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理论物理通讯：英文版

Separability of Solutions to a Schr?dinger Equation

王文华, 曹怀信, 郭志华, 余保民

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

We discuss separability of solutions to a Schr?dinger equation that describes a composite quantum system and give some kinds of Hamiltonians H(t) such that the solution to Schr?dinger equation induced by H(t) is separable at any time provided that it is separable at t = 0. For example, we prove that if the Hamiltonian H is time-independent and equals to the product PA■PB of two projections on the subsystems KAand KB, respectively, then the state |ψ(t) of the composite system starting from a separable initial |ψ(0) = |ψA■|ψB is separable for all t ∈ [0, T] if and only if either |ψA is an eigenstate of PA, or |ψB is an eigenstate of PB.

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We discuss separability of solutions to a Schr?dinger equation that describes a composite quantum system and give some kinds of Hamiltonians H(t) such that the solution to Schr?dinger equation induced by H(t) is separable at any time provided that it is separable at t = 0. For example, we prove that if the Hamiltonian H is time-independent and equals to the product PA■PB of two projections on the subsystems KAand KB, respectively, then the state |ψ(t) of the composite system starting from a separable initial |ψ(0) = |ψA■|ψB is separable for all t ∈ [0, T] if and only if either |ψA is an eigenstate of PA, or |ψB is an eigenstate of PB.
We discuss separability of solutions to a Schrödinger equation that describes a composite quantum system and give some kinds of Hamiltonians H(t) such that the solution to Schrödinger equation induced by H(t) is separable at any time provided that it is separable at t = 0. For example, we prove that if the Hamiltonian H is time-independent and equals to the product P A ⊗ P B of two projections on the subsystems K A and K B , respectively, then the state |ψ(t)〉 of the composite system starting from a separable initial |ψ(0)〉 = |ψ A 〉 ⊗ |ψ B 〉 is separable for all t ∈ [0, T] if and only if either |ψ A 〉 is an eigenstate of P A , or |ψ B 〉 is an eigenstate of P B .
The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.
Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.
If the Hamiltonian is time-dependent, the solution of the time-dependent Schr dinger equation in general becomes difficult. Quantum zeno effect, named by Misra and Sudarshan, is an inhibition of induced or spontaneous transition between quantum states by frequent measurements, so that the system remains in its initial state throughout a long time interval. The frequent measurements decouple the interaction between the system and the environment, and suppress the evolution of quantum state. The phenomenon of Berry phase was first discovered by Pancharantnam in 1956, independently by Longuet-Higgins in 1958, and more generally by Berry in 1984. The Landau-Zener formulas is to calculate the transition probability from an instantaneous lower energy eigenstate at infinite past to an instantaneous upper energy eigenstate at infinite future.