Transition Dipoles and Dipole Strengths

The matrix element that we denote as fiba in Eq. (4.8c) is called a 

To see that the dipole moment of the superposition state oscillates with time, consider the same states at time t = (1 /2)h/ Eb — Ea). For the particle in a box, the energy of the second eigenstate is four times that of the first, Eb = AEa (Eq. 2.24), so ll2)hl Eb — Ea) = /6)h/Ea- The individual wavefunctions at this time have both real and imaginary parts. For the lower-energy state, we find by using the relationship exp(—/0) = cos (0) — i sin (0) that 

For the higher-energy state, which oscillates four times more rapidly, 

This differs from the superposition state at zero time in that it depends on the difference between t/ aCx) and instead of the sum (Fig. 4.3B). Inspection of the electron density function in Fig. 4.3 shows that the electric dipole of the superposition state has reversed direction relative to the orientation at 

The wavefunction of the superposition state returns to its initial shape at t = h/ Ei, — Ea), when the phases of the time-dependent parts of Ta and are 2jr/3 and 8jt/3, respectively. The spatial part of Ea + I b thus oscillates between symmetric and antisymmetric combinations of -ijja and ipi, tpa + 4 b and V a — ipb) with a period of h/ Eb — Ea), and the electric dipole oscillates in concert. 

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