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What is the difference between the potential $V$ and the effective potential $V_{eff}$?

Some times when solving problems, an effective potential $V_{eff}$ is defined and its usually equal to the potential $V$ plus other terms. My question why is this potential called an effective potential.

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As wiki says "The effective potential (also known as effective potential energy) is a mathematical expression combining multiple (perhaps opposing) effects into a single potential." Basically the concept of the effective potential simplifies the equations of motion and simplifies their analysis.

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The effective potential is the potential of interaction you measure between two (or more) emergent physical objects when you forget (or "trace over" in the jargon) certain degrees of freedom of a more detailed model.

If you take two pinned charges in vacuum for instance, they will interact with a "bare" Coulomb interaction in $\sim 1/r$.

If you put these two charges in a sea of mobile positive and negative charges, they will polarize that medium by interacting with a bare Coulomb potential with all the mobile charges and, if we trace over the degree of freedom of this sea of charges, everything is as if, the two pinned charges interacted with a short ranged screened Coulomb potential in $\sim e^{-\kappa r}/r$.

Now, as far as I am concerned, every potential is "effective" in one sense or another. Even the bare Coulomb potential I was talking about has parts of its magnitude that originates from polarizing the vacuum itself, so I don't think you should be too puzzled about it; there is nothing very deep about it...besides the fact that all potentials are effective in some sense.

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