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

Abstract

We discuss two different approaches for splitting the wavefunction 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 to find 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 offcenter 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 non-adiabatic insertion of an asymmetric barrier to induce excitations to the first excited state of the full box. As the barrier height 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 non-adiabatic, not perfectly centered barrier, rather than the idealized adiabatic and central barrier normally assumed.

Deep reinforcement learning for quantum Szilard engine optimization

“…The number of times the agent chooses an action per protocol is given by N t , and the time-step is therefore dt = T /N t . The environment that the agent acts in is the quantum mechanical SPB, with initial state |c 1 | 2 = 1 and |c n>1 | 2 = 0, and time evolution given by the Schrödinger equation i∂ t |ψ(t) = H(t) |ψ(t) , which we solve as in [25]. The sequential process for one episode is then 1.…”

“…The initial state is therefore given by |c 1 (0)| 2 = 1, and the goal is to construct a protocol α(t), which brings us to a final state where |c 1 (T )| 2 = |c 2 (T )| 2 = 1/2, where T is the duration of the protocol. More details on how the instantaneous eigenstates are calculated, and how the time evolution of the total wave function is numerically solved, is given in [25].…”

“…Since this is an intrinsic property of the system, it is impossible to find protocols that avoids excitations for d → 0. For d = 0, the ground state of the left and right compartment constitute a doubly degenerate global ground state, and to achieve an equal splitting of the wave function, one has to insert the barrier adiabatically [25].…”

“…This will in general decrease the efficiency of the quantum Szilard engine, since the measurement only determines which side the particle is found, not its exact eigenstate. However, there is one special way of obtaining exact splitting of the wave function without losing any information in the measurement, for the asymmetric Szilard engine [25]: If we insert the barrier in such a way that the total wave function is a superposition of only the first and second eigenstate at the time of measurement, i.e. |Ψ = (|ψ 1 + |ψ 2 ) / √ 2, the which-side measurement does not result in any information loss since the second eigenstate becomes the ground state of the smaller compartment.…”

Análise da repentina duplicação da largura de um poço quadrado infinito 1-dim: um resultado surpreendente à primeira vista

“…Ele mostrou que, se uma parede divisória é introduzida muito lentamente (aproximação adiabática), a partícula sempre será encontrada no interior da parte maior após a divisória ser introduzida completamente, inviabilizando assim a discussão da não localidade [29][30][31]. V. B. Sφrdal e J. Bergli [32] contornaram o problema da assimetria mostrando que uma introdução assimétrica e não adiabática da parede divisória pode conduzir a probabilidades iguais (50%) de encontrar a partícula em uma das partes. S. J. van Enk [33] considerou o caso de duas caixas idênticas, com uma partícula no estado fundamental em cada caixa.…”