v0=26;
r0=5.35;
α=0.34;
v1=42;
r1=2.8;
rc=1.3;
z1=2;
z2=28;
e=1.6 (10^(-19));

vc[r1_]:=(((z1 z2 (e^2))/(2 rc)))(3-((r1^2)/(rc^2)))/;r1≤rc;
vc[r1_]:=((z1 z2 (e^2))/r1)/;r1>rc;
v[r_]:=-(v0/((1+Exp[((r-r0)/α)])))+v1 Exp[-((((r/r1)))^2)]+vc[r];

Plot[v[r],{r,0,10}]

rate=D[v[r],r];
Plot[rate,{r,0,10}]

eqn=NDSolve[{ψ''[r]+v[r]ψ[r]==0,ψ[0]==0,ψ'[0]==1},ψ[r],{r,0,10}]
Plot[ψ[r]/.eqn,{r,0,10},Axes->True]