Stationary perturbation theory is concerned with finding the changes in the discrete energy levels and the changes in the corresponding energy eigenfunctions of a system, when the Hamiltonian of a system is changed by a small amount.  Let H=H₀+H’=H₀+lW.  H₀ is the unperturbed Hamiltonian whose eigenvalues E₀^p and eigenstates |fⁱ_p> are known. Let {|fⁱ_p>}denote an orthonormal eigenbasis of H₀, H₀|fⁱ_p>= E₀^p|fⁱ_p>.  Here i denotes the degeneracy.  Assume that the matrix elements of H’ in the eigenbasis of H₀ are small compared to the matrix elements of H₀.

<fⁱ_p|H'|fⁱ_p>   <<    <fⁱ_p|H₀|fⁱ_p>=E₀^p.

We write H’=lW with l<<1 and <fⁱ_p|W|fⁱ_p>»E₀^p.

We are looking for the eigenvalues E(l) and the eigenstates |y(l)> of H(l)=H₀+lW.

H(l)|y_p>=E^p(l)|y_p> or, keeping the dependence on l in mind without specifically writing it down, H|y_p>=E^p|y_p>.

Since lW is small, we assume that E and |y> can be expanded as a power series in l.

E^p=E₀^p+lE₁^p+l²E₂^p+... ,
|y_p>=|y_p⁰>+l|y_p¹>+l²|y_p²>+... .

We may then write

(H₀+lW)(|y_p⁰>+l|y_p¹>+l²|y_p²>+...)

= (E₀^p+lE₁^p+l²E₂^p+...)(|y_p⁰>+l|y_p¹>+l²|y_p²>+...).

This equation is must be valid over a continuous range of l.  Therefore we equate coefficients of equal powers of l on both sides to obtain a series of equations that represent successively higher orders of the perturbation.

(H₀-E₀^p)|y_p⁰>=0 implies that |y_p⁰> is a linear combination of unperturbed eigenfuctionns |fⁱ_p> with the corresponding eigenvalue E₀.  We choose <y_p⁰|y_p⁰>=1. |y_p^s> is not uniquely defined.  We can add an arbitrary multiple of |y_p⁰> to each |y_p^s> without affecting the left hand side of the above equations.  Most often this multiple is chosen so that <y_p⁰|y_p^s>=0.  The perturbed ket is then not normalized.

We then have

0                            =                       

To calculate the energy to sth order, we only need to know the state vector to order s-1.

First-order perturbation theory for non-degenerate levels

Consider a particular non-degenerate eigenvalue E₀ⁿ of H₀. H₀|f_n>=E₀ⁿ|f_n>.  The other eigenvalues of H may or may not be degenerate.  We have |y_n⁰>=|f_n> and E₁ⁿ=<f_n|W|f_n>.

The first-order energy correction therefore is lE₁ⁿ=<f_n|H’|f_n>.

We have Eⁿ=E₀ⁿ+<f_n|H’|f_n>+O(l²).

First-order eigenvector corrections:

(H₀-E₀^p)|y_p¹> =( E₁^p-W)|y_p⁰> = (E₁^p-W)|f_p>. |y_p⁰> is an eigenstate of the unperturbed Hamiltonian.  We may expand

in terms of the basis vectors |fⁱ_p'>.  In the expansion b_p=0 because <y_p⁰|y_pⁱ>=0.

Multiply from the left by <fⁱ_p''|.

Therefore

Second-order perturbation theory for non-degenerate levels

Second-order energy corrections:

Since we have found the expression for the state vector to first order, we can now find the expression for the energy to second order.

Let H=H₀+H’=H₀+lW.  In practice, after having derived the perturbation expansion, we often set l=1 and let H’=W be small.

Problems:

Calculate the first-order shift in the ground state of the hydrogen atom caused by the finite size of the proton. Assume the proton is a uniformly charged sphere of radius r = 10^-13cm. The ground state wave function of the hydrogen atom is and the Bohr constant is a₀=0.53´10^-10m.

(a)   Calculate the first-order energy shift of the nth eigenvalue due to the perturbation.

(b)  Write out the first three non vanishing terms for the first-order perturbation expansion of the ground state in terms of the unperturbed eigenfunctions of the infinite well.

(c)  Calculate the second-order energy shift for the ground state.