Frederick Kramer

2022-07-10

$\hslash \approx 0$ and the spread of QM wave function
Is there a direct mathematical method to show that if a quantum wave funtion is initially sharply localized, then it will stay sharply localized if $\hslash \approx 0$? In that case the Ehrenfest theorem implies the transition from quantum mechanics to classical mechanics.
Of course, we are dealing with the propagation of a wave function, but let's not mess with path integrals. Thus, the structure of the general solution of Schrödinger equation should imply the result - if possible.

Mekjulleymg

The propagator for a one-dimensional free particle is, for example,
$K\left({x}^{\prime }-x,{t}^{\prime }-t\right)=\sqrt{\frac{m}{2\pi \mathrm{i}\hslash \left({t}^{\prime }-t\right)}}\mathrm{exp}\left(-\frac{m\left({x}^{\prime }-x{\right)}^{2}}{2\pi \mathrm{i}\hslash \left({t}^{\prime }-t\right)}\right).$
Meaning that $\psi \left({x}^{\prime },{t}^{\prime }\right)=\int K\left({x}^{\prime }-x,{t}^{\prime }-t\right)\phantom{\rule{thinmathspace}{0ex}}\psi \left(x,t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}x$. Here, $m$ is the particle mass.
In plain(er) English, time evolution of a free particle comes down to a convolution with a Gaussian kernel with width proportional to $\hslash$. Mathematically, you can take $\hslash \to 0$, the Gaussian function becomes a $\delta$ function and the wave function does not spread any more. Physically that means that the spread is negligible if $\hslash$ is very small compared to $\frac{m{L}^{2}}{T}$, where $L$ and $T$ are the physically relevant length and time scales, respectively.
If you want to take the classical limit properly, you need decoherence, as discussed in other answers.

ttyme411gl

The problem with trying to understand the spread of the wavefunction in the classical limit by taking $\hslash \approx 0$ or taking the limit $\hslash \to 0$ is that in reality $\hslash \ne 0$. Taking the real non-zero value of $\hslash$there are cases of perfectly ordinary objects for which Ehrenfest's theorem doesn't imply anything like classical behaviour. Rather the wavefunction of that object spreads out a lot over time. The classical limit is actually a result of decoherence and information being copied out of 'classical' objects into the environment.