenvalues and eigenvectors of operators.
  Find eigenvalues and eigenvectors of operators.
 • Calculate solutions for quantum arrays.
  Calculate solutions for quantum arrays.
 • Prove that rows and columns are orthonormal.
  Prove that rows and columns are orthonormal.
 • Prove probability preservation
  Prove probability preservation
 • Prove unitarity of matrices.
  Prove unitarity of matrices.
 • Postulates of Quantum Mechanics. Examples and 
  Postulates of Quantum Mechanics. Examples and 
  interpretations.
  interpretations.
 • Properties of unitary operators
  Properties of unitary operators
                Unitary Transformations
                Unitary Transformations
  • A matrix (or linear operator) U is unitary iff its inverse 
      A matrix (or linear operator) U is unitary iff its inverse 
                                        1        †
                                        1        †
      equals its adjoint:  U  = U
      equals its adjoint:  U  = U
  • Some properties of unitary transformations (UT):
      Some properties of unitary transformations (UT):
       – Invertible, bijective, one-to-one.
           Invertible, bijective, one-to-one.
       – The set of row vectors is orthonormal.
           The set of row vectors is orthonormal.
       – The set of column vectors is orthonormal.
           The set of column vectors is orthonormal.
       – Unitary transformation preserves vector length: 
           Unitary transformation preserves vector length: 
               |U| = | |
                        
               |U   | = |    |
            • Therefore also preserves total probability over all states:
               Therefore also preserves total probability over all states:
                                            2           (s )2
                                                      i
                                                      i
       – UT corresponds to a change of basis, from one orthonormal basis 
           UT corresponds to a change of basis, from one orthonormal basis 
          to another.
           to another.
       – Or, a generalized rotation of  in Hilbert space
           Or, a generalized rotation of  in Hilbert space
                                      Who an when invented all this stuff??
  A great 
  A great 
 breakthrough
 breakthrough