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Electromagnetic transitions Transitions in general: Fermi’s Golden Rule Dipole approximation Transition selection rules

Transitions in general: Fermi’s Golden Rule Dipole ...tulej/Spectroscopy_related_aspects/Lecture25... · Electromagnetic transitions Transitions in general: Fermi’s Golden Rule

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Page 1: Transitions in general: Fermi’s Golden Rule Dipole ...tulej/Spectroscopy_related_aspects/Lecture25... · Electromagnetic transitions Transitions in general: Fermi’s Golden Rule

Electromagnetic transitions

Transitions in general: Fermi’s Golden RuleDipole approximation

Transition selection rules

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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 eigenstatesare solutions of the corresponding stationary Schrödinger equation. The eigenstatesof Hamiltonian are usually “seen” by observing electromagnetic radiation emitted orabsorbed during a transition between two eigenstates. The fact that suchtransitions 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 thedegrees of freedom of the electromagnetic field. A Hamiltonian able to describeelectromagnetic 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 time-

dependent perturbation theory which is expounded in the following section.

Comments:

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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. ThisHamiltonian 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-dependentwave function ψ(t):

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The coefficients cn

(t) in this expansion are time dependent, because the timeevolution of the eigenstates of ˆH0 is, due to the perturbation ˆW , not given bythe exponential functions alone.

The initial condition that the system be in the eigenstate φi of ˆH0 at timet = 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 φfof ˆH0 is:

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In order to calculate the coefficients cn(t) we insert the expansion in the time- dependent Schrödinger equation and obtain using

If we multiply from the left with the φ*m

becomes a system ofcoupled ordinary differential equations for the coefficients cn

(t):

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We can formally integrate the equations:

To first order in the matrix elements of the perturbing operator ˆW , thecoefficients cn

(t) are given by

Inserting the initial conditions we obtain an expression for the transition amplitude cf (t) to the final state φf :

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If the perturbing operator ˆW , and hence the matrix element Wfi , do notdepend 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:

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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 unperturbedbasis doesn’t cause transitions. Then Ei and Ef are not only the eigenvaluesof the unperturbed Hamiltonian ˆH0 in the initial and final state respectively,but they are also the expectation values of the full Hamiltonian ˆH = ˆH0 + ˆWin 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 stateφi to all possible final states φf we must integrate over an infinitesimal energyrange around Ei :

Density of final states

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The formula is Fermi’s famous Golden Rule; it gives the probabilityper unit time for transitions caused by a time-independent perturbing operatorin first-order perturbation theory.

The precise definition of the density ρ(Ef ) of final states φf depends onthe normalization of the final states. Consider for example a free particle in aone-dimensional box of length L. The number of bound states (normalized tounity) 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.

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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 atomand an electromagnetic field we need a Hamiltonian encompassing the atomicdegrees 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 emittedor 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.

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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 fromthe electron or nuclear spins such as EPR and NMR. At short wavelength, i.e., for X- and -rays, the dipole approximation breaks down because λ< d.

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Selection rules for the electric dipole approximations

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