We discuss two different approaches for splitting the wave function of a single-particle box (SPB) into two equal parts. Adiabatic insertion of a barrier in the center of a SPB in order to make two compartments which each have probability 1/2 of finding the particle in it is one of the key steps for a Szilard engine. However, any asymmetry between the volume of the compartments due to an off-center insertion of the barrier results in a particle that is fully localized in the larger compartment, in the adiabatic limit. We show that rather than exactly splitting the eigenfunctions in half by a symmetric barrier, one can use a nonadiabatic insertion of an asymmetric barrier to induce excitations to only the first excited state of the full box. As the barrier strength goes to infinity the excited state of the full box becomes the ground state of one of the new boxes. Thus, we can achieve close to exact splitting of the probability between the two compartments using the more realistic nonadiabatic, not perfectly centered barrier, rather than the idealized adiabatic and central barrier normally assumed.
The ground state of a particle confined to two two-dimensional square boxes connected with a narrow channel is calculated. It is shown that the presence of the channel results in the splitting of the otherwise degenerate ground state. When the width of the channel is small compared to the wavelength of the particle, the wavefunction in the channel is exponentially suppressed, very much like in the case of tunnelling through a potential barrier. Moreover, in the limit of long and narrow channels, any small asymmetry in the shape of the boxes results in the localization of the particle.
Wave mechanics of a particle in 1-D box (size
=d) is critically analyzed to reveal its untouched aspects. When the particle rests in its ground state, its zero-point force (
Fo) produces non-zero strain by modifying the box size from
d to
d′=d+Δd in all practical situations where the force (
Fa) restoring
d is not infinitely strong. Assuming that
Fa originates from a potential
∝x2 (
x being a small change in
d), we find that: (i) the particle and strained box assume a mutually bound state (under the equilibrium between
Fo and
Fa) with binding energy
ΔE=−εo′Δd/d′ (with
εo′=h2/8md′2 being the ground state energy of the particle in the strained box), (ii) the box size oscillates around
d′ when the said equilibrium is disturbed, (iii) an exchange of energy between the particle and the strained box occurs during such oscillations, and (iv) the particle, having collisional motion in its excited states, assumes collisionless motion in its ground state. These aspects have desired experimental support and proven relevance for understanding the physics of widely different systems such as quantum dots, quantum wires, trapped single particle/ion, clusters of particles, superconductors, superfluids, {\it etc.} It is emphasized that the physics of such a system in its low energy states can be truly revealed if the theory incorporates
Fo and related aspects.
In quantum information processing, one often considers inserting a barrier into a box containing a particle to generate one bit of Shannon entropy. We formulate this problem as a one-dimensional Schr\"odinger equation with a time-dependent $\ensuremath{\delta}\text{-function}$ potential. It is a natural generalization of the particle in a box, a canonical example of quantum mechanics, and we present analytic and numerical investigations on this problem. After deriving an exact Volterra-type integral equation, composed of an infinite sum of modes, we show that approximate formulas with the lowest-frequency modes correctly capture the qualitative behavior of the wave function. If we take into account hundreds of modes, our numerical calculation shows that the quantum adiabatic theorem actually gives a very good approximation even if the barrier height diverges within finite time, as long as it is sufficiently longer than the characteristic time scale of the particle. In particular, if the barrier is slowly inserted at an asymmetric position, the particle is localized by the insertion itself, in accordance with a prediction of the adiabatic theorem. On the other hand, when the barrier is inserted quickly, the wave function becomes rugged after the insertion because of the energy transfer to the particle. Regardless of the position of the barrier, the fast insertion leaves the particle unlocalized so that we can obtain meaningful information by a which-side measurement. Our numerical procedure provides a precise way to calculate the wave function throughout the process, from which one can estimate the amount of this information for an arbitrary insertion protocol.