17 research outputs found
Yield--Optimized Superoscillations
Superoscillating signals are band--limited signals that oscillate in some
region faster their largest Fourier component. While such signals have many
scientific and technological applications, their actual use is hampered by the
fact that an overwhelming proportion of the energy goes into that part of the
signal, which is not superoscillating. In the present article we consider the
problem of optimization of such signals. The optimization that we describe here
is that of the superoscillation yield, the ratio of the energy in the
superoscillations to the total energy of the signal, given the range and
frequency of the superoscillations. The constrained optimization leads to a
generalized eigenvalue problem, which is solved numerically. It is noteworthy
that it is possible to increase further the superoscillation yield at the cost
of slightly deforming the oscillatory part of the signal, while keeping the
average frequency. We show, how this can be done gradually, which enables a
trade-off between the distortion and the yield. We show how to apply this
approach to non-trivial domains, and explain how to generalize this to higher
dimensions.Comment: 8 pages, 5 figure
Interference Energy Spectrum of the Infinite Square Well
Certain superposition states of the 1-D infinite square well have transient
zeros at locations other than the nodes of the eigenstates that comprise them.
It is shown that if an infinite potential barrier is suddenly raised at some or
all of these zeros, the well can be split into multiple adjacent infinite
square wells without affecting the wavefunction. This effects a change of the
energy eigenbasis of the state to a basis that does not commute with the
original, and a subsequent measurement of the energy now reveals a completely
different spectrum, which we call the {interference energy spectrum} of the
state. This name is appropriate because the same splitting procedure applied at
the stationary nodes of any eigenstate does not change the measurable energy of
the state. Of particular interest, this procedure can result in measurable
energies that are greater than the energy of the highest mode in the original
superposition, raising questions about the conservation of energy akin to those
that have been raised in the study of superoscillations. An analytic derivation
is given for the interference spectrum of a given wavefunction with
known zeros located at points . Numerical simulations
were used to verify that a barrier can be rapidly raised at a zero of the
wavefunction without significantly affecting it. The interpretation of this
result with respect to the conservation of energy and the energy-time
uncertainty relation is discussed, and the idea of alternate energy eigenbases
is fleshed out. The question of whether or not a preferred discrete energy
spectrum is an inherent feature of a particle's quantum state is examined.Comment: 26 Pages, 5 Figure