Sanity check : the harmonic oscillator

hbar=1;m=1;inf=4;ω=2.;En=ω(2+(1/2));V[x_]:=((m (ω^2))/2)(x^2);
En=ω(0+(1/2));sol0=NDSolve[{-((hbar^2)/(2 m))ψ''[x]+V[x] ψ[x]==En ψ[x], ψ[-inf]==0, ψ[0]==1}, ψ, {x, -inf, inf}];

En=ω(1+(1/2));sol1=NDSolve[{-((hbar^2)/(2 m))ψ''[x]+V[x] ψ[x]==En ψ[x], ψ[-inf]==0, ψ'[0]==1}, ψ, {x, -inf, inf}];
En=ω(2+(1/2));sol2=NDSolve[{-((hbar^2)/(2 m))ψ''[x]+V[x] ψ[x]==En ψ[x], ψ[-inf]==0, ψ[0]==1}, ψ, {x, -inf, inf}];
Plot[{Evaluate[ψ[x]/.sol0], Evaluate[ψ[x]/.sol1], Evaluate[ψ[x]/.sol2]},{x,-inf, inf},PlotRange->All]

The V=1-(2/(1+((Tanh[λ x])^2))) potential

hbar=1;m=1;inf=10;λ=1.;En=-0.80994((hbar^2)/(2m (λ^2)));V[x_]:=1-(2/(1+((Tanh[λ x])^2)))
sol=NDSolve[{-((hbar^2)/(2 m))ψ''[x]+V[x] ψ[x]==En ψ[x], ψ[-inf]==0, ψ[0]==1}, ψ, {x, -inf, inf}];
Plot[{Evaluate[ψ[x]/.sol], V[x]},{x,-inf, inf},PlotRange->All]

The V=10( 1-(2/(1+((Tanh[λ x])^2)))) potential

Ground state

hbar=1;m=1;inf=6;λ=1.;En=-14.738583461((hbar^2)/(2m (λ^2)));V[x_]:=10(1-(2/(1+((Tanh[λ x])^2))))
sol=NDSolve[{-((hbar^2)/(2 m))ψ''[x]+V[x] ψ[x]==En ψ[x], ψ[-inf]==0, ψ[0]==5}, ψ, {x, -inf, inf}];
Plot[{Evaluate[ψ[x]/.sol], V[x]},{x,-inf, inf},PlotRange->All]

First excited state

hbar=1;m=1;inf=7;λ=1.;En=-6.35868914((hbar^2)/(2m (λ^2)));V[x_]:=10(1-(2/(1+((Tanh[λ x])^2))))
sol=NDSolve[{-((hbar^2)/(2 m))ψ''[x]+V[x] ψ[x]==En ψ[x], ψ[-inf]==0, ψ'[0]==15}, ψ, {x, -inf, inf}];
Plot[{Evaluate[ψ[x]/.sol], V[x]},{x,-inf, inf},PlotRange->All]

Second excited state

hbar=1;m=1;inf=7;λ=1.;En=-1.771523((hbar^2)/(2m (λ^2)));V[x_]:=10(1-(2/(1+((Tanh[λ x])^2))))
sol=NDSolve[{-((hbar^2)/(2 m))ψ''[x]+V[x] ψ[x]==En ψ[x], ψ[-inf]==0, ψ[0]==5}, ψ, {x, -inf, inf}];
Plot[{Evaluate[ψ[x]/.sol], V[x]},{x,-inf, inf},PlotRange->All]