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Please note before flagging, I do not need help solving as the math is simple algebra. Where I am lost is understanding what the math means and why/how it is applied.

Problem 2.4 from Reif Fundamentals of Statistical and Thermal Physics:

Consider an isolated system consisting of a large number $N$ of virtually non-interacting localized (not translating) particlesof spin $½$. Each particle has magnetic moment $\mu$ which can point either parallel or antiparallel to an applied magnetic field $H$. The energy of the system is $E=-(n_1 - n_2)\mu H$ where $n_1$ is the number of particles with spin parallel and $n_2$ is the number of particles with spin antiparallel to $H$.

a)Consider the energy range $E \rightarrow E+ \delta E$ where $\delta E$ is very small compared to $E$ but is microscopically large, i.e., $\delta E >> \mu H$. What is the total number of states, $\Omega (E)$, of the system lying in this energy range?

I am given from the book that the total number of states $\Omega (E)=\omega(E) \delta E$ where $\omega (E)=\frac{N!}{n_1 ! n_2 !} = \frac{N!}{n_1 ! (N-n_1)!}$. The density of states, $\omega(E)$, is easy enough to understand and calculate. However, I do not understand what $\delta E$ is, how to calculate it, or why I multiply the density of states by it.

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The density of states is defined that way, such that in a continuous distribution (which is valid here because the energy is large), the number of states between $E_1$ and $E_2$ is

$$N = \int_{E_1}^{E_2}dE \ \Omega(E)$$

where $\Omega(E)$ is the number of microstates for energy $E$. In the case of the infinitesimal range $\delta E$, the function varies negligibly over the range, so the (differential) number of states (leaving out mathematical rigor) is simply the DOS multiplied by the range $\delta E$. That's just mathematics. The same goes for charge density in electrostatics, fluid density, etc.

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My guess is that the volume density of states is 1 per unit volume, and the (energy?) probability density is probably N!/(n1!n2!), and then the author took an integral over those times dE, said "these are constant" and threw in the wiggly dE (to replace the integral being multiplied by 1*N!/(n1!n2!)). (This is supposed to be a comment)

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