Quantization of the electric field inside a box

[thariya]

Hello all,

The second quantization of a general electromagnetic field assumes the energy density integration to be performed inside a box in 3D space. Someone mentioned to me recently that the physical significance of the actual volume used is that it should be chosen based on the detector used to measure some attribute of the field; say the electric field in the radial-direction at some point in space. For example, for an electromagnetic source and a detector placed as shown in the attached image, the integration volume(the box) should be chosen as shown. I'm looking to understand why this is the case. Could anyone please point me in the right direction?

thanks,
Tharindu.

 

[eljose79]

Hello Tharindu,

Thank you for bringing up this interesting topic. The choice of integration volume in second quantization of the electromagnetic field is indeed important and is based on the detector used for measurement. Let me explain why this is the case.

In second quantization, we are dealing with a quantized version of the electromagnetic field, where the field is described in terms of creation and annihilation operators for photons. These operators act on a quantum state to create or destroy a photon. In order to calculate the energy density of the field, we need to perform an integration over all possible modes of the field.

Now, the modes of the field depend on the geometry and size of the integration volume. For example, if we have a spherical detector, the modes of the field will be different compared to a cylindrical detector. This is because the boundary conditions for the field are different in each case.

Therefore, in order to accurately calculate the energy density of the field for a specific detector, we need to choose an integration volume that matches the geometry and size of the detector. This ensures that we are including all the relevant modes of the field in the calculation.

I hope this helps in understanding the significance of choosing the integration volume based on the detector used. For further reading, I would recommend looking into the theory of cavity quantum electrodynamics, which deals with the interaction between quantized electromagnetic fields and confined atoms or molecules.

Best of luck in your research!