Is the energy density normalized differently in the quantum case?

[Hypersphere]

Hi all,

This is all in the context of interaction between (two-level) atoms and an electromagnetic field, basically the Wigner-Weisskopf model. In particular, I tried to derive the value of the atom-field interaction constant and show that it satisfied
$$|g_\mathbf{k}|^2=\frac{\omega_\mathbf{k}}{2\hbar \epsilon_0 V} \left( d^2 \cos^2 \theta \right)$$
where $d$ is the dipole moment and $\theta$ is the angle between the dipole moment and the polarization vector.

http://www.stanford.edu/~rsasaki/AP387/chap6 claim that the vacuum field amplitude satisfy the normalization
$$\int \epsilon_0 E^2 d^3r = \frac{\hbar \omega}{2}$$
which does lead to the above form of $|g|^2$, but from classical electrodynamics (eg. eq. (6.106) in Jackson, 3rd ed.) I'm used to defining the energy density of the electric field as
$$u_E=\frac{1}{2} \epsilon_0 E^2$$

Now, the notes seem to use a energy density that is $2u_E$. Is there a good explanation for this, or does it boil down to one of these conventions? Thanks in advance.

 

[Hypersphere]

Actually, the author of those notes probably just switched to a complex field
$$E_V=\sqrt{\frac{\epsilon_0}{2}}E + i\frac{B}{\sqrt{2\mu_0}}$$
in which case the energy density comes out as
$$u=\int |E_V|^2 d^3 r = \int \left( \frac{\epsilon_0}{2}E^2 + \frac{B^2}{2\mu_0} \right) d^3 r$$
as it should.