Attaching mathematical expressions here is problematic. I am attaching the link to the question here.
link: https://physics.stackexchange.com/q/782250/147579
There are errors in the text of the question in the link, that make the statements there meaningless.
In statistical mechanics, the canonical partition function is over all the energy states of the system. It can happen that there exists one state for any given value of the energy; it can happen that there exist many states for any given value of the energy.
If the number of states per allowed value of the energy is countable, the spectrum is discrete.
For the quantum harmonic oscillator, for example, in one dimension, the energy eigenstates are labeled by one integer, E_n=(n+(1/2) (in units of hbaromega), so there is one state per eigenstate; in two dimensions, the energy eigenstates are labeled by two integers; E_(n,m)=(n+m+1), so there is more than one state per eigenstate. Similarly in three dimensions, where the energy eigenstates are labeled by three integers: E_(n,m,l)=(n+m+l+(3/2)).
It can happen that the spectrum is discrete for a classical system. The typical example is the energy of a standing wave, which can take only discrete values, too and, in more than one spatial dimensions, also is ``degenerate''.
It can be shown that the spectrum of an operator on a compact manifold is discrete.
Dear Dr Debamalya Dutta
Yes the DoS can be calculated for a discrete number of states. For small number of atoms or molecules, when they are in a single quantum level, for example the ground energy level, or the first excited state.
The number of states can be discrete, a few ones or even only one, if the DoS is zero it means that the number of states is constant, and belong to the same type of degrees of freedom.
Please check:
Reif, F. 1966. Statistical Physics. Berkeley Physics Course. McGraw-Hill, New York, USA. Volume 5. pp.398.
Lu, T. and Chen, F. 2012. Multiwfn: A multifunctional wave function analyzer. Journal of Computational Chemistry. 33(5):580-592.
DOI: https://doi.org/10.1002/jcc.22885
Article TDOS quantum mechanical visual analysis for single molecules
Kind Regards.
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