Download Transitions between atomic energy levels and selection rules

Electromagnetic transitions
Transitions in general: Fermi’s Golden Rule
Dipole approximation
Transition selection rules
Comments:
The Hamiltonians for hydrogen, hydrogenic ions or N-electron atom describe the
atomic degrees of freedom of a one- or many-electron atom (or ion). Such an atomic
Hamiltonian possesses a spectrum of eigenvalues, and the associated eigenstates
are solutions of the corresponding stationary Schrödinger equation. The eigenstates
of Hamiltonian are usually “seen” by observing electromagnetic radiation emitted or
absorbed during a transition between two eigenstates. The fact that such
transitions occur and that an atom doesn’t remain in an eigenstate of Hamiltonian
forever, is due to the interaction between the atomic degrees of freedom and the
degrees of freedom of the electromagnetic field. A Hamiltonian able to describe
electromagnetic transitions must thus account not only for the atomic degrees of
freedom, but also for the degrees of freedom of the electromagnetic field. An
eigenstate of the atomic Hamiltonian is in general not an eigenstate of the full
Hamiltonian, a system which is in an eigenstate of atomic Hamiltonian at a given time
will evolve and may be in a different eigenstate of atomic Hamiltonian at a later time. If
we look at the interaction between atom and electromagnetic field as a perturbation of
the non-interacting Hamiltonian, then this perturbation causes time dependent
transitions between the unperturbed eigenstates, even if the perturbation itself is time
independent. Such transitions can be generally described in the framework of timedependent perturbation theory which is expounded in the following section.
Transitions in general:
Fermi‘s golden rule
Consider a physical system which is described by the Hamiltonian
but which is in an eigenstate φi of the Hamiltonian ˆH0 at time t = 0. This
Hamiltonian ˆH0 is assumed to differ from the full Hamiltonian ˆH by a “small
perturbation” ˆW . Even if ˆH0 isn’t the exact Hamiltonian, its (orthonormalized)
eigenstates φn,
still form a complete basis in which we can expand the exact time-dependent
wave function ψ(t):
The coefficients cn(t) in this expansion are time dependent, because the time
evolution of the eigenstates of ˆH0 is, due to the perturbation ˆW , not given by
the exponential functions
alone.
The initial condition that the system be in the eigenstate φi of ˆH0 at time
t = 0 is expressed in the following initial conditions for the coefficients cn(t):
At a later time t, the probability for finding the system in the eigenstate φf
of ˆH0 is:
In order to calculate the coefficients cn(t) we insert the expansion in the timedependent Schrödinger equation and obtain using
If we multiply from the left with the φ*m becomes a system of
coupled ordinary differential equations for the coefficients cn(t):
We can formally integrate the equations:
To first order in the matrix elements of the perturbing operator ˆW , the
coefficients cn(t) are given by
Inserting the initial conditions we obtain an expression for the transition
amplitude cf (t) to the final state φf :
If the perturbing operator ˆW , and hence the matrix element Wfi, do not
depend on time, we can integrate directly and obtain:
For large times t, becomes
This means that for large times t the transition probability per unit time, P i→f
becomes independent of t:
It makes sense to assume that the diagonal matrix elements < φi|ˆW |φi > and
<φf |ˆW |φf> vanish, because a perturbing operator diagonal in the unperturbed
basis doesn’t cause transitions. Then Ei and Ef are not only the eigenvalues
of the unperturbed Hamiltonian ˆH0 in the initial and final state respectively,
but they are also the expectation values of the full Hamiltonian ˆH = ˆH0 + ˆW
in the respective states. The delta function for the transition probabilities expresses
energy conservation in the long-time limit. In many practical examples (such as the
electromagnetic decay of an atomic state) the energy spectrum of the final states of
the whole system (in this case of atom plus electromagnetic field) is continuous. In
order to obtain the total probability per unit time for transitions from the initial state
φi to all possible final states φf we must integrate over an infinitesimal energy
range around Ei:
Density of final states
The formula is Fermi’s famous Golden Rule; it gives the probability
per unit time for transitions caused by a time-independent perturbing operator
in first-order perturbation theory.
The precise definition of the density ρ(Ef ) of final states φf depends on
the normalization of the final states. Consider for example a free particle in a
one-dimensional box of length L. The number of bound states (normalized to
unity) per unit energy is
When applying the Golden Rule we have to take care that the density of the
final states and their normalization are chosen consistently.
For classical fields the potentials A(r, t) and Φ(r, t) are real-valued functions.
For a fully quantum mechanical treatment of a system consisting of an atom
and an electromagnetic field we need a Hamiltonian encompassing the atomic
degrees of freedom, the degrees of freedom of the field and interaction term.
The solution of such Hamiltonian is complex and for one who is interesting can find it
in advanced literature on this subject.
Here will only be reviewed results from this theory of the interest to the transition
selection rules.
In most cases of interest, the wave lengths λ of the photons emitted
or absorbed by an atom are much larger than its spatial dimensions. In such
conditions the electromagnetic transitions in atoms or molecules are treated in the
dipole approximation.
In the dipole approximation, the interaction operator can be written as::
Where
are the electric and magnetic dipole operators of the
molecular system, respectively. If the electric and magnetic field strengths vary over the
size of the molecular system, one must also consider the interaction of the
electromagnetic fields with the quadrupoles, octupoles, etc. of the molecules. The
electric dipole interaction is the dominant interaction in the microwave, infrared, visible
and ultraviolet ranges of the electromagnetic spectrum ( large). The magnetic dipole
interaction is used in spectroscopies probing the magnetic moments resulting from
the electron or nuclear spins such as EPR and NMR. At short wavelength, i.e., for Xand -rays, the dipole approximation breaks down because λ< d.
Selection rules for the electric dipole approximations