Position measurements on quantum mechanical particles in bound states

Thomas Schürmann

We consider successive measurements of position and momentum of a single particle. Let P be the conditional probability to measure the momentum k with accuracy ∆k, given a previously successful position measurement q with accuracy ∆q. Several upper bounds for the probability P are determined. For arbitrary, but given accuracies ∆q, ∆k, these bounds refer to the variation of q, k, the state vector ψ of the particle as well as to an infinite set of measuring partitions. A simple bound is given by the inequality P ≤ ∆k∆q h, where h is Planck’s quantum of action. It is nontrivial for all measurements with ∆k∆q < h. The second bound is obtained by considering the Hilbert-Schmidt norm. As our main result the least upper bound of P is determined. All bounds are independent of the order with which the measuring of position and momentum is made.

Stig Stenholm

J. Dittrich

A simplified one-dimensional model of measurement of the position of a quantum particle by two distant detectors is considered. Detectors are modelled by quantum particles bounded in potential wells with just two bound states, prepared in the excited states. Their de-excitation due to the short range interaction with the measured particle is the signal for the detection. In the approximations of short time or small coupling between the particle and the measuring apparatuses, the simultaneous detection of the particle by both detectors is suppressed. The results extend to other models with two-level detectors.

Henry Margenau

Henry Margenau

Ryan Thomas, Matthew Chilcott, Eite Tiesinga +2 more

Quantum mechanical scattering resonances for colliding particles occur when a continuum scattering state couples to a discrete bound state between them. The coupling also causes the bound state to interact with itself via the continuum and leads to a shift in the bound state energy, but, lacking knowledge of the bare bound state energy, measuring this self-energy via the resonance position has remained elusive. Here, we report on the direct observation of self-interaction by using a nano-eV atom collider to track the position of a magnetically-tunable Feshbach resonance through a parameter space spanned by energy and magnetic field. Our system of potassium and rubidium atoms displays a strongly non-monotonic resonance trajectory with an exceptionally large self-interaction energy arising from an interplay between the Feshbach bound state and a different, virtual bound state at a fixed energy near threshold.