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In statistical thermodynamics, the entropy is given by $S = k_B \ln{\Omega}$ where $\Omega$ is the number of possible microscopic states for a given macroscopic system.

But the atom as a system can have several micro states corresponding to the same value of its energy. For example, the hydrogen atom in it's second energetic level can be in any of 4 states. So can we say that a hydrogen atom in a $n=2$ state has more entropy that another in a $n=1$ state?

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Yes, a single atom could have entropy, if the state of the atom is defined as some density operator. For example: \begin{equation}\tag{1} \rho = \frac{1}{2}| \, 2, 1, +1 \rangle \langle 2, 1, +1 \,| + \frac{1}{2} | \, 2, 1, -1 \rangle \langle 2, 1, -1 \,|, \end{equation} where $|\, n, l, m \rangle$ is the micro-state of energy level $E_{n, l, m}$, where \begin{align}\tag{2} n &= 1, 2, \ldots, \infty, &l &= 0, 1, 2, \ldots n - 1, &m &= 0, \, \pm 1, \, \pm 2, \ldots, \, \pm l. \end{align} According to (1), the atom has a probability $\frac{1}{2}$ to be in the state $|\, 2, 1, +1 \rangle$ and probability $\frac{1}{2}$ to be in a state $|\, 2, 1, -1 \rangle$. Thus, the atom has statistical entropy $S = \ln{2}$.

This macro-state isn't the same as a quantum superposition. For example: \begin{equation}\tag{3} | \, \psi \rangle = \frac{1}{\sqrt{2}} |\, 2, 1, +1 \rangle + \frac{1}{\sqrt{2}} |\, 2, 1, -1 \rangle \end{equation} is still a micro-state and it's entropy is 0. In this case $\rho \ne |\, \psi \rangle \langle \psi |$.

The micro-state and macro-state depend on the way the atom has been prepared. (1) and (3) could be the result of some interaction with another system in the past.

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  • $\begingroup$ I am interested in the ideas in your answer. I haven't taken a course or read any book on statistical mechanics. Is entropy in QM discussed there? Can you recommend me a book or resource where I can learn more about this? $\endgroup$
    – user137661
    Nov 4 '19 at 20:38
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    $\begingroup$ @SV, the best book I know on this subject is in French : "Du microscopique au macroscopique", by Roger Balian. Many other books are discussing this. $\endgroup$
    – Cham
    Nov 4 '19 at 20:39
  • $\begingroup$ I found an English version from Springer, do you think it is as good as the French version? $\endgroup$
    – user137661
    Nov 5 '19 at 0:58
  • $\begingroup$ @SV, the English version is probably good, but I haven't saw it. You may give it a try. $\endgroup$
    – Cham
    Nov 5 '19 at 1:19

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