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נשלח:

**00:50 04/01/2020**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

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