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arXiv:0809.4689 · Open Systems & Information Dynamics

Complexity Invariance by Replication in the Quantum Square Well

Ricardo López‐Ruiz, Jaime Sañudo A

Published Nov 30, 2009• 12 citations
arXivHTMLPDF

Abstract

A new kind of invariance by replication of a statistical measure of complexity is considered. We show that the set of energy eigenstates of the quantum infinite square well displays this particular invariance. Then, this system presents a constant complexity for all the energy eigenstates.
DOI
10.1142/s1230161209000311

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Understanding the mechanisms responsible for the equilibration of isolated quantum many-body systems is a long-standing open problem. In this work we obtain a statistical relationship between the equilibration properties of Hamiltonians and the complexity of their eigenvectors, provided that a conjecture about the incompressibility of quantum circuits holds. We quantify the complexity by the size of the smallest quantum circuit mapping the local basis onto the energy eigenbasis. Specifically, we consider the set of all Hamiltonians having complexity CCC, and show that almost all such Hamiltonians equilibrate if CCC is superquadratic in the system size, which includes the fully random Hamiltonian case in the limit $C\ensuremath{\rightarrow}\ensuremath{\infty}$, and do not equilibrate if CCC is sublinear. We also provide a simple formula for the equilibration time scale in terms of the Fourier transform of the level density. Our results are statistical and, therefore, do not apply to specific Hamiltonians. Yet they establish a fundamental link between equilibration and complexity theory.
The ETH ansatz for matrix elements of a given operator in the energy eigenstate basis results in a notion of thermalization for a chaotic system. In this context for a certain quantity - to be found for a given model - one may impose a particular condition on its matrix elements in the energy eigenstate basis so that the corresponding quantity exhibits linear growth at late times. The condition is to do with a possible pole structure the corresponding matrix elements may have. Based on the general expectation of complexity one may want to think of this quantity as a possible candidate for the quantum complexity. We note, however, that for the explicit examples we have considered in this paper, there are infinitely many quantities exhibiting similar behavior.
One examines the infinitely deep quantum cavity, also known as the quantum infinite square well, within the framework of the real Hilbert space. The solutions are considered in terms of complex wave functions, and also in terms of quaternionic wave functions. The complex results reproduce the usual achievements established in the complex Hilbert space, but also extend them to non-stationary solutions, as well as to distorted stationary solutions, different energy spectra, and dislocated observed position. The quaternionic cases further admit the incidence of self-interaction, something that cannot be observed in complex solutions. Therefore, both the complex and quaternionic solutions are more general than previous cases, thus opening the way to further one-dimensional solutions to be researched in the non-relativistic theory.
Certain superposition states of the 1-D infinite square well have transient zeros at locations other than the nodes of the eigenstates that comprise them. It is shown that if an infinite potential barrier is suddenly raised at some or all of these zeros, the well can be split into multiple adjacent infinite square wells without affecting the wavefunction. This effects a change of the energy eigenbasis of the state to a basis that does not commute with the original, and a subsequent measurement of the energy now reveals a completely different spectrum, which we call the interference energy spectrum of the state. This name is appropriate because the same splitting procedure applied at the stationary nodes of any eigenstate does not change the measurable energy of the state. Of particular interest, this procedure can result in measurable energies that are greater than the energy of the highest mode in the original superposition, raising questions about the conservation of energy akin to those that have been raised in the study of superoscillations. An analytic derivation is given for the interference spectrum of a given wavefunction Ψ ( x , t ) with N known zeros located at points s i = ( x i , t i ) . Numerical simulations were used to verify that a barrier can be rapidly raised at a zero of the wavefunction without significantly affecting it. The interpretation of this result with respect to the conservation of energy and the energy-time uncertainty relation is discussed, and the idea of alternate energy eigenbases is fleshed out. The question of whether or not a preferred discrete energy spectrum is an inherent feature of a particle’s quantum state is examined.