Eigenvalues and density of states

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Discussion Overview

The discussion revolves around the relationship between the density of states (DOS) and the eigenvalues of a Hamiltonian. Participants explore the expectations regarding the histogram of eigenvalues and its comparison to an analytical expression for the DOS, focusing on scaling and unit consistency.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant presents an analytical expression for the density of states and compares it to a histogram of eigenvalues, noting discrepancies in scaling.
  • Another participant suggests that a scale factor may be necessary for either the eigenvalues or the DOS, and raises the possibility of unit conversion errors.
  • A participant emphasizes the importance of bin size in determining the units of a histogram, questioning whether the bin width has been appropriately accounted for.
  • There is a clarification that while histograms are unitless, they require division by the bin size to express results in terms of energy or energy density.
  • One participant expresses uncertainty about the correctness of their approach and seeks confirmation on whether the DOS and histogram should match.

Areas of Agreement / Disagreement

Participants express differing views on whether the DOS and histogram should be exact copies, with some suggesting scaling issues and others focusing on the role of bin size in the histogram's representation.

Contextual Notes

Participants mention potential issues with unit conversions and the dependence of histogram units on bin size, which may affect the comparison between the DOS and the histogram.

Who May Find This Useful

Researchers or students working on quantum mechanics, statistical mechanics, or related fields involving density of states and eigenvalue problems.

Niles
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Hi guys

I have an analytical expression f(x) for my density of states, and I have plottet this. Now, I also have a complete list of my Hamiltonians eigenvalues.

When I make a histogram of these eigenvalues, I thought that I should get an exact (non-continuous) copy of my plot of f(x). They have the same form, but they are not to scale, i.e. f(x) has a maximum at ~100, while the histogram has a maximum at ~60.

Is there an error somewhere, or am I wrong to expect that they should be "exact" copies?
 
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I am not sure whether i am correct!
some scale factor for eigenvaues or for DOS?
Also check with units of y-axis (i often get these errors when i convert meV to cm-1)
or your data is convoluted already?
 
Last edited:
The units are the same (I have checked).

So you believe they should be the same?
 
The units of a numerical histogram depend on the bin size chosen. Are you remembering to divide by the width of the bin?
 
Manchot said:
The units of a numerical histogram depend on the bin size chosen. Are you remembering to divide by the width of the bin?

Isn't a histogram unit-less?
 
Last edited:
Yes, a histogram is unitless. That's why you need to divide by the bin size to get it in units of 1/energy (or 1/energy/volume, if that's what you're dealing with).
 
I see. Thanks - I'll play around with it to see if it works. Thanks.
 

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