The transition dipole moment or transition moment, usually denoted for a transition between an initial state, , and a final state, , is the electric dipole moment associated with the transition between the two states. In general the transition dipole moment is a complex vector quantity that includes the phase factors associated with the two states. Its direction gives the polarization of the transition, which determines how the system will interact with an electromagnetic wave of a given polarization, while the square of the magnitude gives the strength of the interaction due to the distribution of charge within the system. The SI unit of the transition dipole moment is the Coulomb-meter (Cm); a more conveniently sized unit is the Debye (D). Definition A single charged particle For a transition where a single charged particle changes state from to , the transition dipole moment is where q is the particle's charge, r is its position, and the integral is over all space ( is shorthand for ). The transition dipole moment is a vector; for example its x-component is In other words, the transition dipole moment can be viewed as an off-diagonal matrix element of the position operator, multiplied by the particle's charge. Multiple charged particles When the transition involves more than one charged particle, the transition dipole moment is defined in an analogous way to an electric dipole moment: The sum of the positions, weighted by charge. If the ith particle has charge qi and position operator ri, then the transition dipole moment is: In terms of momentum For a single, nonrelativistic particle of mass m, in zero magnetic field, the transition dipole moment between two energy eigenstates ψa and ψb can alternatively be written in terms of the momentum operator, using the relationship This relationship can be proven starting from the commutation relation between position x and the Hamiltonian : Then However, assuming that ψa and ψb are energy eigenstates with energy Ea and Eb, we can also write Similar relations hold for y and z, which together give the relationship above. Analogy with a classical dipole A basic, phenomenological understanding of the transition dipole moment can be obtained by analogy with a classical dipole. While the comparison can be very useful, care must be taken to ensure that one does not fall into the trap of assuming they are the same. In the case of two classical point