Self-adjoint operator | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-03-17

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Self-adjoint_operator


00:02:55 1 Bounded self-adjoint operators
00:04:33 2 Symmetric operators
00:04:43 2.1 Subtleties of the unbounded case
00:05:33 2.2 Definition of a symmetric operator
00:08:30 2.3 A simple example
00:11:21 3 Self-adjoint operators
00:11:31 3.1 Definition of a self-adjoint operator
00:14:19 3.2 Essential self-adjointness
00:14:26 3.3 Geometric interpretation
00:14:36 3.4 An example
00:15:11 4 The distinction between symmetric and self-adjoint operators
00:17:36 4.1 Boundary conditions
00:17:50 4.2 Schrödinger operators with singular potentials
00:19:06 5 Spectral theorem
00:19:42 5.1 Statement of the spectral theorem
00:28:46 5.2 Functional calculus
00:35:12 5.3 Resolution of the identity
00:38:35 5.4 Formulation in the physics literature
00:40:43 6 Extensions of symmetric operators
00:44:05 6.1 Self-adjoint extensions in quantum mechanics
00:46:01 7 Von Neumann's formulas
00:50:32 8 Examples
00:58:09 8.1 A symmetric operator that is not essentially self-adjoint
01:00:07 8.2 Constant-coefficient operators
01:02:19 9 Spectral multiplicity theory
01:02:28 9.1 Uniform multiplicity
01:06:03 9.2 Direct integrals
01:12:09 9.3 Example: structure of the Laplacian
01:12:49 10 Pure point spectrum
01:14:56 11 See also
01:16:54 12 Citations
01:18:16 13 References
01:19:14 See also



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SUMMARY
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In mathematics, a self-adjoint operator (or Hermitian operator) on a finite-dimensional complex vector space V with inner product 
  
    
      
        ⟨
        ⋅
        ,
        ⋅
        ⟩
      
    
    {\displaystyle \langle \cdot ,\cdot \rangle }
   is a linear map A (from V to itself) that is its own adjoint: 
  
    
      
        ⟨
        A
        v
        ,
        w
        ⟩
        =
        ⟨
        v
        ,
        A
        w
        ⟩
      
    
    {\displaystyle \langle Av,w\rangle =\langle v,Aw\rangle }
   for all vectors v and w. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.
Self-adjoint operators are used in functional analysis and quantum mechanics.  In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator 
  
    
      
        
          
            
              H
              ^
            
          
        
      
    
    {\displaystyle {\hat {H}}}
   defined by

  
    
      
        
          
            
              H
              ^
            
          
        
        ψ
        =
        −
        
          
            
              ℏ
              
                2
              
            
            
              2
              m
            
          
        
        
          ∇
          
            2
          
        
        ψ
        +
        V
        ψ
        ,
      
    
    {\displaystyle {\hat {H}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi ,}
  which as an observable corresponds to the total energy of a particle of mass m in a real potential field V. Differential operators are an important class of unbounded operators.
The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially r ...