# How Do Electrons “Resonate Against the Restoring Force Of Positive Nuclei?”

While reading "Surface Plasmon Resonance," I came across the following:

"The resonance condition is established when the frequency of light photons matches the natural frequency of surface electrons oscillating against the restoring force of positive nuclei."

1.) What is the "restoring force of positive nuclei"?

2.) How do electrons resonate against the restoring force of the positive nuclei?

I realize that electrons and protons attract each other and that they both behave like waves, but the concept of a "restoring force" eludes me.

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This is too long for a comment, so here it goes:

What causes a valence electron of a large atom to feel an effective attractive potential that is similar to a harmonic oscillator?

I hope people following this forum at least understand that in order to have electrons around nuclei display a stable solution and form an atom, a potential is necessary.

This potential is very complicated since it is the result of a many body problem.

Fortunately we have mathematical tools to help us estimate the potential of a valence electron by the following logic:

The unknown real potential must be well behaved since the atom is stable and therefore it can be expanded in a Taylor series expansion in the variable of interest, call it r, the radial variable for the potential.

The first few terms are dominant in the series for r<1. The potential must be symmetric because no large asymmetries have been observed in atoms, thus the coefficient of the r term must be zero. The first power that is the strongest in the expansion ( by the form of a Taylor expansion) is the r^2 which is the function of a harmonic oscillator potential, well studied with the Schrodinger equation.

In a first approximation in quantum mechanics, all attractive symmetric potentials have as a dominant term the harmonic oscillator functional form thus one can use the language of a "restoring force" , because a spring is a harmonic oscillator and the terminology carries over.

To address the "resonate" part: a harmonic oscillator restores i.e. pulls back the particle and it does so even in classical physics with a certain frequency ( think of a spring again). In quantum mechanics there still is a frequency associated with the position of the electron's orbital. When the disturbance is of the same frequency as the frequency of the potential solution for the electron one can have a resonance phenomenon.

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I have a solid background in math (Calc 1-3, DE) and almost a minor in physics. Is there a book or source that you would recommend for an in-depth look at this topic? – Dale Apr 1 '13 at 20:54
What leads us to believe that the r^2 value of the Taylor series is dependent on a harmonic oscillator potential? – Dale Apr 1 '13 at 21:04
The potential for the harmonic oscillator classically goes as x^2 see en.wikipedia.org/wiki/Harmonic_oscillator. Look at the Hamiltonian (K energy + potential energy) in the link in my answer. They both go like x^2 . There are references in the links but the harmonic oscillator in quantum mechanics will be found in all first books. Relativistic field theory books start with the harmonic oscillator (at least the one I have rom my student days 50 years ago) Even the frontier research on strings has a multidimensional harmonic oscillator as its basis to start with. – anna v Apr 2 '13 at 4:48
"both go like x^2" refers to both classical and quantum. – anna v Apr 2 '13 at 4:55

The effective potential that a valence electron of a large atom sees vaguely resembles a harmonic oscillator. The "restoring force" is just the valence electron's attraction to the positively charged parent ion.

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What causes a valence electron of a large atom to feel an effective attractive potential that is similar to a harmonic oscillator? – Dale Apr 1 '13 at 3:08
@JoeHobbit: It's an attractive potential from the nucleus combined with the exchange interaction from the core electrons. You get a minimum, and anything with a minimum is a harmonic oscillator to second order. – Dan Apr 1 '13 at 8:50