A very common result is that the density of momentum states in a cubic volume is $\displaystyle\frac{V}{(2\pi\hbar)^3}$ in momentum space. How does this result extend to arbitrary volumes? Are there any nice examples of volumes that are endowed with an interesting set of a momentum eigenstates?

  • $\begingroup$ Semiclassically, it works for an arbitrary finite volume $V$ with a sufficiently nice boundary. Are you asking about exact quantum mechanical results? $\endgroup$
    – Qmechanic
    Jan 14, 2020 at 13:32
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    $\begingroup$ You might be interested to read about Weyl's formula: en.wikipedia.org/wiki/Weyl_law . This is basically the name by which this result for density of states is known for mathematicians. $\endgroup$ Jan 14, 2020 at 14:37
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    $\begingroup$ An interesting result from 1982 Density of states on fractals : “fractons” by S. Alexander & R. Orbach. $\endgroup$
    – A.V.S.
    Jan 15, 2020 at 16:17
  • $\begingroup$ @AleksandrArtemev that’s perfect, thanks! $\endgroup$ Jan 16, 2020 at 22:06

1 Answer 1


Okay, so, for a free particle, with the hamiltonian $$ H = - \frac{\hbar^2}{2m} \Delta $$ living in a cavity of volume $V$ (for any shape of the cavity) number of energy eigenstates with energy less than $E$ has the following asymptotic $$ N(E) \sim \frac{V}{(2\pi \hbar)^d} \omega(\sqrt{2mE}),\, E \gg \frac{\hbar^2}{2ma^2} $$ Here $\omega(R)$ is a volume of $d$-dimensional ball of radius $R$, $a$ is a characteristic size of a cavity. This is a property of Laplacian eigenvalues called asymptotic Weyl formula, or "Weyl law": https://en.wikipedia.org/wiki/Weyl_law . By assuming $E=p^2/2m$, one gets the usually used density of states in momentum space $\frac{V}{(2\pi \hbar)^d}$. The formula is valid in any dimension.

Although in the formulated problem one assumes Dirichlet boundary condition (with wavefunction equal to zero on the boundary), the leading asymptotic is independent of them (which is why people usually use periodic boundary conditions for simplicity and the result is right). This property can also be generalized for nonzero external potential in a pretty natural way (see Wiki article for the result).

The fact that this result is an asymptotic for large $E$ motivates this result being called "semiclassical". Unfortunately, I wasn't able to find any kind of "quantum-mechanical proof" to this formula that would derive it using semi-classical approximation; in physical literature it sometimes is loosely referred to as multidimensional analogue of Bohr-Sommerfeld rule, although I don't really feel this is the case, as Bohr-Sommerfeld rule speaks of a contour integral, rather than an integral over volume in momentum space.

The specifics of a cavity shape and boundary conditions become important for energies $E \sim \frac{\hbar^2}{2ma^2}$; the first correction term is proportional to the area of boundary of a cavity and has different signs for Dirichlet or Neumann b.c. . Maybe this correction term could be interpreted as some kind of "boundary states", though I am not sure about that.


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