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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

[tex]|g_\mathbf{k}|^2=\frac{\omega_\mathbf{k}}{2\hbar \epsilon_0 V} \left( d^2 \cos^2 \theta \right)[/tex]

where [itex]d[/itex] is the dipole moment and [itex]\theta[/itex] is the angle between the dipole moment and the polarization vector.

These notes claim that the vacuum field amplitude satisfy the normalization

[tex]\int \epsilon_0 E^2 d^3r = \frac{\hbar \omega}{2}[/tex]

which does lead to the above form of [itex]|g|^2[/itex], 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

[tex]u_E=\frac{1}{2} \epsilon_0 E^2 [/tex]

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

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# Is the energy density normalized differently in the quantum case?

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