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arXiv:1812.04381 · Physical Review A

Quantum particle in a split box: Excitations to the ground state

Vegard B. Sørdal, Joakim Bergli

Published Feb 20, 2019• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.99.022121

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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.
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.
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.
An adiabatic cycle involving the δ-function potential is shown to excite Bose particles confined in a one-dimensional box.We consider a cycle, during which a wall described by a δ potential is applied and its strength and position are slowly varied.When the system is initially prepared in the equilibrium ground state, the adiabatic cycle brings all bosons into the first excited one-particle state, leaving the system in a nonequilibrium state.The absorbed energy during the cycle is proportional to the number of bosons.
We determine the low-energy properties of a trapped Bose gas split in two by a potential barrier over the whole range of barrier heights and asymmetry between the wells. For either weak or strong coupling between the wells, our two-mode theory yields a two-site Bose-Hubbard Hamiltonian with the tunneling, interaction, and bias parameters calculated simply using an explicit form of two mode functions. When the potential barrier is relatively low, most of the particles occupy the condensate mode and our theory reduces to a two-mode version of the Bogoliubov theory, which gives a satisfactory estimate of the spatial shape and energy of the lowest collective excitation. When the barrier is high, our theory generalizes the standard two-site Bose-Hubbard model into the case of asymmetric modes, and correctly predicts a full separation of the modes in the limit of strong separation of the wells. We provide explicit analytic forms for the number squeezing and coherence as a function of particle number and temperature. We compare our theory to other two-mode theories for bosons in a double well and discuss their validity in different parameter regimes.
This chapter discusses the problem of one particle in a one-dimensional box free of the potential energy inside the box, that is, no external force is exerted on the particle inside the box. The corresponding Schrödinger equation is given along with the boundary conditions which yields an elliptic second-order differential equation and the solutions are given next: the wave function, quantum numbers and corresponding energy. The normalized wave function is also given which is similar to that of the one-particle harmonic oscillator. Afterwards, the Hamiltonian function of the particle in a bidimensional box is given along with its wave function that is the product of two wave function of the particle in a one-dimensional box and the energy is the sum of two energy components, x and y, of the particle in a one-dimensional box. Finally, there is a couple of exercises to plot the wave functions of a rectangular box to find the probability density of the wave function for a symmetrical box. (164 words).