Harmonic Oscillator examples of Variance Minimization

In 1d, V = x², Ñ = ¶/¶x, m_i=m (reduced mass of nuclei measured in electron mass units)

 

                Equation 2.7 of Variance Minimization htm simplifies to

(1.1)

 

Ground state

                Try

(1.2)

So that (1.1) becomes

(1.3)

This is independent of x if a=Ö(m/2), making E = 1/Ö(2m).  So that for the ground state

              (1.4)

Testing this

(1.5)

So that

(1.6)

First excited state

                Try

(1.7)

So that (1.1) becomes

(1.8)

There is a plus and an equal minus on the 1/x².

(1.9)

 

This is consatnt for a = Ö(m/2), yielding E =3/Ö(2m), in accordance with E_n =(n+1/2)*2