The wave function between 0 and L/2:

psi1  =  A Sin[k x]

Up to a plus or minus sign, the wave function between L/2 and L:

psi2  =  A Sin[k (L - x)]

Normalize:

\(eq1\  = \ 
      Integrate[psi1^2, {x, 0, L/2}]\  + \ 
          Integrate[psi2^2, 
				{x, L/2, L}] == 1\)

Substitute Normalization back into original functions:

psi1a  =  psi1 /. Solve[eq1, A][[2]]

psi2a  =  psi2 /. Solve[eq1, A][[2]]

Case I:  Psi 1 = -Psi 2 (notice derivative in deriative of psi2):

eq2  = 
      Simplify[Simplify[ 
            D[-psi2a, x]  -   D[psi1a, x] /. 
              x -> L/2]  ==  (2 m S/hbar^2 psi1 /. 
              x -> L/2)]

This is only true if k L/2 = n Pi or k = 2 n Pi/L.  These are the normal odd-excited state wave functions for a particle in a box of size L.  These functions are not changed by the addition of the new potnetial.

Case II:  Psi 1 = Psi 2

eq3  = 
      Simplify[Simplify[ 
            D[psi2a, x]  -   D[psi1a, x] /. 
              x -> L/2]  ==  (2 m S/hbar^2 psi1a /. 
              x -> L/2)]

This is true if :

eq4  =  
      Tan[k L/2]/(k L/2)  ==  -hbar^2 L/2/m/S

Plot[Tan[x]/x, {x, 0,  4  Pi}]

On this graph: -hbar^2 L/(2 m S) is a horizontal line.  Where it intersects with Tan[x]/x is a permissible energy level.

LimIts:  S-->0  --> Tan[x]/x -->- Infinity  --> x->(2 n + 1)/2 Pi

kL/2 = (2 n +1)/2 Pi  or k = (2 n + 1) Pi / L

Limit S -->- Infinity  ---> Tan[x]/x--> 0  ---> x-> m Pi

Thus

k L/2 = m Pi or k = 2 m Pi/L  --- The same energy levels as the wave function in the 
opposite direction.

Additional plots:

eq5  =  Tan[x]/x == -ss

root1 = FindRoot[Evaluate[eq5 /. ss -> 20], {x, 2}]

Plot[
      Evaluate[If[xx < 1, Sin[x*xx], Sin[x*(2 - xx)]] /. root1], {xx, 0, 
        2}, PlotRange -> All]

root2 = FindRoot[Evaluate[eq5 /. ss -> 2], {x, 2}]

Plot[
      Evaluate[If[xx < 1, Sin[x*xx], Sin[x*(2 - xx)]] /. root2], {xx, 0, 
        2}, PlotRange -> All]

root3 = FindRoot[Evaluate[eq5 /. ss ->  .2], {x, 2}]

Plot[
      Evaluate[If[xx < 1, Sin[x*xx], Sin[x*(2 - xx)]] /. root3], {xx, 0, 
        2}, PlotRange -> All]

root4 = FindRoot[Evaluate[eq5 /. ss ->  .002], {x, 2}]

Plot[
      Evaluate[If[xx < 1, Sin[x*xx], Sin[x*(2 - xx)]] /. root4], {xx, 0, 
        2}, PlotRange -> All]