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Quantum Studies Mathematics and Foundations

A simple application of the time evolution operator in the solution of a paradox in quantum mechanics

Daniel H. T. Franco, L. S. Lima

Published Jul 28, 2017• 1 citation
Publisher
DOI
10.1007/s40509-017-0117-y

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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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There is a widespread belief in the quantum physical community, and textbooks used to teach quantum mechanics, that it is a difficult task to apply the time evolution operator on an initial wavefunction. Because the Hamiltonian operator is, generally, the sum of two operators, then it is not possible to apply the time evolution operator on an initial wavefunction ψ( x , 0), for it implies using terms like . A possible solution is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wavefunction. However, the exponential operator does not directly factorize, i.e. . In this study we present a useful procedure for 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 a particle subject to a constant force and a harmonic oscillator. Also, we discuss an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.

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The conventional tacit assumption according to which quantum processes take place in the Newtonian time topology is abandoned. The chronotopological space is based on the interaction proper time neighborhood. It is disconnected, satisfies the separation axioms of the topological space [Formula: see text], and is developed and used to solve some problems related to quantum field theory. In chronotopology the time evolution operator, [Formula: see text], breaks down into the alternative [Formula: see text] = either [Formula: see text]. The unitary, U, and the reduction, R, dynamics are implemented by means of two time evolution operators, [Formula: see text] and [Formula: see text], derived from a quantized version of Gel'fand's theory of the generalized random and infinitely divisible fields. As an application the solution of the paradox of Schrödinger's cat is presented.

International Journal of Modern Physics A•Nov 20, 1998•1 citation
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.
There is a widespread belief in the quantum physical community, and textbooks used to teach quantum mechanics, that it is a difficult task to apply the time evolution operator on an initial wavefunction. Because the Hamiltonian operator is, generally, the sum of two operators, then it is not possible to apply the time evolution operator on an initial wavefunction ψ( x , 0), for it implies using terms like . A possible solution is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wavefunction. However, the exponential operator does not directly factorize, i.e. . In this study we present a useful procedure for 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 a particle subject to a constant force and a harmonic oscillator. Also, we discuss an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.
The conventional tacit assumption according to which quantum processes take place in the Newtonian time topology is abandoned. The chronotopological space is based on the interaction proper time neighborhood. It is disconnected, satisfies the separation axioms of the topological space [Formula: see text], and is developed and used to solve some problems related to quantum field theory. In chronotopology the time evolution operator, [Formula: see text], breaks down into the alternative [Formula: see text] = either [Formula: see text]. The unitary, U, and the reduction, R, dynamics are implemented by means of two time evolution operators, [Formula: see text] and [Formula: see text], derived from a quantized version of Gel'fand's theory of the generalized random and infinitely divisible fields. As an application the solution of the paradox of Schrödinger's cat is presented.