Quantum time evolution after position measurement

Question

Consider a free particle with hamiltonian $\hat{H}=\frac{\hat{p}^2}{2m}$ and propagator $\hat{U}(t) = e^{-\frac{i}{\hbar}\hat{H}t}$:

we can compute the time evolution of a position wavefunction as:

$$ \langle x|\psi(t)\rangle = \int_{-\infty}^\infty dp\langle x|p\rangle \langle p | \psi(t)\rangle = ... = \sqrt{\frac{m}{2\pi\hbar i t}}\int_{-\infty}^\infty dx'\psi(x',t=0)e^{i\frac{(x-x')^2}{2\hbar t}m} $$

However, when trying to conceptually play with this equation, I stumble on a problem: By the postulates, after a position measurement on the free particle, its state collapse into $|x_0\rangle$ and has position wavefunction: $\langle x|x_0\rangle = \delta(x-x_0)$. Intuitively, one expects that the time evolution of a measured free particle disperses the wavefunction again into a gaussian, until further measurement or interaction. However, the time evolution of the dirac delta, doesn't seem to yield such a result:

By setting $|\psi(t=0)\rangle = |x_0\rangle$, and then trying to time evolve this initial state, we get:

$$ \psi(x,t) = \sqrt{\frac{m}{2\pi\hbar i t}}\int_{-\infty}^{\infty}dx'\delta(x'-x_0)e^{i\frac{(x-x')^2}{2\hbar t}m} = \sqrt{\frac{m}{2\pi\hbar i t}}e^{i\frac{(x-x_0)^2}{2\hbar t}m} $$

But this is an unphysical! As the position probability density $|\psi(x,t)|^2 = \frac{m}{2\pi\hbar t}$. Which is constant over all space...

Answer 1

This question reveals that the basis states of the position and momentum operators are not really proper states. They are completely non-normalizable, and they thus do not represent a valid probability density. They are a useful mathematical concept as in the Dirac formulation of state-space quantum mechanics, but not a particularly valid physical one. This is also reflected in the fact that if you were to prepare a quantum system in an eigenstate of the position operator then the standard deviation of the position would be 0, in flagrant violation of the Heisenberg uncertainty principle that $\sigma_x \sigma_p \geq \frac{\hbar}{2}$. This means that no physical system can ever be prepared in an exact eigenstate of the position operator (or the momentum operator for that matter). Instead, you could consider preparing the system in some initial Gaussian state with a narrow spread over position. This would allow you to properly propagate the dynamics and observe the "spreading" phenomenon that you desired.

Answer 2

What could be the physical meaning of "setting $|\psi(t=0)\rangle = |x_0\rangle$"? Assume that this means to "see", using light, that the particle is, with certainty, in a (nearly) exact location. That light would then have (nearly) zero wavelength and thus (nearly) infinite energy.

The interaction with that (nearly) infinite energy light disturbs the particle so greatly that it can (nearly) instantly end up (nearly) anywhere, over all space, as you say. Is the extremely quick loss of the knowledge of the position ("constancy" of the spatial distribution) really that surprising or unphysical, given your (already rather tall) assumptions?

You could construct a different apparatus, of course. But measurements are never gentle. Especially if they are too precise.