# Dipole approximation in acceleration form

• kknull
In summary, the conversation discusses the dipole approximation in a hydrogenic atom and the proof that <b| [p,V] | a> = <b| grad(V) |a> or <b| V grad |a> = 0. The suggested method for proving this uses commutators and expanding V as a power series.
kknull
hi,

I have 2 localized states in a hydrogenic atom.. we're in dipole approimation.
I have to proof that

<b| [p,V] | a> = <b| grad(V) |a>
or equally:
<b| V grad |a> = 0

(this is the dipole approximation in acceleration form)

someone can help me?

page 323 eq (8.22)

thanks :)

Last edited by a moderator:
first work out the commutators:

[p,x]=?

[p,x^2]=?

[p,x^3]=?

and so on...

Then expand V as a power series and use the above commutators to prove the relation.

## 1. What is the dipole approximation in acceleration form?

The dipole approximation in acceleration form is a simplification of the full electromagnetic field equations used to describe the behavior of charged particles. It assumes that the acceleration of a charged particle is much smaller than the speed of light, allowing for a linear approximation of the electric and magnetic fields.

## 2. Why is the dipole approximation in acceleration form useful?

The dipole approximation in acceleration form is useful because it allows for a simpler and more manageable way to analyze the behavior of charged particles in electromagnetic fields. It also provides a good approximation for many practical applications, such as in laser-matter interactions and atomic physics.

## 3. How is the dipole approximation in acceleration form derived?

The dipole approximation in acceleration form is derived by taking the first-order Taylor expansion of the full electromagnetic field equations, assuming small acceleration and neglecting terms of higher order. This results in a set of simplified equations that can be solved to describe the motion of charged particles in electromagnetic fields.

## 4. What are the limitations of the dipole approximation in acceleration form?

The dipole approximation in acceleration form is limited by the assumption that the acceleration of charged particles is much smaller than the speed of light. This means that it is not accurate for high-energy particles or in situations where the acceleration is significant, such as in particle accelerators.

## 5. How can the dipole approximation in acceleration form be improved?

The dipole approximation in acceleration form can be improved by including higher-order terms in the expansion, which can account for larger accelerations and higher energies. Additionally, more advanced techniques such as numerical simulations can be used to accurately model the behavior of charged particles in complex electromagnetic fields.

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