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In terms of physical quantities, the $|\psi \rangle$ of quantum mechanics and the microstate $St(q,p)$ of statistical mechanics are both a vector, and the microstate of statistical mechanics can be considered as a vector with a size of $3N$, while $| \psi\rangle$ is also a vector, although it's size is unknown.
When the correspondence between the number of microstates and the volume of the phase space is proved by one-dimensional harmonic oscillators to be $\Omega = \dfrac{\Gamma}{\hbar}$ relational proof, the number of energy eigenvalues is regarded as the number of microstates, then whether the number of $\{St\}$ is the same as the number of $\{|\psi\rangle\}$?

$N$ is the number of particles. $\Omega$ is the number of microstates. $\Gamma$ is the volume of accessible area of phase space.

Roger V.
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Zhao Dazhuang
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  • They are both called "state" but are different things. A classical particle can be in (q,p) because it follows a well defined trajectory in the phase space, but in quantum mechanics this is not the case (many trajectories are possible, this is the path integral interpretation). – Quillo Sep 16 '22 at 08:18
  • @Quillo So I ask whether the two are one-to-one correspondence, or the number is the same？ – Zhao Dazhuang Sep 16 '22 at 08:21
  • not at all, there is no simple relation between the two objects, they just share the same name. Similarly, you also have a "thermodynamic state", that, again, has little to do with the QM state or the phase space state. – Quillo Sep 16 '22 at 08:34
  • @Quillo The thermodynamic state and the state of the phase space are the macroscopic state and the microscopic state respectively. Their relationship is one-to-many. Is there a similar relationship between quantum mechanics and phase space? For example, $|\psi(\mathbf{r}) |^2$ should have a one-to-one correspondence with $\rho(\mathbf{r})$ of the microstate. – Zhao Dazhuang Sep 16 '22 at 08:40
  • "thermo" is a macrostate, that can be obtained from an ensemble of both "QM" or "classical" microstates. However, the relation between the classical state and the QM one is not straightforward at all, and you have to look for the "phase space formulation of QM" https://en.wikipedia.org/wiki/Phase-space_formulation – Quillo Sep 16 '22 at 08:54

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What is the difference between the state $| \psi \rangle$ of quantum mechanics and the microscopic state $St(q,p)$ of statistical mechanics?

Apples and oranges.

The $| \psi \rangle$ of quantum mechanics and the $St(q,p)$ of statistical mechanics use mathematics, vectors , tensors and even complicated functions of special relativity four vectors, but they describe observations through a different mathematical process.

In physics theories, extra axioms are imposed so as to pick up those mathematical solutions which will have correct units and obey experimentally observed laws. These are called laws in classical mechanics and electrodynamics, and postulates and principles in quantum mechanics; the objective is to have a theory for physical observations that fits them and, important , is predictive.

It is true that statistical mechanics relies a lot on probability, but the probability postulate of quantum mechanics describes a probable location for a particle that is biased by the wavefunction, a solution of the appropriate quantum differential equation. The distribution is not statistical, also there is no "number of $| \psi \rangle$, $ψ$ is a continuous mathematically function on which creation and annihilation operators act in order for a particle to appear at (x,y,z,t).

anna v
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  • What I'm asking is the number of ${\psi}$, which is the number of bases, this exists – Zhao Dazhuang Sep 16 '22 at 10:13
  • @ZhaoDazhuang the number of classical states and the number of quantum states depend on the system at hand. For a particle, they are both uncountable. The mathematical relation between the two is involved, as anna v is saying in her answer: see https://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution – Quillo Sep 16 '22 at 11:42
  • @Quillo Thank you very much – Zhao Dazhuang Sep 18 '22 at 07:42