Perturbation Theory Examples

Example: Infinite square well with step perturbation

The wave function.   Use  n_ to make  n  an  function (index)

(Subscript[ψ, n_])=(Sqrt[(2/a)])Sin[((n π x)/a)];

Verify orthonormality of the wafve functions.

Table[Integrate[(Subscript[ψ, m])(Subscript[ψ, n]),{x,0,a}],{m,1,3},{n,1,3}]

Calculate the matrix elements of the perturbation.

Table[Integrate[(Subscript[ψ, m])(Subscript[V, 0])(Subscript[ψ, n]),{x,0,a/2}],{m,1,7},{n,1,7}]

Example: generic 3-state matrix system

In this system, we have already converted the Hamiltonian and perturbation potential into matrix elements.  The eigenfunctions are simply the standard unit vectors--the Hamiltonian is diagonal in this basis.

H=(({{(Subscript[λ, 1]), 0, 0}, {0, (Subscript[λ, 2]), 0}, {0, 0, (Subscript[λ, 3])}}));V=(({{α, δ, ϵ}, {δ, β, ζ}, {ϵ, ζ, γ}}));

The first order energy is just the diagonal matrix elements of V.  To get the new wave functions, divide each off-diagonal element of V by (Subscript[E, m])-(Subscript[E, n]), with 1 on the diagonals.  Each column is a new wave function.

U=(({{1, (δ/((Subscript[λ, 1])-(Subscript[λ, 2]))), (ϵ/((Subscript[λ, 1])-(Subscript[λ, 3])))}, {(δ/((Subscript[λ, 2])-(Subscript[λ, 1]))), 1, (ζ/((Subscript[λ, 2])-(Subscript[λ, 3])))}, {(ϵ/((Subscript[λ, 3])-(Subscript[λ, 1]))), (ζ/((Subscript[λ, 3])-(Subscript[λ, 2]))), 1}}));

We can verify that the new basis U is orthonormal to first order in V:

Simplify[U\[Transpose].U]/.(δ^2)->0/.(ϵ^2)->0/.(ζ^2)->0/.δϵ->0/.ϵδ->0/.ζϵ->0

Now we verify that the total Hamiltonian is diagonal in this new eigenbasis.  The diagonal elements are just α+(Subscript[λ, 1]), etc., as before.  Note that (δ^2)->0 , etc.  Thus the following matrix is actually diagonal to first order:

Simplify[U\[Transpose].(H+V).U]/.(δ^2)->0/.(ϵ^2)->0/.(ζ^2)->0

Finally we get the second order energy by multiplying on the right by U, and taking the diagonal entries.

Simplify[(H+V).U]