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Communications in Theoretical Physics

Separability of Solutions to a Schrödinger Equation

Wenhua Wang, Huaixin Cao, Zhihua Guo, Baomin Yu

Published Aug 1, 2014• 0 citations
Publisher

Abstract

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 .

DOI
10.1088/0253-6102/62/2/06

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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 .
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.
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.
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This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
Problems in which Schrodinger's equation with timedependent hamiltonian is applied to one system should rather be considered as representing cases of two systems, the second acting appreciably on the first but not inversely; the systems as a whole is then considered to be isolated and, consequently, subject to treatment by means of a Schrodinger equation with time-independent hamiltonian. The composition and decomposition of systems is dealt with by the method of statistical operators of v. Neumann. Both methods of dealing with the problem are shown to lead in good approximation to the same results, but only for times fulfilling a given inequality. (auth)