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

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.

Figure 1

Schematic of the three first eigenfunction and energies for a symmetric box for three different values of the barrier strength $\alpha \left(t\right)$. (a) Initial state of the system, before the barrier is inserted. (b) At an intermediate time before $\alpha \left(t\right)\to \infty$. We see that when the barrier is inserted at the center of the box it hits the nodes of the antisymmetric eigenfunctions, and therefore there are no excitations to this state [see Eq. (6)]. (c) The limit when $\alpha \left(t\right)\to \infty$. The total wave function is symmetric about the barrier, and the probability of finding the particle in either compartment is 1/2.

Figure 2

Same as Fig. 1, but for an asymmetric box. (a) The initial state of the system, identical to Fig. 1. Only now the eigenfunction of the first excited state is nonzero at the point we insert the barrier, allowing excitations from the ground state. From the intermediate time step in (b) to the final state in (c) the eigenfunction of the ground and first excited states evolves such that it is approximately zero in the smaller and larger compartments, respectively.

Figure 3

Energy levels as a function of time. We see that the odd energy levels approach the evens as the strength of the barrier increases, and the final spacing between them decreases with the magnitude of the asymmetry.

Figure 4

Dependence of the ratio $\frac{\langle {\psi }_1\left(t\right)\left|\widehat{\delta }\left(x\right)\right|{\psi }_m\left(t\right)\rangle }{E_m-E_1}$ on the strength of the barrier $\alpha$. Its magnitude gives us an indication of the coupling between the ground state and the higher excited states. We see that the coupling between the ground state and the first excited state remains substantial for high values of $\alpha$, while all the others decay quickly. This indicates that we can induce transitions between those two levels without exciting higher states when $\alpha$ is large. This plot was obtained with $\epsilon =0.1$.

Figure 5

Probability of finding the particle in the largest compartment (solid lines), as a function of the barrier insertion rate constant $A$ and the asymmetry parameter $\epsilon$. The probability to excite levels higher than the first excited state is shown in the dashed lines.