Two- and Three-Dimensional Electron Traps — Question 8

Question 8

Field polarization selects which confinement index can change in a dipole transition

The three-dimensional box eigenfunctions factor into normalized one-dimensional functions $u_{n_x}(x)u_{n_y}(y)u_{n_z}(z)$. A weak x-polarized field couples through x. In one dimension the off-diagonal position matrix element is nonzero only for opposite-parity indices.

1. Factor the dipole matrix element and state the index selection rules.

2. From state (1,1,1), compare transitions to (2,1,1) and (2,2,1), including whether another linear polarization can permit the latter in first-order dipole coupling.

Question 8 - Solution

Field polarization selects which confinement index can change in a dipole transition

(a) Method and calculation

$$\langle r_xr_yr_z|x|n_xn_yn_z\rangle=x_{r_xn_x}\delta_{r_yn_y}\delta_{r_zn_z}.$$$$\boxed{r_y=n_y,\quad r_z=n_z,\quad r_x+n_x\text{ odd}}\quad\text{for an off-diagonal transition}.$$

(b) Interpretation and further calculation

$$\boxed{(1,1,1)\to(2,1,1)\text{ allowed for x polarization}}.$$$$\langle221|(e_xx+e_yy+e_zz)|111\rangle=0\Rightarrow\boxed{111\to221\text{ dipole-forbidden}}.$$

Physical check

Each linear coordinate operator can change only its own factor. Energy matching alone does not guarantee a dipole transition.