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arXiv:1611.07129 · Physical Review A

Particle in a box with a time-dependent δ-function potential

Seung Ki Baek, Su Do Yi, Minjae Kim

Published Nov 21, 2016• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.94.052124

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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.

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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 \ensuremathδ-function\ensuremath{\delta}\text{-function}\ensuremathδ-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 particle in an expanding/contracting 1-dimension box is revisited in action-angle like variables with direct thermodynamic interpretation. An angle dependent potential is proposed accurately describing the mechanical behavior while also capturing thermodynamic evolution -- entropy production -- within a canonical Hamiltonian framework. Heat transfer at constant volume is analyzed, and the derived thermal conductance matches the universal quantum of heat conductance GQG_{Q}GQ​ in the quantum limit. Having a Hamiltonian scheme interpretable in thermodynamic terms, a Schrödinger-like wave equation is formulated whose wavefunction solutions contain the information about the entropy evolution. The results show exact agreement with 'classical' results for non abrupt changes. Finally, comparisons with a pure quantum mechanical treatment of the wave function in an expanding box confirm consistency in quasi-static regimes and reveal adiabaticity breakdown under far-from-equilibrium conditions.
The time-dependent Schrödinger equation with a time-dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wavefunction at the origin, one may derive the wavefunction everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the potential lead to the conservation of the normalization of the probability density.
This paper can serve as an illustration for a course on quantum mechanics. It gives a non-trivial example of the time evolution of a system, which includes a particle in the potential of two δ-traps: one of which is in the fixed position, and the other moves with constant velocity. The numerical solution of the Schrödinger equation represents the behaviour of the vector of state in quite a transparent way; one can see rough pictures of the processes of charge transfer and ionization. The numerical procedure is explained, and an exercise can be carried out in constructing other time-dependent systems, containing δ-potentials.
The time dependent Schroedinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density.
Note (March 8, 2022) von Neumann's entropy may be written in Shannon's form for the eigenvalues of the density operator. If the density operator is written as Sum over j mj |j><j| with the set |j> not being orthonormal, then von Neumann's entropy would not be Shannon's entropy -Sum over j mj ln(mj) because the set mj are not the eigenvalues of the density matrix. In the literature, Shannon's entropy, however, is sometimes used with the weights of the projection states i.e. mj weights of |j>. For example, see: Kak, S. The Transactional Nature of Quantum Informationhttps://arxiv.org/pdf/0907.2465.pdf Note: (March 10, 2022) As an example of a state linked to the "fictitious" P(x)P(p) consider a single particle in a box with infinite potential walls. Outside the box W(x)=0 so P(x)=0, but P(p)=a(p)a(p) is nonzero for all p. Thus, one needs to consider P(x)P(p) together to make sense of the region outside of the box. Using the argument that one cannot know p and x exactly and so should consider P(x) and P(p) separately seems to lead to considerations of P(p) being nonzero i.e changes of measuring particle momenta outside the box which constradicts P(x)=0 outside. In (1) it is argued that von Neumann’s density operator d = Sum over i bi |i><i| = Sum over j mj |j><j> where in one case |i> represents an orthonormal basis and in the other of |j> a different nonorthogonal basis is used. The problem is that von Neumann entropy -Sum over i bi ln(bi) and -Sum over j mjln(mj) are not the same, they depend on the projection operator used. The author of (1) goes on to argue that historically it was suggested this might mean that a quantum state in quantum statistical mechanics is not defined clearly. We wish to consider the case of a pure bound state for which we have defined S=entropy=Sp+Sx where Sp=-Integral dp P(p) ln(P(p)) and Sx = - Integral dx P(x) ln(P(x)). According to quantum mechanics, one may not know p and x exactly, but S=Sp+Sx is equivalent to using P(p)P(x) in Shannon’s entropy formula. (Alternatively, one may use P(p/x)P(x).) As a result, the so-called state (x,p) associated with P(p)P(x) assumes knowledge of p and x exactly and so is not a real quantum state perhaps in the manner suggested in (1). In order to consider this issue, we argue that a quantum bound problem may be thought of in two ways: the classical mechanics type picture, and the statistical mechanical type one. The first approach is consistent with using the wavefuntion W(x) in the time-independent Schrodinger equation. P(p) and P(x) do not appear, but this is still a statistical picture because KE(x) = {Sum over p pp/2m a(p) exp(ipx)}/W(x). KE(x)+V(x)=E, nevertheless, has the form of a classical mechanical particle. In other words, it has one total E at any x and a potential value V(x) at x. In fact, KE(x) may be considered to represent a vrms(x) which is the velocity of the classical particle, although this is a purely mathematical construct. One may define P(p/x) = a(p)exp(ipx)/W(x) which is complex and allows for interference among exp(ipx) so this picture is very different from that of usual classical probabilities. The second picture of quantum mechanics involves P(x) or P(p) which behave as classical probabilities (i.e. are real, positive and sum to 1). For example: Operator ave = < W | B | W> is associated with this picture (as is von Neumann’s density matrix). One may write energy as: E= Sum over p P(p)pp/2m + Sum over x V(x) P(x), but this equation does not lead to solutions. One must use the Schrodinger to equation to solve for W(x) and find a(p) from W(x)=Sum over p a(p)exp(ipx) and then form P(x)=W(x)W(x) and P(p)=a(p)a(p). Nevertheless quantum statistics, in order to be linked with classical statistical mechanics, seems to be formulated in terms of P(p) and P(x). For example the diagonal elements of von Neumann’s density matrix contain this type of probability. Given that quantum mechanics may be viewed in two different statistical manners, (A) in terms of W(x), the Schrodinger equation and P(p/x) or (B) in terms of P(p) and P(x), a formulation of quantum statistical mechanics based on P(p) and P(x) may lead to unusual results. In particular, P(p) and P(x) represent exact p and x values, but according to quantum theory one cannot know these at the same time. Thus talking of P(p) and P(x) (in a classical way) seems to mean referring to a quantum state that is not really defined, perhaps one of the states noted in (1). This may not be a problem, however, if the underlying statistics follows from W(x) and a(p) and P(p/x) for which p and x are defined, but are linked to complex probabilities.
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.