The Quantum Backflow Phenomenon

It has been known since at least 1969, that there exist quantum states of a free
particle on the line with purely positive momentum that exhibit probability transfer
towards the negative half-line. This phenomenon, the quantum backflow effect, is
the subject of this thesis. The largest amount of backflow any state can exhibit over
a single time interval, the Bracken–Melloy constant, was initially calculated to be
cBM ≈ 0.04.

In the first half of this thesis, we consider the question of how much quantum
backflow states can exhibit over disjoint time intervals. This is done by formulating
the question in terms of the spectra of bounded operators. When considering more
than one disjoint time interval, we discuss a new phenomenon where a state exhibits
more probability transfer in the same direction as its momentum than any classical
state. We call this effect quantum overflow. Given a number M of disjoint time
intervals, we give bounds on the maximum amount of backflow and overflow a state
can exhibit over any M many intervals. The limiting cases in which two disjoint
intervals merge into one is particularly studied. Finally, we show plots of the time
t = 0 momentum space wavefunction of states exhibiting multiple quantum backflow
and quantum overflow.

The second half of the thesis contains a detailed numerical investigation into the
single backflow problem. Based on previous conjectures, we construct the maximum
backflow state from a given set of normalised states closely resembling the sinc
function. The Bracken—Melloy constant is then bounded from below by the largest
solution to a generalised eigenvalue problem. Using a multi-step numerical procedure,
we solve a high precision large dense generalised eigenvalue problem. From this
we obtain new numerical approximation to the backflow state and the best known
rigorous lower bound of the Bracken—Melloy constant.