Schrodinger Equation
      
        We take Schrodinger’s equation as 
        one of the postulates of quantum 
        mechanics
      
        Schrodinger himself just “figured it 
        out”
      
        Thus there is no formal proof
          We rely on comparison of its 
          predictions with experiment to validate 
          it
      
        But we’ll briefly try to motivate it 
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             Schrodinger Equation
       
          We’d like the quantum wave equation
           To be consistent with de Broglie-Einstein 
            relations
           To be consistent with E = T+V = 
              2
            p /2m+V
           To be linear in Ψ(x,t)
              This means if Ψ  and Ψ  are solutions, then 
                               1       2
               c Ψ  + c Ψ  is a solution
                1  1    2 2
           To have traveling wave solutions for a 
            free particle (the case where V(x,t)=0)
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           Schrodinger Equation
      The first two assumptions can be 
        combined into2
                   Ep V
                       2m
                        2k2
                          V
                         2m
      The third assumption means that the 
        wave equation can only contain terms 
        like Ψ or its derivatives (no constants or 
        higher order powers)
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              Schrodinger Equation
       Recall some of our solutions to the 
          classical wave equation
                       sin(kx t) or eikxt
       Note that 2
                           gives a factor of k2
                       x2
                        gives a factor of 
                       t
       Thus we might guess a wave equation 
          that looks like           2
                      V
                          t       x2
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           Schrodinger Equation
      We could evaluate the constants α and β 
        using the exponential free particle 
        solution and find
             (x,t)     2 2(x,t)
           i   t    2m    x2   V(x)(x,t)
      But we normally take Schrodinger’s 
        equation as one of the postulates of 
        quantum mechanics
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