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picture1_Quantum Mechanics Ppt 78430 | Phys242 Lec20


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File: Quantum Mechanics Ppt 78430 | Phys242 Lec20
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 ...

icon picture PPT Filetype Power Point PPT | Posted on 05 Sep 2022 | 3 years ago
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           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 
                                                 2
             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)
                                                             3
           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)
                                                   4
              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
                                                               5
           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
                                                    6
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...Schrodinger equation we take s as one of the postulates quantum mechanics himself just figured it out thus there is no formal proof rely on comparison its predictions with experiment to validate but ll briefly try motivate d like wave be consistent de broglie einstein relations e t v p m linear in x this means if and are solutions then c a solution have traveling for free particle case where first two assumptions can combined into ep k third assumption that only contain terms or derivatives constants higher order powers recall some our classical sin kx eikxt note gives factor might guess looks could evaluate using exponential find i normally...

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