Solving the Paradox of Hydrogen Atom Partition Function

In summary, the conversation discussed the apparent paradox of the partition function of the hydrogen atom diverging, despite the fact that the Bohr energies of the atom follow a 1/n^2 pattern. The resolution to this paradox is that the partition function only considers occupied states, and since there is only one electron occupying a state in the hydrogen atom, the partition function only consists of a single term. This makes the use of statistical methods unnecessary and the situation can be better understood by using the grand canonical ensemble instead of the canonical ensemble.
  • #1
Entropia
1,474
1
Hello people,

Somebody asked me the following. Anybody want to give it a go?


"Consider the following.

We know, from elementary quantum mechanics, that the Bohr energies
of the hydrogen atom go as (-E_0 / n^2), where n is, of course, the
principal quantum number.

We also know, from elementary statistical mechanics, that the
partition function of a system is the sum of exp (-E / k T), over all states.

Taken together, one can easily demonstrate that the partition
function of the hydrogen atom actually diverges.

How can this be? What is the resolution to this apparent paradox?"
 
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  • #2
1/n^2 diverges?
 
  • #3
For the actual hydrogen atom, shouldn't you have Es = constant * n^2. For the atom's electrons, you have this 1/n^2 thing. I don't think the grand partition function wouldn't diverge in this case, unless you had and infinite number of electrons which isn't possible. Maybe I'm wrong, but it might be that you have to use the grand canonical ensemble for this system instead of the canonical ensemble.
 
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  • #4
I think you guys mix up some things here...

Everything you write is OK except the combination of them. Indeed the energy levels of Hydrogen go like 1/n^2; and indeed the partition function is a sum over states, but it is a sum over OCCUPIED states. There is only one electron occupying a state at any time here, so in this case the partition function consists of a single term. There really is no point in using statistical methods, when you have a single particle (or two for that matter) instead of a near infinite amount like 10^23 (as is the case in solid matter).
 

1. What is the Paradox of Hydrogen Atom Partition Function?

The Paradox of Hydrogen Atom Partition Function refers to the discrepancy between the calculated partition function of a hydrogen atom using the classical statistical mechanics approach and the actual observed partition function. The classical approach predicts a partition function of 1, while the observed value is approximately 1.75.

2. Why is this paradox important in science?

This paradox is important because it highlights the limitations of classical statistical mechanics in accurately predicting the behavior of a system at the atomic level. It also challenges our understanding of the fundamental laws of physics and calls for further research and development of more accurate theories.

3. How can this paradox be solved?

There are several proposed solutions to this paradox, including the use of quantum statistical mechanics, taking into account the effects of electron spin, and considering the influence of the surrounding environment on the hydrogen atom. Further research and experimentation are needed to determine the most accurate solution.

4. What are the potential implications of solving this paradox?

If this paradox is solved, it could lead to a better understanding of the behavior of atoms and molecules at the atomic level. This could have practical applications in fields such as material science, chemistry, and quantum computing.

5. What current research is being done to solve this paradox?

There are ongoing studies and experiments being conducted to address this paradox, including using advanced theoretical models and conducting experiments with highly accurate measurements. Collaborations between scientists from different fields, such as physics and chemistry, are also being pursued to find a comprehensive solution.

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