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.