Derivation of time-dependent Schrödinger equation from De Broglie hypothesisIn a quantum mechanics script I'm reading, the Schrödinger equation is "derived" (more precisely, motivated) by the De Broglie hypothesis. It starts at λ = 2 π h p ω = E h then takes the partial derivatives of the wave Ψ ( x , t ) = Ψ ( 0 , 0 ) e 2 π i x λ − i t ω (1) ∂ ∂ x Ψ ( x , t ) = i p h Ψ ( x , t ) ∂ ∂ t Ψ ( x , t ) = − i E h Ψ ( x , t )With the non-relativistic free particle E = 1 2 m p 2 one gets (2) i h ∂ ∂ t Ψ ( x , t ) = E Ψ ( x , t ) = p 2 2 m Ψ ( x , t ) From there, they miraculously get the time-dependent Schrödinger equation. I cannot understand this step. If I insert formula (1) for p 2 in (2), I get something with ( ∂ ∂ x Ψ ) 2 , which is not the second partial derivative of Ψ with respect to x.Any hints?

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Derivation of time-dependent Schrödinger equation from De Broglie hypothesisIn a quantum mechanics script I&#039;m reading, the Schrödinger equation is &quot;derived&quot; (more precisely, motivated) by the De Broglie hypothesis. It starts at  λ  =            2      π      h        p    ω  =      E    h  then takes the partial derivatives of the wave   Ψ  (  x  ,  t  )  =  Ψ  (  0  ,  0  )      e                            2          π          i          x                λ            −      i      t      ω                          (1)                              ∂                      ∂            x                          Ψ        (        x        ,        t        )        =                              i            p                    h                Ψ        (        x        ,        t        )                  ∂          ∂      t        Ψ  (  x  ,  t  )  =  −  i      E    h    Ψ  (  x  ,  t  )With the non-relativistic free particle   E  =      1          2      m            p    2   one gets                    (2)                    i        h                  ∂                      ∂            t                          Ψ        (        x        ,        t        )        =        E        Ψ        (        x        ,        t        )        =                              p            2                                2            m                          Ψ        (        x        ,        t        )            From there, they miraculously get the time-dependent Schrödinger equation. I cannot understand this step. If I insert formula (1) for       p    2   in (2), I get something with   (      ∂          ∂      x        Ψ      )    2  , which is not the second partial derivative of   Ψ with respect to   x.Any hints?

Finnegan Zimmerman · Accepted Answer

You&#039;re looking for some form of differential operator that will take your plane wave  Ψ  (  x  ,  t  )  =  Ψ  (  0  ,  0  )  exp  ⁡      [                  2        π        i        x            λ        −    i    ω    t    ]  and will return       p    2    Ψ  (  x  ,  t  ). As you&#039;ve noticed, you can apply a space derivative to get  −  i  h      ∂          ∂      x        Ψ  (  x  ,  t  )  =  p  Ψ  (  x  ,  t  )  .To get another   p, you can apply the space derivative again to get            (      −      i      h              ∂                  ∂          x                    )        2    Ψ  (  x  ,  t  )  =  −  i  h      ∂          ∂      x        p  Ψ  (  x  ,  t  )  =      p    2    Ψ  (  x  ,  t  )  ,but as you note you could also square the whole thing, to get            (      −      i      h              ∂                  ∂          x                    Ψ      (      x      ,      t      )      )        2    =      p    2    Ψ  (  x  ,  t      )    2    .The reason we take the former version and not the latter is that the second form introduces an extra factor of   Ψ  (  x  ,  t  ), which would make the wave equation nonlinear and introduce all sorts of terrible, horrible nightmares.In the end, the argument is heuristic, and you shouldn&#039;t read all that much into it. The Schrödinger equation (particularly once you introduce a potential) is a postulate in its own right and cannot be derived flawlessly from first principles. You can only offer some justification for its form, which is what you&#039;re trying to do.In detail, the argument you&#039;re making is of this form: if we believe the de Broglie relation   p  =  h      /    λ and if we believe the Planck relation   E  =  ℏ  ω, both of which give us the particle properties in terms of the wave properties, and if the matter wave is a plane wave of the form   Ψ  (  x  ,  t  )  =  Ψ  (  0  ,  0  )  exp  ⁡      [                  2        π        i        x            λ        −    i    ω    t    ]  , then a wave equation of the form  i  ℏ            ∂      Ψ              ∂      t        =  −            ℏ      2              2      m                          ∂        2            Ψ              ∂              x        2            reproduces the correct classical relationship between the energy and the momentum,   E  =      p    2        /    2  mThat&#039;s all that the argument can claim. In particular,it doesn&#039;t rule out alternative ways of arriving at that energy-momentum relation (which is, in the end, the dispersion relation for the wave),it doesn&#039;t say anything about what happens when there is a potential, andit doesn&#039;t say anything about wave states that are not plane waves.The miracle, of course, is that when you disregard all three of those objections, you introduce the potential as a pretty big kludge, and you look for the non-plane-wave solutions for some complicated problem like the hydrogen atom, you get something that matches pretty exactly with experiment. This is the real justification for the Schrödinger equation: an a posteriori justification based on its success in replicating experimental results, instead of an a priori justification based on some over-arching set of first principles.

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