Quantum Mechanics Boundary conditions

[MaxJ]

Homework Statement

below

Relevant Equations

below

For this problem,

The solution is,

I have a doubt about Step number 3 about boundary conditions. Someone maybe be able to solve that doubt?

Kind wishes

 

[Orodruin]

I have a doubt about Step number 3 about boundary conditions. Someone maybe be able to solve that doubt?

Not unless you tell us what your actual doubt is. The math is straightforward, essentially plug-and-chug.

 

[MaxJ]

Not unless you tell us what your actual doubt is. The math is straightforward, essentially plug-and-chug.

Sir, bless you.

Doubt is matching wave functions give A + B = C + D and how they get their expression for derivatives (also in step 3).

 

[Steve4Physics]

Doubt is matching wave functions give A + B = C + D and how they get their expression for derivatives (also in step 3).

It would be more conventional (and clearer) to consider 2 separate wave-functions, one for each region: $\psi_1$ for $x<0$ and $\psi_2$ for $x \ge 0$.

Each wave-function contains 2 terms, representing waves moving in the +x and -x directions in that region.

So, adapting your solution’s notation:
$\psi_1(x) = A e^{ikx} + Be^{-ikx}$
$\psi_2(x) = C e^{iqx} + De^{-iqx}$

At $x=0$ we require that $\psi_1 =\psi_2$. Simply evaluate $\psi_1$and $\psi_2$ at $x=0$ and equate them.

Similarly for the dervatives.