Atomic dipole moment for a superposition state

[DivGradCurl]

Hi all,

I'm trying to understand how to calculate the time dependent expectation value of the atomic dipole moment for a superposition state, and I have a good guess to check with you. Say we have
$$\psi = \frac{1}{\sqrt{2}} \left[ \psi _{100} + \psi _{310} \right]$$
at t = 0. Then, for t > 0:
$$\Psi (t) = \frac{1}{\sqrt{2}} \left[ \psi _{100} e^{-iE_1 t/\hbar} + \psi _{310} e^{-iE_3 t/\hbar} \right] = \frac{1}{\sqrt{2}} \left[ \psi _{100} e^{-i\omega_1 t} + \psi _{310} e^{-i\omega_3 t} \right]$$
Given that the time dependent expectation value of the atomic dipole moment is defined as
$$\langle \vec{d}(t) \rangle = -e \langle \vec{r} (t) \rangle$$
I proceed as follows:
$$\langle \vec{d}(t) \rangle = -e \langle \Psi (t) | \mbox{ } \vec{r} \mbox{ } | \Psi (t) \rangle$$
$$\langle \vec{d}(t) \rangle = -e \langle \left( \frac{1}{\sqrt{2}} \psi _{100} e^{-i\omega_1 t} + \frac{1}{\sqrt{2}} \psi _{310} e^{-i\omega_3 t} \right) | \mbox{ } \vec{r} \mbox{ } | \left( \frac{1}{\sqrt{2}} \psi _{100} e^{-i\omega_1 t} + \frac{1}{\sqrt{2}} \psi _{310} e^{-i\omega_3 t} \right) \rangle$$
$$\langle \vec{d}(t) \rangle = -e \langle \left( \frac{1}{\sqrt{2}} \psi _{100} e^{-i\omega_1 t} \right) | \mbox{ } \vec{r} \mbox{ } | \left( \frac{1}{\sqrt{2}} \psi _{310} e^{-i\omega_3 t} \right) \rangle -e \langle \left( \frac{1}{\sqrt{2}} \psi _{310} e^{-i\omega_3 t} \right) | \mbox{ } \vec{r} \mbox{ } | \left( \frac{1}{\sqrt{2}} \psi _{100} e^{-i\omega_1 t} \right) \rangle$$
$$\langle \vec{d}(t) \rangle = -e \left( \frac{1}{\sqrt{2}} e^{+i\omega_1 t} \right) \langle \psi _{100} | \mbox{ } \vec{r} \mbox{ } | \psi _{310} \rangle \left( \frac{1}{\sqrt{2}} e^{-i\omega_3 t} \right) -e \left( \frac{1}{\sqrt{2}} e^{+i\omega_3 t} \right) \langle \psi _{310} | \mbox{ } \vec{r} \mbox{ } | \psi _{100} \rangle \left( \frac{1}{\sqrt{2}} e^{-i\omega_1 t} \right)$$
$$\langle \vec{d}(t) \rangle = -e \left( \frac{1}{\sqrt{2}} e^{+i\omega_1 t} \right) \langle 100 | \mbox{ } \vec{r} \mbox{ } | 310 \rangle \left( \frac{1}{\sqrt{2}} e^{-i\omega_3 t} \right) -e \left( \frac{1}{\sqrt{2}} e^{+i\omega_3 t} \right) \langle 310 | \mbox{ } \vec{r} \mbox{ } | 100 \rangle \left( \frac{1}{\sqrt{2}} e^{-i\omega_1 t} \right)$$
$$\langle \vec{d}(t) \rangle = -e \left( \frac{1}{\sqrt{2}} e^{+i\omega_1 t} \right) \langle 10 | \mbox{ } r \mbox{ } | 31 \rangle \langle 00 | \mbox{ } \hat{r} \mbox{ } | 10 \rangle \left( \frac{1}{\sqrt{2}} e^{-i\omega_3 t} \right) -e \left( \frac{1}{\sqrt{2}} e^{+i\omega_3 t} \right) \langle 31 | \mbox{ } r \mbox{ } | 10 \rangle \langle 10 | \mbox{ } \vec{r} \mbox{ } | 00 \rangle \left( \frac{1}{\sqrt{2}} e^{-i\omega_1 t} \right)$$
$$\langle \vec{d}(t) \rangle = -e \left( \frac{1}{\sqrt{2}} e^{+i\omega_1 t} \right) (0.517 a_0) \left( 0, 0, \frac{1}{\sqrt{3}}\right) \left( \frac{1}{\sqrt{2}} e^{-i\omega_3 t} \right) -e \left( \frac{1}{\sqrt{2}} e^{+i\omega_3 t} \right) (0.517 a_0) \left( 0, 0, \frac{1}{\sqrt{3}}\right) \left( \frac{1}{\sqrt{2}} e^{-i\omega_1 t} \right)$$
$$\langle \vec{d}(t) \rangle = -e \frac{1}{2} e^{+i\omega_1 t} e^{-i\omega_3 t} (0.517 a_0) \left( 0, 0, \frac{1}{\sqrt{3}}\right) -e \frac{1}{2} e^{+i\omega_3 t} e^{-i\omega_1 t} (0.517 a_0) \left( 0, 0, \frac{1}{\sqrt{3}}\right)$$
$$\langle \vec{d}(t) \rangle = \left[ -e \frac{1}{2} (0.517 a_0) \left( 0, 0, \frac{1}{\sqrt{3}}\right) \right] \left( e^{+i\omega_1 t} e^{-i\omega_3 t} + e^{+i\omega_3 t} e^{-i\omega_1 t}\right)$$
$$\langle \vec{d}(t) \rangle = \left[ -e \frac{1}{2} (0.517 a_0) \left( 0, 0, \frac{1}{\sqrt{3}}\right) \right] \left( e^{+i(\omega_1 - \omega_3) t} + e^{-i(\omega_1 - \omega_3) t} \right)$$
$$\langle \vec{d}(t) \rangle = \left[ -e \frac{1}{2} (0.517 a_0) \left( 0, 0, \frac{1}{\sqrt{3}}\right) \right] 2\cos \left[(\omega_1 - \omega_3) t \right]$$
$$\fbox{\displaystyle\langle \vec{d}(t) \rangle = -e (0.517 a_0) \left( 0, 0, \frac{1}{\sqrt{3}}\right) \cos \left[(\omega_1 - \omega_3) t \right]}$$

Is this correct? Thanks!

 

[Orodruin]

I did not check your math but the approach is reasonable.