occur at This suggests that the pair of particles that are bound and separated by a distance act as a single quasiparticle with angular momentum . If we scale the lengths as , then the interaction energy becomes where The value of the at which the energy is minimum, , is independent of the ratio . However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when When the ratio differs from 1, then the energy minimum is higher (Figure 3). Therefore, for even values of total momentum, the lowest energy occurs when (Figure 4) or where the total angular momentum is written as When the total angular momentum is odd, the minima cannot occur for The lowest energy states for odd total angular momentum occur when or and which also appear as series for the filling factor in the fractional quantum Hall effect. Charge density spread over a wave function The charge density is not actually concentrated in a delta function. The charge is spread over a wave function. In that case the electron density is The interaction energy becomes where is a confluent hypergeometric function or Kummer function. In obtaining the interaction energy we have used the integral (see ) As with delta function charges, the value of in which the energy is a local minimum only depends on the total angular momentum, not on the angular momenta of the individual currents. Also, as with the delta function charges, the energy at the minimum increases as the ratio of angular momenta varies from one. Therefore, the series and appear as well in the case of charges spread by the wave function. The Laughlin wavefunction is an ansatz for the quasiparticle wavefunction. If the expectation value of the interaction energy is taken over a Laughlin wavefunction, these series are also preserved. Magnetostatics Darwin interaction in a vacuum A charged moving particle can generate a magnetic field that affects the motion of another charged particle. The static version of this effect is called the Darwin interaction. To calculate this, consider the electrical currents in space generated by a moving charge with a comparable expression for . The Fourier transform of this current is The current can be decomposed into a transverse and a longitudinal part (see Helmholtz decomposition). The hat indicates a unit vector. The last term disappears because which results from