Register to reply

Is the energy density normalized differently in the quantum case?

by Hypersphere
Tags: case, density, differently, energy, normalized, quantum
Share this thread:
Hypersphere
#1
Dec7-13, 11:26 PM
P: 184
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
[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.
Phys.Org News Partner Physics news on Phys.org
'Squid skin' metamaterials project yields vivid color display
Scientists control surface tension to manipulate liquid metals (w/ Video)
Simulation method identifies materials for better batteries
Hypersphere
#2
Dec10-13, 11:36 AM
P: 184
Actually, the author of those notes probably just switched to a complex field
[tex]E_V=\sqrt{\frac{\epsilon_0}{2}}E + i\frac{B}{\sqrt{2\mu_0}}[/tex]
in which case the energy density comes out as
[tex]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[/tex]
as it should.


Register to reply

Related Discussions
Sum of signal and its probability density (special case...) Set Theory, Logic, Probability, Statistics 1
Dealing with normalized quantum functions Calculus & Beyond Homework 3
At what redshift does energy density in matter equal energy density in radiation? Cosmology 5
Normalized function and minimum energy Quantum Physics 4
Q:What is the analogue of the excitation vector in the case of mass density? General Physics 2