Schrödinger picture | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-24

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture


00:02:36 1 Background
00:04:29 2 The time evolution operator
00:04:39 2.1 Definition
00:05:44 2.2 Properties
00:08:36 2.3 Differential equation for time evolution operator
00:14:36 3 Summary comparison of evolution in all pictures
00:14:51 4 See also



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SUMMARY
=======
In physics, the Schrödinger picture (also called the Schrödinger representation) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures.
In the Schrödinger picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. For time evolution from a state vector 
  
    
      
        
          |
        
        ψ
        (
        
          t
          
            0
          
        
        )
        ⟩
      
    
    {\displaystyle |\psi (t_{0})\rangle }
   at time t0 to a state vector 
  
    
      
        
          |
        
        ψ
        (
        t
        )
        ⟩
      
    
    {\displaystyle |\psi (t)\rangle }
   at time t, the time-evolution operator is commonly written 
  
    
      
        U
        (
        t
        ,
        
          t
          
            0
          
        
        )
      
    
    {\displaystyle U(t,t_{0})}
  , and one has

  
    
      
        
          |
        
        ψ
        (
        t
        )
        ⟩
        =
        U
        (
        t
        ,
        
          t
          
            0
          
        
        )
        
          |
        
        ψ
        (
        
          t
          
            0
          
        
        )
        ⟩
        .
      
    
    {\displaystyle |\psi (t)\rangle =U(t,t_{0})|\psi (t_{0})\rangle .}
  In the case where the Hamiltonian of the system does not vary with time, the time-evolution operator has the form

  
    
      
        U
        (
        t
        ,
        
          t
          
            0
          
        
        )
        =
        
          e
          
            −
            i
            H
            (
            t
            −
            
              t
              
                0
              
            
            )
            
              /
            
            ℏ
          
        
        ,
      
    
    {\displaystyle U(t,t_{0})=e^{-iH(t-t_{0})/\hbar },}
  where the exponent is evaluated via its Taylor series.
The Schrödinger picture is useful when dealing with a time-independent Hamiltonian H; that is, 
  
    
      
        
          ∂
          
            t
          
        
        H
        =
        0
      
    
    {\displaystyle \partial _{t}H=0}
  .