Electric Dipole Moment

One of the more important results from our framework for quantum mechanics is that we can develop descriptions of other physical quantities (e.g., $\mathbf{L}$) based on our more fundamental operators (e.g., $\hat{x}$ and $\hat{p}$). Now that we have developed a model for the Hydrogen atom, we can begin to explore quantities like the electric dipole moment:

\[\mathbf{d} = q\mathbf{r} = q(x\mathbf{i}+y\mathbf{j}+z\mathbf{k})\]

In this activity, we will set up the integral needed to calculate the expectation value of this electric dipole moment, $\langle \mathbf{d} \rangle$.

Consider the state $\ket{210}$.

  1. Have a look at the density plots in McIntyre (Fig. 8.7 on page 267). Given the desnity plot for $\ket{210}$, what do you expect for $\langle \mathbf{d} \rangle$? How can you tell?
  2. Write the position representation of this state $\langle x \vert 210 \rangle$. Then write it in Cartesian coordinates.
  3. Time evolve the state. If you were to calculate $\langle \mathbf{d} \rangle = \langle 210 \vert \mathbf{d} \vert 210 \rangle$, do you expect the result to be time dependent? Why or why not?
  4. Set up the necessary integrals to compute $\langle \mathbf{d} \rangle$. Notice that you will have the sum of three integrals as you are computing the expectation value of a vector quantity.
  5. Look carefully at the integrals. One of the things we can exploit in these kinds of calculations is if the integrand is an odd function and the domain of the integral is even (symmetric), then the contributions from either side of the integral cancel and the integral is zero. Do you notice anything like that in your integrals?