Is this interpretation logical? (Perturbation theory)

[ani4physics]

Hi all. Just have a quick question on perturbation theory. Let's consider a molecule in ground electronic state. If a time-independent external perturbation acts on the molecule, the average electronic energy is going to change. From time-independent perturbation theory, we know that

<E> = E(0) + lambda . E(1) + (lamdba)^2 . E(2) + ...

Where lambda is the external parameter which determines the strength of the perturbation. E(0) is the unperturbed energy of teh ground state. If we substract E(0) from <E>, we get,

delta E = lambda . E(1) + (lamdba)^2 . E(2) + ...

Now my question is: when the perturbation acts, the net energy change of the molecule is delta E. If we consider the external perturbation as "an external force", can we interpret delta E as the net work done by the external force on the molecule?

Please let me know if this interpretation is correct. Happy holidays to everyone.

 

[ani4physics]

Any answer will be greatly appreciated guys.

Let me rephrase the question in a different way: Let's consider and external electric field acting on an atom. We know that the field will polarize the electronic cloud and will induce dipole moment in it. How do we proceed to calculate how much work is required by the external field in order to polarize the atom.

 

[Bob S]

Let's consider and external electric field acting on an atom. We know that the field will polarize the electronic cloud and will induce dipole moment in it. How do we proceed to calculate how much work is required by the external field in order to polarize the atom.

If you include classical electric polarizability of gas atoms in electromagnetic fields, look up Rayleigh scattering of sunlight in our atmosphere. The only "work" or energy "loss" is due to classical dipole radiation. See

http://en.wikipedia.org/wiki/Rayleigh_scattering

[added] The Clausius Mosotti relation extends this discussion to the polarizability of solids and liquids. See

http://en.wikipedia.org/wiki/Clausius–Mossotti_relation

Bob S

 

[ani4physics]

Thanks for the reply. Still that does not answer my question. Let me again rephrase the queston:

Let's consider an electric field acting on a H atom. The operator for the force acting on the electron is F(i).e(ri), where i denotes the elecron, ri is the coordinate of electron i, and F(i) is the external field on electron i. The external force will modify the ground state electronic wave function, and the wave function will be,

psi(0) + lambda psi(1) + ...,

where psi(0) is the unperturbed wave function, psi(1) is the first-order correction, and so on. lambda determines the strength of the field.

Now, how much work is done by the external force in order to change the wave function from psi(0) to psi(0) + lambda psi(1) + ...?

 

[Bob S]

Electric dipole excitation is discussed in

http://en.wikipedia.org/wiki/Electric_dipole_transition

http://farside.ph.utexas.edu/teaching/qm/lectures/node64.html

http://webpages.ursinus.edu/lriley/courses/p212/lectures/node40.html

The lowest excited state of the H atom is the 2p state, 10.2 eV above the 1s ground state. It can be excited by a 10.2 eV (1215 Angstrom) photon, and decays via a 2p-->1s transition to the 1s state with a lifetime of ~ 1.6 nsec. So the "work" required to raise the electron to the 2p state is 10.2 eV.

Bob S

 

[Bob S]

A dc electric field can strip the "outer" electron off the hydrogen atom negative ion (1 proton plus 2 electrons). The binding energy of the outer electron is ~ 0.75 eV, and there are no other bound states. The Lorentz transform of a magnetic field for a moving ion in the lab ("magnetic stripping") is an electric field in the ion rest frame. See

G. M. Stinson, W. C. Olsen, W. J. McDonald,
P. Ford, D. Axen, and E. W. Blackmore, "Elec-
tric Dissociation of H- Ions by Magnetic
Fields," Nucl. Instrum. Meth. 74, 33 (1969).

and "NEUTRALIZATION OF H^- BEAMS BY MAGNETIC STRIPPING" in

http://accelconf.web.cern.ch/AccelConf/p81/PDF/PAC1981_2704.PDF

Bob S

 

[Bob S]

Here is a paper that gives the electric field dependence of the bound "outer" electron lifetime in the negative hydrogen atom ion. The outer electron is bound by about 0.75 eV, and the tunneling lifetime through the 1/r Coulomb barrier depends on the applied electric field. The plots on pages 9 and 12 show the tunneling lifetimes for electric fields ranging from about 2 to 7 MV/cm. The tunneling lifetimes range from about 10^-4 seconds to 10^-10 seconds. The measurements are compared to the Fowler-Nordheim formula for the electron tunneling lifetime through a 1/r Coulomb barrier with an added electric field.

http://www-bd.fnal.gov/pdriver/H-workshop/jason.pdf

Bob S