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Derivative of Unitary is not always -iHU

A S Muddu

Published Dec 23, 2020• 0 citations
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Abstract

$i\dot{U} = HU$ (or $\dot{U} = -iHU$) is the equation that is said to govern the evolution of a unitary matrix $U$ given the Hamiltonian $H$ of the system. This equation is said to hold true even if the Hamiltonian is time dependent. We show iU = HU (or $\dot{U} = -iHU$) is the equation that is said to govern the evolution of a unitary matrix $U$ given the Hamiltonian $H$ of the system. This equation is said to hold true even if the Hamiltonian is time dependent. We show in this paper that $i\dot{U} = HU$ may not always hold for time dependent Hamiltonians.this paper that $i\dot{U} = HU$ may not always hold for time dependent Hamiltonians.

DOI
10.22541/au.160873315.53937359/v1

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•Jan 1, 1996•4 citations
iU˙=HUi\dot{U} = HUiU˙=HU (or U˙=−iHU\dot{U} = -iHUU˙=−iHU) is the equation that is said to govern the evolution of a unitary matrix UUU given the Hamiltonian HHH of the system. This equation is said to hold true even if the Hamiltonian is time dependent. We show iU = HU (or U˙=−iHU\dot{U} = -iHUU˙=−iHU) is the equation that is said to govern the evolution of a unitary matrix UUU given the Hamiltonian HHH of the system. This equation is said to hold true even if the Hamiltonian is time dependent. We show in this paper that iU˙=HUi\dot{U} = HUiU˙=HU may not always hold for time dependent Hamiltonians.this paper that iU˙=HUi\dot{U} = HUiU˙=HU may not always hold for time dependent Hamiltonians.
I point out that if one defines the operator UR(t)U_R(t)UR​(t) as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of UR(t)U_R(t)UR​(t) as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator HHH but an operator H′H'H′ which differs from HHH if the metric operator is time-dependent. In the latter case this effective Hamiltonian H′H'H′ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
The time-evolution operator obtained from the fractional-time Schrödinger equation (FTSE) is said to be nonunitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the nonunitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.
The dynamics of a quantum system is governed by the Hamiltonian H^.If∣Ψ⟩ \hat H.If|\Psi \rangleH^.If∣Ψ⟩ is a quantum state of the system in an abstract Hilbert space representation, its time-evolution is given by −ℏi∂∂t∣Ψ⟩=H^∣Ψ⟩ - \frac{\hbar }{i}\frac{\partial }{{\partial t}}|\Psi \rangle = \hat H|\Psi \rangle−iℏ​∂t∂​∣Ψ⟩=H^∣Ψ⟩ (1.1)
The time-evolution operator for an explicitly time-dependent Hamiltonian is expressed as the product of a sequence of unitary operators. These are obtained by successive time-dependent unitary transformations of the Hilbert space followed by the adiabatic approximation at each step. The resulting adiabatic product expansion yields a generalization of the quantum adiabatic approximation. Furthermore, it leads to an infinite class of exactly solvable models. PACS number: 03.65.Bz Consider the dynamics of a quantum mechanical system whose Hamiltonian H = H(τ) is explicitly time-dependent. The evolution of a state vector |ψ(τ) 〉 is governed by the Schrödinger equation: i¯h | ˙ ψ(τ) 〉 = H(τ) |ψ(τ) 〉 , |ψ(0) 〉 = |ψ0 〉 , (1) where a dot means a time-derivative. Alternatively, one has |ψ(τ) 〉 = U(τ)|ψ0〉, where i τ − H(t) dt U(τ) = T e ¯h 0 (2) is the time-evolution operator. Here T denotes the time-ordering operator. The purpose of this article is to express U(τ) as the product of a sequence of unitary operators each of