Derivations for Schrodinger's equations for potential step

[space-time]

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:

As we know, the normal Schrödinger equation is:

(-ħ²/2m) (∂²Ψ/∂x²) + 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 V₀ for x ≥ 0

the Schrödinger equations are:

(∂²Ψ₁/∂x²) + K₁²Ψ₁(x) = 0

and
(∂²Ψ₂/∂x²) + K₂²Ψ₂(x) = 0

where K₁ = squrt(2mE) / ħ and K₂ = squrt(2m(E- V₀)) / 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 Schrödinger get these step potential equations from the original one? The step potential equations don't even seem to have the (-ħ²/2m) term in front. Can someone please explain to me why step potentials seem to have different Schrödinger equations than the normal one?

 

[Matterwave]

This is just basic algebra, giving $K_1$ as an example, and you should work out $K_2$ for yourself:

$$\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi=E\psi$$ $$\frac{\partial^2}{\partial x^2}\psi=-\frac{2m}{\hbar^2}E\psi$$ $$\frac{\partial^2}{\partial x^2}\psi+\frac{2mE}{\hbar^2}\psi=0$$ $$\frac{\partial^2}{\partial x^2}\psi+K_1^2\psi=0$$

 

[space-time]

This is just basic algebra, giving $K_1$ as an example, and you should work out $K_2$ for yourself:

$$\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi=E\psi$$ $$\frac{\partial^2}{\partial x^2}\psi=-\frac{2m}{\hbar^2}E\psi$$ $$\frac{\partial^2}{\partial x^2}\psi+\frac{2mE}{\hbar^2}\psi=0$$ $$\frac{\partial^2}{\partial x^2}\psi+K_1^2\psi=0$$

I see. Thank you. It is just simple algebra. This does bring up the question however of why that was necessary in the first place. Why didn't he just leave the equation as it originally was (since these other versions are just algebraically manipulated versions of the original)? After all, the original version is just as simple to solve as the other versions. I have a hypothesis as to why this manipulation was necessary (it has to do with being able to normalize the solutions). I am not 100% sure however that this is the reason.

 

[Matterwave]

I see. Thank you. It is just simple algebra. This does bring up the question however of why that was necessary in the first place. Why didn't he just leave the equation as it originally was (since these other versions are just algebraically manipulated versions of the original)? After all, the original version is just as simple to solve as the other versions. I have a hypothesis as to why this manipulation was necessary (it has to do with being able to normalize the solutions). I am not 100% sure however that this is the reason.

No, you can solve it without doing any algebraic manipulations. And this has nothing to do with normalization. But if you do these manipulations, it just makes things a little simpler when you go ahead and solve it.

 

[jtbell]

When you finally start writing out the solutions and manipulating them algebraically, would you rather wrestle with things like $e^{ik_1 x}$ and $e^{ik_2 x}$ or with things like $e^{i\sqrt{2mE}x/\hbar}$ and $e^{i\sqrt{2m(E-V_0)}x/\hbar}$ ?