Classical limit of quantum mechanics in terms of center of mass

Question

When we say that quantum mechanics reproduces classical mechanics at macroscopic scales, is it a statement about the center of mass of a macroscopic system?

More specifically, let $\psi (x)$ be the wavefunction of the CoM co-ordinate of a macroscopic system. Let the CoM position and momentum operator be $X$ and $P$. Then, is the claim that:

$$\frac{d}{dt} \langle \psi|P|\psi \rangle =\left(-\frac{dV}{dx}\right)_{x= (\langle \psi |X|\psi \rangle)}$$

$$\frac{d}{dt} \langle \psi |X|\psi \rangle =\frac{1}{m}\langle \psi |P|\psi \rangle$$

Is this what we claim when we say quantum mechanics reproduces classical mechanics? If no, what is the claim?

If yes, how can one derive this fact about the CoM wavefunction? These equations are almost the same as Ehrenfest's theorem.

I think maybe these equations rely on the claim that the fluctuations of the CoM wavefunction about the mean are small. Then how can one show that the fluctuations of the CoM wavefunction about the mean are always small for macroscopic systems? How exactly do the fluctuations become small just because there are too many particles?

Answer 1

Classical limit may mean several things in quantum mechanics:

• Ehrenfest theorem shows that the averaged quantum equations of motion have the same form as the classical equations of motion. In case of an oscillator this readily reduces to the center-of-mass motion. However, for a more complex potentials we have $$ \langle V(x)\rangle = V(\langle x\rangle) + \left[\langle V(x)\rangle - V(\langle x\rangle)\right]. $$ Showing that the correction in the right-hand-side is small requires making certain assumptions about the wave function used for averaging, which are covered by the correspondence principle.
• WKB/quasiclassical approximation Closely related way to see the link between classical and quantum mechanics is that the most probably trajectories for particles are the classical ones (in the path integral language these correspond to the extremum of the action). Again, this happens only under certain conditions: one can think of an example of a two-well potential, where the behavior would become classical only if the barrier is very thick and we can neglect the tunneling.
• Decoherence Both bullets above didn't really deal with macroscopic objects in thermodynamic sense: an object may be big and still behave as a quantum one. In case of many particle systems or when a quantum object is coupled to an environment, the classical behavior for averages is achieved due to decoherence, i.e., the energy and momentum exchange with other particles/environment. In this case one can arrive at the center-of-mass equations of motion, but this requires that the thermodynamic averaging is done alongside the quantum mechanical averaging (see also Is quantum mechanics applicable to only small things? and Is there an equivalence between the creation of a wavefunction and the collapse of a wavefunction?).

Example
As a simple way to get some intuition, one could consider a Gaussian wave packet $$ \psi(x)=\frac{1}{(2\pi\sigma^2)^{1/4}}e^{-\frac{(x-x_0)^2}{4\sigma^2}}, |\psi(x)|^2=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-x_0)^2}{2\sigma^2}}, \langle x\rangle = \int dx x|\psi(x)|^2 = x_0. $$ Then $$ \langle V(x)\rangle = V(x_0)\langle e^{\log V(x)-\log V(x_0)}\rangle \approx V(x_0)\langle e^{\alpha(x-x_0)+\frac{1}{2}\beta (x-x_0)^2}\rangle = V(x_0)\frac{1}{\sqrt{1+\sigma^2\beta}}e^{\frac{\sigma^2\alpha^2}{1+\sigma^2\beta}},\\ \alpha = \frac{d}{dx}\log V(x)|_{x=x_0},\beta = \frac{d^2}{dx^2}\log V(x)|_{x=x_0}. $$ The center-of-mass description then holds for $$ \sigma^2|\beta| = \sigma^2\left|\frac{d^2}{dx^2}\log V(x)|_{x=x_0}\right|\ll 1,\\ \sigma^2\alpha^2 = \sigma^2\left|\frac{d}{dx}\log V(x)|_{x=x_0}\right|^2\ll 1, $$ i.e., when the potential is smooth on the characteristic scale of the wave function change.