What is the volime of a k-state?

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Discussion Overview

The discussion centers around the concept of the volume of a k-state in the context of the free gas model and its implications in quantum mechanics. Participants explore the meaning of this volume in relation to wave vectors and phase space, as well as the intuition behind it.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant notes that in the free gas model, the vector k is defined as k=π*n/L, and proposes that the volume of a k-vector is (2π)^3/V.
  • Another participant questions the meaning of a vector having a volume and seeks clarification on this concept.
  • A different participant suggests that the volume relates to how many k-values can fit in a given box or length, indicating a possible connection to the quantization of phase space.
  • One participant references an external link to further illustrate the concept, acknowledging the complexity and potential for misunderstanding.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the meaning of the volume of a k-vector, and while there are some shared ideas about its relation to fitting k-values in a box, no consensus is reached on the interpretation of this volume.

Contextual Notes

The discussion highlights the ambiguity surrounding the definition of volume in this context, as well as the dependence on the interpretation of phase space and wave vectors.

Ruizhe
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In free gas model, the vector k=π*n/L, and the volume of a k-vector is (2π)^3/V.
But I don´t understand the meaning of the volume. Why could a vector have a volume? What is the meaning of it?
 
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Ruizhe said:
But I don´t understand the meaning of the volume. Why could a vector have a volume? What is the meaning of it?

"quantum" of the phase space,
4c08301fca6e4c550c98b44938138143.png
. In 3 dimension,

actually the quantum encloses a volume element in phase space and V is the volume spanned by the wave vector
 
It comes from how many "k"'s you can fit in a given box, or length, or... At least I believe that's what you're referring to. From this information, you can effectively figure out the "volume" of a k-vector. It seems rather counter intuitive.
They somewhat touch on it here:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node88.html
and I'll probably botch the explanation, but you can see it in action in the link above.
 
Thanks a lot!
 
BiGyElLoWhAt said:
It comes from how many "k"'s you can fit in a given box, or length, or... At least I believe that's what you're referring to. From this information, you can effectively figure out the "volume" of a k-vector. It seems rather counter intuitive.
They somewhat touch on it here:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node88.html
and I'll probably botch the explanation, but you can see it in action in the link above.
Thanks
 

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