How Does Semiclassical State Counting Relate to Quantum Energy Eigenstates?

In summary, the conversation discusses a claim about the measure of a set related to energy eigenstates in a quantized model. The result is known as the Quantum Correction For Thermodynamic Equilibrium and is justified through a proof. The link provided in the conversation may have been accidentally sent to the wrong thread.
  • #1
jostpuur
2,116
19
Long time ago I encountered a claim that if you fix some energy interval [itex][E_A,E_B][/itex], the measure of the set

[tex]
\{(x,p)\;|\;E_A\leq H(x,p)\leq E_B\}
[/tex]

where [itex]H(x,p)[/itex] is some classical Hamiltonian, is going to be approximately proportional to the number of energy eigenstates contained in the energy interval in the quantized model. It could be that you had to divide this measure by [itex]\hbar[/itex], and that's where the approximate number would come from. Or perhaps by some power of [itex]\hbar[/itex] depending on the dimension?

Do you know this result, and does it have a recognizable name? How is it justified (proven)?
 
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  • #3
I don't have ability to download that pdf file, but based on the title I would guess that that reply got accidentally sent to a wrong thread. I had recently opened another thread where I requested information about Boltzmann distribution in quantum mechanical setting:

https://www.physicsforums.com/threads/boltzmann-with-degenerate-levels.902321/

When a PF user has multiple tabs opened in his or her browser, I guess it can happen that a message gets accidentally sent to a wrong thread?
 

1. What is semiclassical state counting?

Semiclassical state counting is a method used in statistical mechanics to calculate the number of energy states in a system. It combines classical and quantum mechanics by treating the particles as classical objects, but taking into account their quantum mechanical properties.

2. Why is semiclassical state counting important?

Semiclassical state counting is important because it allows us to understand the behavior of complex systems at the molecular level. It also provides a way to calculate thermodynamic properties of these systems, such as entropy and free energy, which are crucial for understanding physical and chemical processes.

3. How does semiclassical state counting work?

Semiclassical state counting involves dividing the system into small cells and assigning a phase space volume to each cell. The number of states in each cell is then determined by considering the quantum mechanical properties of the particles, such as their spin and energy levels. The total number of states is then calculated by summing over all cells.

4. What are the limitations of semiclassical state counting?

One limitation of semiclassical state counting is that it is only applicable to systems with a large number of particles, where quantum effects can be averaged out. It also assumes that the system is in thermal equilibrium, which may not always be the case. Additionally, it does not take into account interactions between particles, which can be important in some systems.

5. How is semiclassical state counting used in research?

Semiclassical state counting is used in a variety of research areas, including condensed matter physics, quantum chemistry, and cosmology. It is often used to study the properties of materials, such as the electronic structure of solids, or to predict the behavior of gases and liquids at different temperatures and pressures. It has also been applied in studies of black holes and the early universe.

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