I'm working with the Schrodinger equation − ℏ 2 2 m ∂ 2 ψ ∂ x 2 = E ψand trying to use the Euler Method to approximate a wavefunction. The SEQ can be rewritten as ψ ″ = − 2 m E ℏ 2 ψMy question is 2-fold: How do I rewrite the SEQ as a set of first order differential equations? Am I using Euler Method correctly?My attempt:I think it is this, but I'm not sure:Let w = ψ ′ and w ′ = ψ ″ . Now we have w ′ = − 2 m E ℏ 2 ψand to use Euler method it is w i + 1 = w i + h w ′ → ψ i + 1 ′ = ψ i ′ + h ψ i ″ ψ i + 1 = ψ i + h w → ψ i + 1 = ψ i + h ψ i + 1 ′

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I&#039;m working with the Schrodinger equation            −              ℏ        2                    2      m                          ∂        2            ψ              ∂              x        2              =  E  ψand trying to use the Euler Method to approximate a wavefunction. The SEQ can be rewritten as      ψ    ″    =  −            2      m      E              ℏ      2        ψMy question is 2-fold: How do I rewrite the SEQ as a set of first order differential equations? Am I using Euler Method correctly?My attempt:I think it is this, but I&#039;m not sure:Let   w  =      ψ    ′   and       w    ′    =      ψ    ″  . Now we have      w    ′    =            −      2      m      E              ℏ      2        ψand to use Euler method it is      w          i      +      1        =      w    i    +  h      w    ′    →      ψ          i      +      1        ′    =      ψ    i    ′    +  h      ψ    i    ″        ψ          i      +      1        =      ψ    i    +  h  w  →      ψ          i      +      1        =      ψ    i    +  h      ψ          i      +      1        ′

marktje28 · Accepted Answer

It&#039;s convenient to rewrite the original equation as two first order ones. Namely,      φ    ′        (              x            )    =  −      k          2        ψ      (              x            )      ,        ψ    ′        (              x            )    =  φ      (              x            )      ;        k          2        =                    2        m        E                    h                  2                                                                            d                                                                                  d                                      x                                                                                                                      (                                                                    ψ                                  (                                                        x                                                      )                                                            φ                                  (                                                        x                                                      )                                                                                    )                                                    =                                                            (                                                                                                                                                                          0                                                                                                                                                                            1                                                                                                                                                                                                                    −                                                          k                                                              2                                                                                                                                                                                                                                      0                                                                                                                                                              )                            ⏟                                                                                            A                                                                                                                                                                                             (                                                                                                  ψ                                              (                                                                          x                                                                        )                                                                                    φ                                              (                                                                          x                                                                        )                                                                                                                        )                                                                                  ⏟                                                                          Ψ                                  (                                                        x                                                      )                                                                            ⟹                Ψ                            ′                          (                                x                              )                =                  A                Ψ                  (                                x                              )                                                                                              Ψ                                  n                  +                  1                                            =                              Ψ                                  n                                            +              h                                            A                                            Ψ                                  n                                                          ,                            E              u              l              e              r

I'm working with the Schrodinger equation − <mi clas

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