Variational Principle in Practice

Recipe:

1. Define our system (Hamiltonian)
2. Propose trial wave function family (parametrized wave function, basis set)
3. Minimize parameters

Harmonic Oscilator

Ground State

Trial wave function family

γ[x_,α_]:=Exp[-x^2α]
Manipulate[Plot[γ[x,α],{x,-10,10}],{α,-3,3}]

Integrate[γ[x,α]γ[x,α],{x,-Infinity,Infinity},Assumptions->α>0]

ψ[x_,α_]:=Exp[-x^2α]/Sqrt[((Sqrt[(π/2)])/(Sqrt[α]))]

Hψ

H[x_,ψ_]:=-ℏ^2/(2m)D[ψ,{x,2}]+1/2 m x^2ω^2ψ

H[x,ψ[x,α]]

<H>= <ψ|H|ψ>

Integrate[ψ[x,α]H[x,ψ[x,α]],{x,-Infinity,Infinity},Assumptions->α>0]

H[α_]=((m (ω^2))/(8 α))+((α (ℏ^2))/(2 m))

Solve[D[H[α],α]==0,α]

H[((m ω)/(2 ℏ))]

ψ[x,((m ω)/(2 ℏ))]//FullSimplify

(First) Excited State

κ[x_,α_]:=x Exp[-x^2α]

Manipulate[Plot[κ[x,α],{x,-5,5}],{α,.1,5}]

Integrate[κ[x,α]κ[x,α],{x,-Infinity,Infinity},Assumptions->α>0]

ξ[x_,α_]:=x Exp[-x^2α]/Sqrt[((Sqrt[(π/2)])/(4 (α^(3/2))))]

Integrate[ξ[x,α]H[x,ξ[x,α]],{x,-Infinity,Infinity},Assumptions->α>0]

Solve[D[((3 m (ω^2))/(8 α))+((3 α (ℏ^2))/(2 m)),α]==0,α]

HH[α_]:=((3 m (ω^2))/(8 α))+((3 α (ℏ^2))/(2 m))

HH[((m ω)/(2 ℏ))]

Reference: Quantum mechanics, Cohen (vol II, p. 1148)

Homework:
1. Read  Cohen (vol II, p. 1148)  together with the notebook
2. Try non-exact wave function that is proposed there.
3. Try another odd function for the first exited state (sin(x)Exp[-x^2α])
4. Computationa physics, Thijssen, chapter 3.
5. Introduction to Quantum mechanics, Griffiths, chapter 7.