Let's write Schrodinger equation applied to ψ(x) :
−2mℏ2ψ′′+Vψ=Eψ
First calculate ψ′′ :
ψ′′=Ne−x/x0(x02n(n−1)(x0x)n−2−2x02n(x0x)n−1+x021(x0x)n)
ψ′′=Ne−x/x0(x0x)n(x021−xx02n+x2n(n−1))=ψ(x021−xx02n+x2n(n−1))
Now we substitue this expression into the Schrodinger equation :
(2m−ℏ2(x021−xx02n+x2n(n−1))+V)ψ=Eψ
As ψ=0 everywhere except for x=0, we can simplify by ψ and find
2m−ℏ2(x021−xx02n+x2n(n−1))+V=E
V=E+2mℏ2(x021−xx02n+x2n(n−1))
As V→0 when ∣x∣→+∞ , we find that
E=−2mx02ℏ2
V=2mℏ2(−xx02n+x2n(n−1))
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