Why is the de Broglie equation as well as the Schrodinger equation is correct for massive particle?Starting from special relativity, here I see the de Broglie approximation is valid only if m 0 = 0Derivation: E 2 = P 2 C 2 + m 0 2 C 4 . Here we put Plank-Einstein relation E = h ν = h C λ . Finally, λ = h P 2 + m 0 2 C 2 ( 1 )If m 0 = 0 then λ = h p (de Broglie approximation).Furthermore we know the Schrodinger equation was derived by assuming that the de Broglie approximation is true for all particles, even if m 0 ≠ 0. But if we take special relativity very strictly then this approximation looks incorrect.In addition, if we try to derive the Schrodinger equation from the exact relation found in '1', we find completely different equation. For checking it out, lets take a wave function- Ψ = A e i ( 2 π λ x − ω t ) = A e i ( P 2 + m 0 2 C 2 ℏ x − E ℏ t ) (putting λ from '1', h 2 π = ℏ and E = ℏ ω).Then, ∂ 2 Ψ ∂ x 2 = − P 2 + m 0 2 C 2 ℏ 2 Ψ = − E 2 C 2 ℏ 2 Ψ ⟹ E 2 Ψ = − C 2 ℏ 2 ∂ 2 Ψ ∂ x 2 ( 2 )Again, ∂ Ψ ∂ t = − i E ℏ Ψ ⟹ E Ψ = − i ℏ ∂ Ψ ∂ t Here we see operator E = − i ℏ ∂ ∂ t ⟹ E 2 = − ℏ 2 ∂ 2 ∂ t 2 ⟹ E 2 Ψ = − ℏ 2 ∂ 2 Ψ ∂ t 2 ( 3 )Combining (2) and (3) we find the differential equation: ∂ 2 Ψ ∂ x 2 = 1 C 2 ∂ 2 Ψ ∂ t 2 It is the Maxwell's equation, not the well known Schrodiner equation!Therefore for the Schrodinger equation to exist, the de Broglie approximation must hold for m 0 ≠ 0. I see a clear contradiction here. Then why is the Schrodinger equation correct after all?

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Why is the de Broglie equation as well as the Schrodinger equation is correct for massive particle?Starting from special relativity, here I see the de Broglie approximation is valid only if       m    0    =  0Derivation:      E    2    =      P    2        C    2    +      m    0    2        C    4  . Here we put Plank-Einstein relation   E  =  h  ν  =  h      C    λ  . Finally,  λ  =      h                  P        2            +              m        0        2                    C        2                (  1  )If       m    0    =  0 then   λ  =      h    p   (de Broglie approximation).Furthermore we know the Schrodinger equation was derived by assuming that the de Broglie approximation is true for all particles, even if       m    0    ≠  0. But if we take special relativity very strictly then this approximation looks incorrect.In addition, if we try to derive the Schrodinger equation from the exact relation found in &#039;1&#039;, we find completely different equation. For checking it out, lets take a wave function-  Ψ  =  A      e          i      (                        2          π                λ            x      −      ω      t      )        =  A      e          i      (                                    P            2                    +                      m            0            2                                C            2                          ℏ            x      −              E        ℏ            t      )         (putting   λ from &#039;1&#039;,       h          2      π        =  ℏ and   E  =  ℏ  ω).Then,                     ∂        2            Ψ              ∂              x        2              =  −                    P        2            +              m        0        2                    C        2                    ℏ      2        Ψ  =  −            E      2                      C        2                    ℏ        2              Ψ    ⟹        E    2    Ψ  =  −      C    2        ℏ    2                      ∂        2            Ψ              ∂              x        2                (  2  )Again,             ∂      Ψ              ∂      t        =  −  i      E    ℏ    Ψ    ⟹    E  Ψ  =  −  i  ℏ            ∂      Ψ              ∂      t      Here we see operator   E  =  −  i  ℏ      ∂          ∂      t          ⟹        E    2    =  −      ℏ    2              ∂      2              ∂              t        2                ⟹        E    2    Ψ  =  −      ℏ    2                      ∂        2            Ψ              ∂              t        2                (  3  )Combining (2) and (3) we find the differential equation:                    ∂        2            Ψ              ∂              x        2              =      1          C      2                          ∂        2            Ψ              ∂              t        2            It is the Maxwell&#039;s equation, not the well known Schrodiner equation!Therefore for the Schrodinger equation to exist, the de Broglie approximation must hold for       m    0    ≠  0. I see a clear contradiction here. Then why is the Schrodinger equation correct after all?

Kayleigh Mendez · Accepted Answer

Your relation in (1) is wrong, because for massive particles   λ  ν  =  c does not hold. I will show that if you insert the correct relation between   λ and   ν, you recover the de Broglie relation, whether the particle is massive or massless.To see why this is true, take the usual expression for energy,      E    2    =      p    2        c    2    +      m    2        c    4  and substitute the usual Planck-Einstein relations,   E  =  ℏ  ω and   p  =  ℏ  k. From this we get      ℏ    2        ω    2    =      ℏ    2        k    2        c    2    +      m    2        c    4  which reduces to  k  =                    (                  ω          c                )            2        −                  (                              m            c                    ℏ                )            2      When we take the product   λ  ν  =            2      π        k        ω          2      π        =      ω    k  , we get  λ  ν  =      ω    k    =      ω                            (                      ω            c                    )                2            −                        (                                    m              c                        ℏ                    )                2              =      1                            (                      1            c                    )                2            −                        (                                    m              c                                      ℏ              ω                                )                2              =      c          1      −                        (                                    m                              c                2                                                    ℏ              ω                                )                2            Now note that            m              c        2                    ℏ      ω        =            m              c        2              E    =            m              c        2                    γ      m              c        2              =      1    γ    =      1    −          β      2      which, when we substitute in the above, gives us  λ  ν  =      c    β  Now that we know how to substitute   λ for   ν, let&#039;s proceed with your derivation. Starting with       E    2    =      p    2        c    2    +      m    2        c    4   and taking the Planck-Einstein relation   E  =  h  ν  =            h      c              β      λ      , we get the following:                    h        2                    c        2                            β        2                    λ        2              =      p    2        c    2    +      m    2        c    4  which reduces to:  λ  =      h          β                        p          2                +                  m          2                          c          2                    Inserting the standard definition again,       E    c    =            p      2        +          m      2              c      2      , we get  λ  =            h      c              β      E      and inserting       E    p    =            γ      m              c        2                    γ      β      m      c        =      c    β  , we end up with  λ  =      h    p  which is the de Broglie relation. It&#039;s not an approximation; it&#039;s exact. There were no approximations in the above derivation.

Why is the de Broglie equation as well as the Schrodinger equation is correct for massive particle?

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