Limit as $x_1 \to x_0$ for the propagator of the harmonic oscillator

Question

Consider a non-relativistic particle of mass $m$, moving along the $x$-axis in a potential $V(x) = m\omega^2x^2/2$. use path-integral methods to find the probability to find the particle between $x_1$ and $x_1 + dx_1$ if the particle is at $x_0$ at time $t = 0$.

One finds the propagator to be $$P(x_1, t_1; x_0, 0) = \sqrt{{{m\omega}\over{2\pi i\hbar \sin \omega t_1}}}e^{{{im\omega}\over{2\hbar \sin\omega t_1}}((x_0^2 + x_1^2)\cos\omega t_1 - 2x_0x_1)}$$and the requisite probability to be $$\left|\int P(x_1, t_1; x_0, 0)\psi(x_1)\,dx_1\right|^2$$for some wavepacket $\psi$ localized between $x_1$, $x_1 + dx_1$. In the limit $x_1 \to x_0$, we have$$\lim_{(x_1 - x_0) \to 0} P(x_1, t_1; x_0, 0) = \sqrt{{m\omega}\over{2\pi i\hbar \sin\omega T}}e^{{{im\omega}\over{\hbar\sin\omega T}}(x_0^2\cos\omega T - x_0^2)}.$$

My question is, what is the physical significance of taking this limit $x_1 \to x_0$?

Answer 1

The probability density $$\lim_{x_1\to x_0} P(x_1, t_1; x_0, 0)$$ is basically the probability density for a particle at position $x_0$ to be back at that position after a time $t_1$ has passed.

Answer 2

As $T = t_1 - t_0 \to 0$, we have$$\begin{align}\lim_{t_1 -~ t_0~ \to~ 0} \langle x_1, t_1\,|\,x_0, t_0\rangle &= \lim_{T~ \to~ 0}\left({{m\omega}\over{2\pi i\sin\omega T}}\right)^{1/2}\text{exp}\left[{{im\omega}\over{2\sin\omega T}}\left(\left(x_1^2 + x_0^2\right)\cos \omega T - 2x_0x_1\right)\right]\\ &=\lim_{\epsilon ~= ~iT/m~ \to~ 0} {1\over{\sqrt{2\pi\epsilon}}}\text{exp}\left[-{{(x_1 - x_0)^2}\over{2\epsilon}}\right] = \delta(x_1 - x_0).\end{align}$$This is just as expected from the orthonormality of $\hat{x}(t)$ eigenvalues at $t = t_0$.

Answer 3

Ideologically speaking, the absolute value of the propagator squared

$$\tag{1} |K(x_f,t_f;x_i,t_i)|^2 \mathrm{d}x_f ~=~ \frac{m\omega}{2\pi\hbar\sin\omega \Delta t}\mathrm{d}x_f, \qquad \Delta t~:=~t_f-t_i~>~0,$$

is the probability that a harmonic oscillator starting at $t_i$ in position $x_i$ will finish within the position interval $[x_f,x_f+\mathrm{d}x_f]$ at time $t_f$.

In particular OP can study the case $x_i=x_f$, i.e. the probability of returning to the same position in a given time $\Delta t$.

Counter-intuitively, according to eq. (1), the probability does not depend on the start and end positions $x_i$ and $x_f$ at all! This foresees the fact that the notion of absolute (as opposed to relative) probabilities of the Feynman kernel $K(x_f,t_f;x_i,t_i)$ cannot be maintained on a non-compact position space, cf. e.g. Ref. 1 and this Phys.SE post.

In general, the probabilistic interpretation of eq. (1) only holds for short times $\Delta t\ll \tau$, where $\tau$ is some characteristic time scale of the system.

References:

1. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, 1965.