I have been studying potential steps and barriers as well as reflection and transmission coefficients and how to derive them. Most of it makes sense to me except for one thing:(adsbygoogle = window.adsbygoogle || []).push({});

As we know, the normal Schrodinger equation is:

(-ħ^{2}/2m) (∂^{2}Ψ/∂x^{2}) + v(x)Ψ = EΨ

For a step potential however, my book and web resource both say that for the boundary conditions:

v(x)= 0 for x < 0 and V_{0}for x ≥ 0

the Schrodinger equations are:

(∂^{2}Ψ_{1}/∂x^{2}) + K_{1}^{2}Ψ_{1}(x) = 0

and

(∂^{2}Ψ_{2}/∂x^{2}) + K_{2}^{2}Ψ_{2}(x) = 0

where K_{1}= squrt(2mE) / ħ and K_{2}= squrt(2m(E- V_{0})) / ħ

(the plus sign in the second one changes into a minus when the particle doesn't have enough energy to overcome the step).

Where/How exactly did Schrodinger get these step potential equations from the original one? The step potential equations don't even seem to have the (-ħ^{2}/2m) term in front. Can someone please explain to me why step potentials seem to have different Schrodinger equations than the normal one?

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# Derivations for Schrodinger's equations for potential step

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