Estimate of a solution of Schroedinger equation for a free particle

Question

Let $\psi(x,t)$ be a solution of the Schroedinger on the line $$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$ One assumes that $\psi(x,0)$ "behaves well" as $|x|\to \infty$, e.g. square integrable on the line. If necessary one may impose some extra assumptions.

For a fixed bounded interval $[a,b]$ I would be interested to estimate the behaviour of $$\int_a^b|\psi(x,t)|^2dx$$ when $t\to +\infty$. The above integral is expected to decay as $t\to +\infty$, and the question is to give a more explicit estimate. Eventually I would like to show that $$\int_{0}^{+\infty}dt\int_a^b|\psi(x,t)|^2dx<\infty.$$

Answer 1

For the free evolution specifically, in relation to Mateusz's comment: on the Fourier side you can write the solution as (for $m = -1/2$; you can rescale/invert time to get other scalings)

$$ \phi(t,x) = \frac{1}{\sqrt{2\pi}} \int e^{it\xi^2} e^{ix\xi} \hat{\phi}_0(\xi) ~\mathrm{d\xi} $$

where $\hat{\phi}_0$ is the Fourier transform of the initial data. Stationary phase tells you that locally uniformly in $y$ you have (notice also that the limit is independent of $y$)

$$ \lim_{t\to\infty} t^{1/2} \phi(t,y) = \frac{e^{i\pi/4}}{\sqrt{2}} \hat{\phi}_0(0) = \frac{e^{i\pi/4}}{2\sqrt{2}} \int \phi_0(x) ~\mathrm{d}x $$

Similarly, you have that if $\hat{\phi}_0(0)$ has vanishing first $k$ derivatives, then locally uniformly

$$ \lim_{t\to\infty} t^{1/2 + k+1} \phi(t,y) \propto \hat{\phi}_0^{(k+1)}(0) \propto \int x^{k+1} \phi_0(x) ~\mathrm{d}x $$

This estimate holds, in particular, for all Schwartz initial data.

Answer 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 $t\gg (4/\pi)^{1/3}m\Delta_{x}^{2}$.

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.