Dipole excitation response function - physical interpretation

[crock88]

Hi everyone, I'm a new member but it's not the first time I look at the forum.
Well, I don't know if this is the right section to post my question. I think it is related to quantum mechanics interpretation too. Anyway, let's have a look at my problem.

I've computed cross section for photon scattering on a nucleus. What I get is a kinematic factor (not really interesting) and a dynamic part which is substantially what I call response function. This response function is proportional to matrix elements of nucleus currents between initial and final states. Then I performed a multipolar expansion and until now it's all ok.
Consider in particular the dipolar term, i.e.

R(\omega)\propto<\psi_f|\hat{D}|\psi_i>

where we have the expectation value of the dipole operator between an initial and final state.
Now, what is the physical interpretation of the dipole operator acting between these two states?
This response function describes just how the system responds to a perturbation induced by the photon field. So computing it I would say that we are going to see how the system changes when I perturb it with a photon of given energy \omega[\itex]. But where the dipole operator enters in this? I would say the same if instead of the dipole operator i would have the quadrupole operator.<br /> <br /> What does the dipole operator do? I start with an initial wavefunction, I act on it with the dipole operator and then i want to see the overlap of this state with a final one. How is the wavefunction modified? Is it really modified? Can I still use a wavefunction interpretation? <br /> <br /> P.S. I&#039;ve seen already a similar discussion on this forum but the answers did not convinced me.<br /> <br /> Thanx for the attention!

 

[andrien]

A transition can take place via dipole operator acting between the states.Sometimes dipole transitions are forbidden then it take place via higher order like quadrupole transitions.

 

[crock88]

A transition can take place via dipole operator acting between the states.Sometimes dipole transitions are forbidden then it take place via higher order like quadrupole transitions.

I know this, but I don't "see" it. What does "take place via dipole operator acting between the states" mean? I know this is the way one usually says and then there are selection rules and so as you say some transitions are forbidden.

 

[andrien]

I know this, but I don't "see" it. What does "take place via dipole operator acting between the states" mean? I know this is the way one usually says and then there are selection rules and so as you say some transitions are forbidden.

Introduction of dipole operator takes place while you evaluate the transition rate with e^ikx factor set equal to 1.It belongs to quantum theory of radiation where when you evaluate the matrix element between two states by taking the interaction hamiltonian b/w the two states where interaction contains part(with one photon transition only) the term like A.p.You put the plane wave form for A(vector potential) which yields term like <B|p.ε^α e^-ikx|A> for the transition.we utilise a series expansion for exponential term,with the approximation e^-ikx=1 known as dipole approximation because in this case you have
<B|p.ε^α|A> or <B|p|A>.ε^α.Now we have [p²,x]=-2ih^-p,which applied to <B|p|A> gives<B|[H₀,x](im/h^-)|A>=E_B-E_A/h^-(imx_BA).so you can see that the matrix element is reduced to finding it for x.Higher order calculations are more complicated say for quadrupole you get a second rank traceless tensor with x_ix_j-δ_ij/3(|x|²) taken.The selection rules for dipole transitions can be inferred using only group theory.However all selection rules can be obtained from wigner eckart theorem which should lead to non vanishing clebsch gordon coefficient coefficient gives the selection rules for dipole,quadrupole,M1 transition etc.For more higher orders it is rather beneficial to invoke vector spherical harmonics.