System is h/2pi=1   m=1   V=4   a=4
Found(energies):
(   0.25989340956674017,1)  
(    1.0332205648117216,1)  
(    2.2929464172906138,1)  
(    3.8961698088235814,1)  
The wavefunctions are:
psi0[x_]:=Sin[0.72096242560446954784 x] /; x<=4 
psi0[x_]:=-0.00000000142178524277*10^-6 Exp[2.73499783928004580602 x]+ 14372.93680518333167128731 Exp[-2.73499783928004580602 x]  /; x>4 
psi1[x_]:=Sin[1.43751213199174188162 x] /; x<=4 
psi1[x_]:=0.00000010264665592613*10^-6 Exp[2.43588974922440959503 x]+ -8662.44382109256548574194 Exp[-2.43588974922440959503 x]  /; x>4 
psi2[x_]:=Sin[2.14146978371893625948 x] /; x<=4 
psi2[x_]:=-0.00017768805667158955*10^-6 Exp[1.84773027398989508896 x]+ 1227.44836079462675115792 Exp[-1.84773027398989508896 x]  /; x>4 
psi3[x_]:=Sin[2.79147624343234612709 x] /; x<=4 
psi3[x_]:=4268.46534556375900137937*10^-6 Exp[0.45569768745609967597 x]+ -6.26301738337165936343 Exp[-0.45569768745609967597 x]  /; x>4 

(*where*) a=4;

Plot[psi0[x],{x,0,2*a}]