Converting a summation into an integration

In summary, the conversation discusses the integration volume for converting a summation in reciprocal space. The volume is defined as ##V_k## and can vary depending on the system being studied. In the case of a Fermi gas, it would be the volume of the Fermi sphere. When using a delta function to restrict the values of k, the integration volume is redefined as ##\frac{2\Omega}{(2\pi)^3}##, taking into account rotational symmetry and the number of energy states. The document provided further explains the concept of density of states and how it relates to the integration volume.
  • #1
Amentia
110
5
Hello,

I want to convert a summation in reciprocal space and I am unsure about the integration volume. I have started with the formula:

$$\sum_{\vec{k}} \rightarrow \frac{V_{k}}{(2\pi)^{3}}\int\int\int \mathrm{d}V_{k}$$

where:

$$\mathrm{d}V_{k} = k^{2}\mathrm{d}k \sin{\theta_{k}}\mathrm{d}\theta_{k}\mathrm{d}\phi_{k}$$

Here k can go from 0 to infinity, so what should be the volume ##V_{k}##?

My second question is when I perform the integration with a delta function which restricts k to a finite set of values, should I redefine the volume a posteriori?

$$\frac{V_{k}}{(2\pi)^{3}}\int\int\int \mathrm{d}V_{k}f(\vec{k})\delta(k-k_{0})$$

Here should we have ##V_{k}= k_{0}4\pi## or even a "2D volume" ##V_{k}= 4\pi##? While the volume was supposed to be already defined before we make use of the delta function.

I hope my questions are clear!
 
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  • #2
Q1: ##V_k## is the volume of the system. For a Fermi gas it would be the volume of the Fermi sphere.
Q2: Due to rotational symmetry, the preceding factor of your integral would become ##\frac{2\Omega}{(2\pi)^3}##. The 2 comes in if you have a Fermi gas(two energy states, spin up and spin down) the ##\Omega=4\pi## is the solid angle (scattering cross section) of the sphere of volume ##V_k##. Please see https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/3/67057/files/2018/09/density_of_states-vjqh7n.pdf for a detailed explanation.
 
  • #3
Thank you for the document. In equation (10), they get rid of ##V_{d}## by defining the density of states. But what if it was defined wihout dividing by this volume? And then assume I have a real system with a finite volume ##V = L_{x}L_{y}L_{z}##. Can I relate ##V_{d}## to V? E.g. ##V_{d}=(2\pi)^{3}/V## so that I will have in front of my integral:

$$\frac{V_{d}\Omega}{(2\pi)^{3}} = \frac{\Omega}{V} = \frac{4\pi}{L_{x}L_{y}L_{z}} $$

where I have assumed that the function inside the integral does not depend on the angles?
 

Related to Converting a summation into an integration

What is the difference between a summation and an integration?

A summation is a mathematical operation that involves adding a sequence of numbers together, while an integration is a mathematical operation that involves finding the area under a curve.

Why would someone want to convert a summation into an integration?

Converting a summation into an integration allows for more precise calculations and can be used to solve more complex mathematical problems.

What are the steps for converting a summation into an integration?

The steps for converting a summation into an integration include: determining the limits of the summation, writing the summation in sigma notation, finding the general term of the summation, and using integration rules to convert the summation into an integration.

Can any summation be converted into an integration?

No, not all summations can be converted into integrations. The summation must have a finite number of terms and follow a specific pattern in order to be converted into an integration.

What are some real-world applications of converting a summation into an integration?

Converting a summation into an integration is commonly used in physics and engineering to calculate complex systems and solve differential equations. It is also used in economics and finance to model and predict trends and patterns.

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