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The evolution of free wave packets addresses the problem of the late-time spreading of a wave packet $\psi(x,t)$ that at $t=0$ has momentum distribution $\phi(p)$, average position $\bar{x}$, and position variance $\Delta_x$: $$\psi(x,t)= \sqrt{\frac{m}{ it}}\, \exp \big[\frac{im}{2\hbar t}(x^2-\bar{x}^{2})\big]\phi \big(\frac{m}{t}(x-\bar{x})\big)+\delta \psi$$ $$|\delta \psi|^{2}\leqslant\sqrt{m^{3}/\pi \hbar^{3}t^{3}}\Delta_{x}^{2}. $$ For $|\delta \psi |$ to be small compared with $\psi$ we require $|\delta \psi |^{2}\ll1/2\Delta_{x}$, which leads to $t\gg (4/\pi)^{1/3}m\Delta_{x}^{2}$.

If I'm not mistaken, I would conclude that the large-$t$ decay of the survival probability $\int_a^b |\psi(x,t)|^2\,dx$ may be as slow as $\propto 1/t$, so the integral $\int_0^\infty dt$ may diverge logarithmically. See this example in the cited paper for the case of an initial square wave packet where this slow decay applies:

\begin{equation} \psi(x,0)= \begin{cases} 1/\sqrt{a}, &|x|<a/2 \\ 0, &|x|\geq a/2.\end{cases} \end{equation}

\begin{equation} \label{eq:phiRect} \phi(p)=\sqrt{\frac{a}{2\pi \hbar}}\frac{\sin (a p/2\hbar)}{a p/2\hbar} \end{equation} and therefore, for $t\gg ma^2 /\hbar$, \begin{equation} \label{eq:appRect} \psi(x,t)\approx \sqrt{\frac{a m}{2\pi i\hbar t}}\, \exp \left(\frac{i m x^2 }{ 2\hbar t}\right)\frac{\sin (a m x / 2\hbar t)}{a m x / 2\hbar t}. \end{equation} with a logarithmically diverging $\int_0^\infty \int_0^1 |\psi(x,t)|^2\,dt$.

The evolution of free wave packets addresses the problem of the late-time spreading of a wave packet $\psi(x,t)$ that at $t=0$ has momentum distribution $\phi(p)$, average position $\bar{x}$, and position variance $\Delta_x$: $$\psi(x,t)= \sqrt{\frac{m}{ it}}\, \exp \big[\frac{im}{2\hbar t}(x^2-\bar{x}^{2})\big]\phi \big(\frac{m}{t}(x-\bar{x})\big)+\delta \psi$$ $$|\delta \psi|^{2}\leqslant\sqrt{m^{3}/\pi \hbar^{3}t^{3}}\Delta_{x}^{2}. $$ For $|\delta \psi |$ to be small compared with $\psi$ we require $|\delta \psi |^{2}\ll1/2\Delta_{x}$, which leads to $t\gg (4/\pi)^{1/3}m\Delta_{x}^{2}$.

The large-$t$ decay of the survival probability $\int_a^b |\psi(x,t)|^2\,dx$ may therefore be as slow as $\propto 1/t$, so the integral $\int_0^\infty dt$ may diverge logarithmically. See this example in the cited paper for the case of an initial square wave packet where this slow decay applies:

\begin{equation} \psi(x,0)= \begin{cases} 1/\sqrt{a}, &|x|<a/2 \\ 0, &|x|\geq a/2.\end{cases} \end{equation}

\begin{equation} \label{eq:phiRect} \phi(p)=\sqrt{\frac{a}{2\pi \hbar}}\frac{\sin (a p/2\hbar)}{a p/2\hbar} \end{equation} and therefore, for $t\gg ma^2 /\hbar$, \begin{equation} \label{eq:appRect} \psi(x,t)\approx \sqrt{\frac{a m}{2\pi i\hbar t}}\, \exp \left(\frac{i m x^2 }{ 2\hbar t}\right)\frac{\sin (a m x / 2\hbar t)}{a m x / 2\hbar t}. \end{equation} with a logarithmically diverging $\int_0^\infty \int_0^1 |\psi(x,t)|^2\,dxdt$.

The above is in response to the OP where the case of zero potential is requested. For the special case that there is an infinitely high potential wall at $x=a$ and a delta-function tunnel barrier at $x=b$, the survival probability has been calculated exactly in An Exact Solution to the Time-dependent Schrodinger Equation for a Model One-dimensional Potential. The survival probability $\int_a^b |\psi(x,t)|^2\,$ decays for large times as $1/t^3$. The paper argues that this power law decay is generic for escape from a potential well.

The evolution of free wave packets addresses the problem of the late-time spreading of a wave packet $\psi(x,t)$ that at $t=0$ has momentum distribution $\phi(p)$, average position $\bar{x}$, and position variance $\Delta_x$: $$\psi(x,t)= \sqrt{\frac{m}{ it}}\, \exp \big[\frac{im}{2\hbar t}(x^2-\bar{x}^{2})\big]\phi \big(\frac{m}{t}(x-\bar{x})\big)+\delta \psi$$ $$|\delta \psi|^{2}\leqslant\sqrt{m^{3}/\pi \hbar^{3}t^{3}}\Delta_{x}^{2}. $$ For $|\delta \psi |$ to be small compared with $\psi$ we require $|\delta \psi |^{2}\ll1/2\Delta_{x}$, which leads to $t\gg (4/\pi)^{1/3}m\Delta_{x}^{2}$.

If I'm not mistaken, I would conclude that the large-$t$ decay of the survival probability $\int_a^b |\psi(x,t)|^2\,dx$ may be as slow as $\propto 1/t$, so the integral $\int_0^\infty dt$ may diverge logarithmically. See for example equation 28 in the cited paper for the case of an initial square wave packet where this slow decay applies.

The evolution of free wave packets addresses the problem of the late-time spreading of a wave packet $\psi(x,t)$ that at $t=0$ has momentum distribution $\phi(p)$, average position $\bar{x}$, and position variance $\Delta_x$: $$\psi(x,t)= \sqrt{\frac{m}{ it}}\, \exp \big[\frac{im}{2\hbar t}(x^2-\bar{x}^{2})\big]\phi \big(\frac{m}{t}(x-\bar{x})\big)+\delta \psi$$ $$|\delta \psi|^{2}\leqslant\sqrt{m^{3}/\pi \hbar^{3}t^{3}}\Delta_{x}^{2}. $$ For $|\delta \psi |$ to be small compared with $\psi$ we require $|\delta \psi |^{2}\ll1/2\Delta_{x}$, which leads to $t\gg (4/\pi)^{1/3}m\Delta_{x}^{2}$.

If I'm not mistaken, I would conclude that the large-$t$ decay of the survival probability $\int_a^b |\psi(x,t)|^2\,dx$ may be as slow as $\propto 1/t$, so the integral $\int_0^\infty dt$ may diverge logarithmically. See this example in the cited paper for the case of an initial square wave packet where this slow decay applies:

\begin{equation} \psi(x,0)= \begin{cases} 1/\sqrt{a}, &|x|<a/2 \\ 0, &|x|\geq a/2.\end{cases} \end{equation}

\begin{equation} \label{eq:phiRect} \phi(p)=\sqrt{\frac{a}{2\pi \hbar}}\frac{\sin (a p/2\hbar)}{a p/2\hbar} \end{equation} and therefore, for $t\gg ma^2 /\hbar$, \begin{equation} \label{eq:appRect} \psi(x,t)\approx \sqrt{\frac{a m}{2\pi i\hbar t}}\, \exp \left(\frac{i m x^2 }{ 2\hbar t}\right)\frac{\sin (a m x / 2\hbar t)}{a m x / 2\hbar t}. \end{equation} with a logarithmically diverging $\int_0^\infty \int_0^1 |\psi(x,t)|^2\,dt$.

The evolution of free wave packets addresses the problem of the late-time spreading of a wave packet $\psi(x,t)$ that at $t=0$ has momentum distribution $\phi(p)$, average position $\bar{x}$, and position variance $\Delta_x$: $$\psi(x,t)= \sqrt{\frac{m}{ it}}\, \exp \big[\frac{im}{2\hbar t}(x^2-\bar{x}^{2})\big]\phi \big(\frac{m}{t}(x-\bar{x})\big)+\delta \psi$$ $$|\delta \psi|^{2}\leqslant\sqrt{m^{3}/\pi \hbar^{3}t^{3}}\Delta_{x}^{2}. $$ For $|\delta \psi |$ to be small compared with $\psi$ we require $|\delta \psi |^{2}\ll1/2\Delta_{x}$, which leads to $t\gg (4/\pi)^{1/3}m\Delta_{x}^{2}$.

The evolution of free wave packets addresses the problem of the late-time spreading of a wave packet $\psi(x,t)$ that at $t=0$ has momentum distribution $\phi(p)$, average position $\bar{x}$, and position variance $\Delta_x$: $$\psi(x,t)= \sqrt{\frac{m}{ it}}\, \exp \big[\frac{im}{2\hbar t}(x^2-\bar{x}^{2})\big]\phi \big(\frac{m}{t}(x-\bar{x})\big)+\delta \psi$$ $$|\delta \psi|^{2}\leqslant\sqrt{m^{3}/\pi \hbar^{3}t^{3}}\Delta_{x}^{2}. $$ For $|\delta \psi |$ to be small compared with $\psi$ we require $|\delta \psi |^{2}\ll1/2\Delta_{x}$, which leads to $t\gg (4/\pi)^{1/3}m\Delta_{x}^{2}$.

If I'm not mistaken, I would conclude that the large-$t$ decay of the survival probability $\int_a^b |\psi(x,t)|^2\,dx$ may be as slow as $\propto 1/t$, so the integral $\int_0^\infty dt$ may diverge logarithmically. See for example equation 28 in the cited paper for the case of an initial square wave packet where this slow decay applies.