Approximation methods

Stationary perturbation theory

Let H=H₀+W. Let be an orthonormal eigenbasis of H₀,  .  Here i denotes the degeneracy.  Let H|y_p>=E^p|y_p>.  Expand E^p and |y_p>.   E^p=E₀^p+E₁^p+E₂^p+... , |y_p>=|y_p⁰>+|y_p¹>+|y_p²>+... .  Stationary perturbation theory yields  E^p=E₀^p,     If |f_p> is not degenerate then |y_p⁰>=|f_p>.

E₁^p and a_pⁱ are found from .  We have to diagonalize the matrix of W in the subspace spanned by the degenerate states    If E₀^p is not degenerate then

  .  

This is the first order energy correction for non-degenerate states.

If E₀^p is not degenerate then 

. 

This is the first order correction to a non-degenerate eigenvector. 

The second order energy correction for non- degenerate states is 

.

Time-dependent perturbation theory

Let H=H₀+W(t). Let be an orthonormal eigenbasis of H₀,  .   Let  .  Assume that at t=0 the system is in the state |f_i>.  The probability of finding the system in the state |f_f> (f¹i) at time t is

in first order time-dependent perturbation theory.

Fermi’s golden rule

Assume there exists a group of states nearly equal in energy  Let  be the density of final states, i.e. is the number of final states in the interval dE characterized by some discrete index b.  Let W(t)=Wexp(±iwt).  Then the transition probability per unit time is given by  .  This is Fermi’s golden rule.

The variational method

Consider an arbitrary physical system with a time-independent Hamiltonian.  Let {E_n} be its eigenvalues and {|f_n} its eigenstates.  H|f_n>=E_n|f_n>.  Let E₀ denote the energy of the ground state.  If |y> is an arbitrary ket in the state space of the system, then   This is the basis for the variational method.  Choose a family of kets |y(a)> which satisfy the boundary conditions and which depend on a number of free parameters a.  Calculate the mean value <H>(a).  Minimize <H>(a) with respect to a.  The minimal value thus obtained constitutes an approximation to the ground state energy E₀ of the system.

Identical particles

Atomic Spectra

LS coupling

We assume that the non-central part of the electrostatic interaction is much bigger than the spin-orbit interaction.  (This is usually true for light multi-electron atoms.)  The electrostatic interaction leads to a splitting of the level corresponding to a given electron configuration into a number of sublevels characterized by different values of the total orbital angular momentum of the electrons, L, and their total spin, S.  The operator for the electrostatic interaction commutes with L = l₁+l₂+l₂+… and S = s₁+s₂+s₃+… .  To each term LS belong (2L+1)(2S+1) states, differing by the values of M_L and M_S.  The spin-orbit interaction leads to a splitting of the term LS into a number of components corresponding to different values of the total angular momentum J.  But it does not completely remove the degeneracy.  Each J component is degenerate with a multiplicity of 2J+1.

S_J(2J+1) = (2L+1)(2S+1).

In the LS coupling scheme, a term is designated by ^2S+1L_J.  2S+1 is called the multiplicity of the term.

Which terms corresponding to a given configuration have the lowest energy?

Hund's Rule (established empirically)