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arXiv:0711.0535

Which operator generates time evolution in Quantum Mechanics?

Miloslav Znojil

Published Nov 5, 2007• 7 citations
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Abstract

We attract attention to an interesting family of quantum systems where the generator $H_{(gen)}$ of time-evolution of wave functions is not equal to the Hamiltonian $H$. We describe the origin of the difference $H_{(gen)}-H$ and interpret it as a carrier of a compressed information about the other relevant observables.

DOI
10.48550/arxiv.0711.0535

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Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

Mostafazadeh, Ali

I point out that if one defines the operator $U_R(t)$ 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 $U_R(t)$ 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 $H$ but an operator $H'$ which differs from $H$ if the metric operator is time-dependent. In the latter case this effective Hamiltonian $H'$ 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.

arXiv:0711.1078•Nov 7, 2007•4 citations

Factorizing the time evolution operator

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There is a widespread belief in the quantum physical community, and in textbooks used to teach Quantum Mechanics, that it is a difficult task to apply the time evolution operator on an initial wave function. That is to say, because the hamiltonian operator generally is the sum of two operators, then it is a difficult task to apply the time evolution operator on an initial wave function, because it implies to apply terms like (A+B)^n. A possible solution of this problem is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wave function. However, the exponential operator does not directly factorize. In this work we present useful ways to factorizing the time evolution operator when the argument of the exponential is a sum of two operators which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed in previous papers published in this Journal. Finally, we discuss the possible insight that can be learned using this approach in teaching Quantum Mechanics.

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This study interprets four major mysteries related to observation in quantum mechanics. These are based on a new interpretation of the energy–time uncertainty relation, which yields time retrogression and suggests that a quantum state’s time evolution may fluctuate if measurement fixes its energy. The resulting time-reversed state is represented by a bra vector. The time reversed bra vector and the time-evolving ket vector interact along the time axis to generate a time-invariant bra–ket state, indicating state determination. The first topic is why observation selects one state from a superposition. Among all time-reversed states, the state with the least energy dispersion reaches farthest into the past, establishing that state while the others disappear. The existence of a single state in the farthest past explains quantum state contraction induced by observation. The second topic is the spatial contraction of the wave packet upon observation. In QFT, the wave functions forming the wave packet arise from vacuum energy fluctuations which disappear through the inner product in the bra–ket state. Consequently, external vacuum energy exerts a compressive force that contracts the wave packet. This mechanism explains both state selection and spatial contraction upon observation. The third topic is long-range correlation without transmission time in quantum entanglement. This study explains that photon states go back into the past and are determined there, accounting for the simultaneous correlation between distant photons. The fourth topic is the origin of randomness in measurement outcomes. This study interprets randomness in energy dispersion as arising from phase differences between measuring-device states. Thus, the state is determined probabilistically yet deterministically through measurement. These interpretations reinforce local-reality and restore determinism in quantum mechanics.

Journal of Modern Classical Physics & Quantum Neuroscience•Sep 15, 2020•0 citations
We attract attention to an interesting family of quantum systems where the generator H(gen)H_{(gen)}H(gen)​ of time-evolution of wave functions is not equal to the Hamiltonian HHH. We describe the origin of the difference H(gen)−HH_{(gen)}-HH(gen)​−H and interpret it as a carrier of a compressed information about the other relevant observables.
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.
There is a widespread belief in the quantum physical community, and in textbooks used to teach Quantum Mechanics, that it is a difficult task to apply the time evolution operator on an initial wave function. That is to say, because the hamiltonian operator generally is the sum of two operators, then it is a difficult task to apply the time evolution operator on an initial wave function, because it implies to apply terms like (A+B)^n. A possible solution of this problem is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wave function. However, the exponential operator does not directly factorize. In this work we present useful ways to factorizing the time evolution operator when the argument of the exponential is a sum of two operators which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed in previous papers published in this Journal. Finally, we discuss the possible insight that can be learned using this approach in teaching Quantum Mechanics.
Let HHH be a self-adjoint operator (a Hamiltonian) on a complex Hilbert space $ \mathcal{H}$
The relativistic free particle system in 1+1 dimensions is formulated as a ``bi-Hamiltonian system''. One Hamiltonian generates ordinary time translations, and another generates (essentially) boosts. Any observer, accelerated or not, sees evolution as the continuous unfolding of a canonical transformation which may be described using the two Hamiltonians. When the system is quantized both Hamiltonians become Hermitian operators in the standard positive definite inner product. Hence, each observer sees the evolution of the wave function as the continuous unfolding of a unitary transformation in the standard positive definite inner product. The result appears to be a consistent single particle interpretation of relativistic quantum mechanics. This interpretation has the feature that the wave function is observer dependent, and observables have non-local character, similar to what one might expect in quantum gravity.
This study interprets four major mysteries related to observation in quantum mechanics. These are based on a new interpretation of the energy–time uncertainty relation, which yields time retrogression and suggests that a quantum state’s time evolution may fluctuate if measurement fixes its energy. The resulting time-reversed state is represented by a bra vector. The time reversed bra vector and the time-evolving ket vector interact along the time axis to generate a time-invariant bra–ket state, indicating state determination. The first topic is why observation selects one state from a superposition. Among all time-reversed states, the state with the least energy dispersion reaches farthest into the past, establishing that state while the others disappear. The existence of a single state in the farthest past explains quantum state contraction induced by observation. The second topic is the spatial contraction of the wave packet upon observation. In QFT, the wave functions forming the wave packet arise from vacuum energy fluctuations which disappear through the inner product in the bra–ket state. Consequently, external vacuum energy exerts a compressive force that contracts the wave packet. This mechanism explains both state selection and spatial contraction upon observation. The third topic is long-range correlation without transmission time in quantum entanglement. This study explains that photon states go back into the past and are determined there, accounting for the simultaneous correlation between distant photons. The fourth topic is the origin of randomness in measurement outcomes. This study interprets randomness in energy dispersion as arising from phase differences between measuring-device states. Thus, the state is determined probabilistically yet deterministically through measurement. These interpretations reinforce local-reality and restore determinism in quantum mechanics.