תשובה לשאלה שנשאלתי

מנהל: dcohen

yoavzig
הודעות: 15
הצטרף: 20:48 23/11/2019

תשובה לשאלה שנשאלתי

Hello Everyone,

In the previous Tirgul I was asked why is the sum over momenta and spin,

$$\sum_{\vec{k},s}=\sum_s \int \frac{V d^3 k}{(2\pi)^3} ?$$

I didn't provide a good enough answer; however now I intend to give one. I didn't specify in the Tirgul that the system of the N
electrons lives in a box of volume V=L^3 with periodic boundary conditions. This implies that the momenta are quantized

$$\vec{k}=\frac{2\pi}{L} \vec{n}$$

where $$\vec{n}$$ is a vector of integer components. The latter follows from the periodicity of the eigen-wavefunctions. As a result, the sum over the integer vectors $$\vec{n}$$ (which I denoted by $$\vec{k}$$ - a poor choice of notations)
"turns into an integral $$\int V \frac{d^3 k}{(2\pi)^3}$$ when the box is large enough so that the "distance
(in momentum space)
between different $$\vec{n}$$ states is small. This allows us to approximate an integer sum by a continuous sum, or an integral.

The same result holds true if the boundary conditions are of Dirichlet type. In that case, $$\vec{k}=\frac{\pi}{L} \vec{n}$$ and $$\vec{n}$$ is a vector of positive integer components. One has to multiply by a factor of 1/8 to relate the sum of positive integer vectors to a sum over all integer vectors (Guy asked about this issue) - and eventually the resulting integral over momenta stays the same as in the periodic boundary conditions case (again the assumption that the box has a large volume is made).

Yoav

yoavzig
הודעות: 15
הצטרף: 20:48 23/11/2019

Re: תשובה לשאלה שנשאלתי

Hello,

I would like to write below about some of the points that were not clear in the last Tirgul on the quantization of the EM field.

First, it is incorrect to write the electric and magnetic fields as integrals over all momenta $$\vec{k}$$ and at the same time declare that we are interested in a wave propagating only in the $$z$$ axis. In particular, it is a mistake to select all the polarization vectors $$\epsilon_{k \lambda} , \epsilon^* _{k \lambda}$$ to lie on the $$x$$ axis.
A right way to treat the polarization vectors is to choose them to be real and

$$e_{\vec{k} 1} \times e_{\vec{k} 2} = \hat{k} ~~~ \text{and}~~~ e_{\vec{k} i} \cdot e_{\vec{k} j} = \delta_{ij}.$$

I calculated the commutation relations [E_i , B_j] in this choice, see the published Tirgul.

Second, I stated that the EM field vanishes on conducting plates. However, this is true inside them; right outside, the boundary conditions for the EM field read (a) the electric field components parallel to the plates vanish and (b) the perpendicular magnetic field vanishes.

This is important for a correct counting of the polarization states. When the plates are of distance d, the mometum of photons propagating perpendicular to the plates is quantized,$$k_z = n\pi/d$$. For nonzero values of n, there are two polarization states, but for n=0 there is only one. The reason is that the parallel electric field obeys the Dirichlet boundary conditions, hence is identically zero for zero momentum perpendicular to the plates. The electric field perpendicular to the plates does not need to satisfy the Dirichlet boundary conditions and it has a single polarization state when n=0.

Hope it is clear...
Yoav