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A question came up when studying the formula for calculating the total number of possible energy states of a particle within a container.

Why is the number of possible energy states independent of the shape of the container and merely dependent on the number of states in only 3 perpendicular container dimensions ($x$, $y$ and $z$), while a momentum vector can be directed at any direction within the container? Shouldn't the length of the container in that same direction as the momentum vector also determine the number of energy states in that direction?

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    $\begingroup$ The number of possible energy states is infinite; why would it be dependent of anything? $\endgroup$
    – Ivan Neretin
    Oct 27, 2018 at 21:54
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    $\begingroup$ Hi Johnny, can you provide more details regarding the model you’re working with, including a citation to a textbook or other reference? It is a lot easier to answer questions of this type when the model is fully laid out. Also, this question is quite similar to your last one, and I would probably recommend editing your last question than posting a new one. $\endgroup$
    – a-cyclohexane-molecule
    Oct 28, 2018 at 11:05
  • $\begingroup$ @a-cyclohexane-molecule I have made an illustration but in 2D momentum space along with the question, found here: dropbox.com/s/wsomw9odrwbrsec/Question.jpg?dl=0 Regarding my last question, the question is down the list by now and I thought it would not be bumped up and noticed by others if it is edited. $\endgroup$
    – Phy
    Oct 29, 2018 at 14:17
  • $\begingroup$ Note that all edits will automatically bump questions. $\endgroup$
    – a-cyclohexane-molecule
    Oct 29, 2018 at 22:23
  • $\begingroup$ @a-cyclohexane-molecule So shall I put my illustration in my previous question and ignore this one for now? $\endgroup$
    – Phy
    Oct 30, 2018 at 21:40

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For a sufficiently small container, the size and shape of the system is significant, e.g. some energy states are prohibited. For example, CdTe quantum dots behave differently from the bulk material.

However, the kinetic theory is based on the assumption that, "The number of molecules is so large that statistical treatment can be applied." The momentum is treated as a vector having x, y and z components and does not depend on the shape of the container (though a cube is convenient for illustration of three orthogonal directions). The equations can be derived using spherical coordinates, and they still must integrate collisions in all directions, so are equivalent.

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  • $\begingroup$ Within a momentum sphere, the total number of energy states is based on the number of states in each of the 3 container length dimensions multiplied by each other and fractionized according to the sphere volume, right? This implies that the density of states within the momentum sphere is homogeneous. But why isn't it so that each momentum vector within that sphere would have a number of states according to the container's length dimension in that SAME direction of that momentum vector, since the number of states is dependent on dimensions in the first place? $\endgroup$
    – Phy
    Oct 28, 2018 at 1:32

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