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Why don't we consider the negative values of $k_x, k_y$ and $k_z$ when we compute the density of states of a 3D infinit square well?

Physics Asked by Nicolas Schmid on April 18, 2021

I watched some derivations of the density of states in a square box of length L with potential $V=0$ for the points inside the box and $V=infty$ outside the box.
Using separation of variable one can get $frac{2mE}{hbar^2} = k_x^2+k_y^2+k_z^2$ with $k_j = frac{pi}{L}n_j$ and $n_j$ an integer.
Defining $k^2 = k_x^2+k_y^2+k_z^2$ and assuming $k$ is very big we can calculate how many different wave functions there are with a $k$ having a length between $k$ and $k+dx$ by taking the volume of the shell of a sphere with radius $k$, which is $4pi k^2dk$ and divide this by the volume $(frac{pi}{L})^3$. We must multiply the result by $2$ since each combination of $k_j$ can have two different spins.
Summing up I would have said that the density of states is:
$g(k)dk = 2frac{4pi k^2dk}{(frac{pi}{L})^3}$

The thing is the solution says that we only consider the positive $n_j$ so we should multiply the whole thing with $1/8$. I don’t understand why wave function with negative $n_j$ should not be taken into account since for example a wave function with $n_x=1, n_y=2, n_z=3$ does not have the same "direction" as a wave with $n_x=-1, n_y=2, n_z=3$ so I don’t get why we shouldn’t count both.

2 Answers

The energy eigenstates take on the form of products of sines/cosines (depending on how you set up the coordinate system). The arguments of these functions will be proportional to $n_i$ for $i=x,y,z$, and so negative values of these quantum numbers will only change the eigenfunctions by a factor of $1$ or $-1$, thus not changing the actual physical state.

Answered by BioPhysicist on April 18, 2021

You only need to consider positive values as the energy does not depend on the sign of any of the components of $vec k$. You then have counted only 1/8 of all possibilities and must multiply by 8.

Answered by my2cts on April 18, 2021

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