The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable ( $\mathit{x}$ , $\hat{v}$ ) , with the commutation relation [ $\mathit{x}$ , $\hat{v}$ ] = $\mathit{i}\hbar$ / $\mathit{m}$ , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion $\mathit{K}$ ( $\mathit{x, v, t}$ ) of the classical system, The comparison with the quantization in the space ( $\mathit{x, p}$ ) is done with the usual Schrödinger’s equation for the Hamiltonian $\mathit{H}$ ( $\mathit{x, p, t}$ ), and with the commutation relation [ $\mathit{x}$ , $\hat{p}$ ] = $\mathit{i}\hbar$ . It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the ( $\mathit{x}$ , $\hat{v}$ ) quantization has fewer oscillations than the ( $\mathit{x}$ , $\hat{p}$ ) quantization, the average energy of the system is higher in ( $\mathit{x}$ , $\hat{p}$ ) quantization than on the ( $\mathit{x}$ , $\hat{v}$ ) quantization, and the Boltzmann- Shannon entropy on the ( $\mathit{x}$ , $\hat{p}$ ) quantization is higher than on the ( $\mathit{x}$ , $\hat{v}$ ) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable ($\mathit{x}$ , $\hat{v}$) , with the commutation relation [$\mathit{x}$ , $\hat{v}$] = $\mathit{i}\hbar$/$\mathit{m}$ , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion $\mathit{K}$($\mathit{x, v, t}$) of the classical system, The comparison with the quantization in the space ($\mathit{x, p}$) is done with the usual Schrödinger’s equation for the Hamiltonian $\mathit{H}$($\mathit{x, p, t}$), and with the commutation relation [$\mathit{x}$ , $\hat{p}$] = $\mathit{i}\hbar$. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the ($\mathit{x}$ , $\hat{v}$) quantization has fewer oscillations than the ($\mathit{x}$ , $\hat{p}$) quantization, the average energy of the system is higher in ($\mathit{x}$ , $\hat{p}$) quantization than on the ($\mathit{x}$ , $\hat{v}$) quantization, and the Boltzmann- Shannon entropy on the ($\mathit{x}$ , $\hat{p}$) quantization is higher than on the ($\mathit{x}$ , $\hat{v}$) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.

Book Publisher International (a part of SCIENCEDOMAIN International)Sep 1, 20231 citation

Quantum elements of time and space

Dennis Marks

Bulletin of the American Physical SocietyJan 29, 20170 citations

Historical Development

P. J. E. Peebles

This chapter presents the origins of quantum mechanics. The story of how people hit on the highly non-intuitive world picture of quantum mechanics, in which the physical state of a system is represented by an element in an abstract linear space and its observable properties by operators in the space, is fascinating and exceedingly complicated. The much greater change from the classical world picture of Newtonian mechanics and general relativity to the quantum world picture came in many steps taken by many people, often against the better judgment of participants. There are three major elements in the story. The first is the experimental evidence that the energy of an isolated system can only assume special discrete or quantized values. The second is the idea that the energy is proportional to the frequency of a wave function associated with the system. The third is the connection between the de Broglie relation and energy quantization through the mathematical result that a wave equation with fixed boundary conditions allows only discrete quantized values of the frequency of oscillation of the wave function (as in the fundamental and harmonics of the vibration of a violin string).This chapter presents the origins of quantum mechanics. The story of how people hit on the highly non-intuitive world picture of quantum mechanics, in which the physical state of a system is represented by an element in an abstract linear space and its observable properties by operators in the space, is fascinating and exceedingly complicated. The much greater change from the classical world picture of Newtonian mechanics and general relativity to the quantum world picture came in many steps taken by many people, often against the better judgment of participants. There are three major elements in the story. The first is the experimental evidence that the energy of an isolated system can only assume special discrete or quantized values. The second is the idea that the energy is proportional to the frequency of a wave function associated with the system. The third is the connection between the de Broglie relation and energy quantization through the mathematical result that a wave equation with fixed boundary conditions allows only discrete quantized values of the frequency of oscillation of the wave function (as in the fundamental and harmonics of the vibration of a violin string).

Princeton University Press eBooksMay 14, 20190 citations

Quantum Uncertainty and Relativity

Sebastiano Tosto

The major challenge of modern physics is to merge relativistic and quantum theories \ninto a unique conceptual frame able to combine the basic statements of the former with \nthe quantization, the non-locality and non-reality of the latter. A previous paper has \nshown that the statistical formulation of the space-time uncertainty allows to describe \nthe quantum systems in agreement with these requirements of the quantum world. The \npresent paper aims to extend the same theoretical model and approach also to the special \nand general relativity.The major challenge of modern physics is to merge relativistic and quantum theories \ninto a unique conceptual frame able to combine the basic statements of the former with \nthe quantization, the non-locality and non-reality of the latter. A previous paper has \nshown that the statistical formulation of the space-time uncertainty allows to describe \nthe quantum systems in agreement with these requirements of the quantum world. The \npresent paper aims to extend the same theoretical model and approach also to the special \nand general relativity.

Apr 1, 20125 citations

The Classical and the Quantum World

CHRISTOF WETTERICH

Fundamental theories of physicsJan 1, 20250 citations

\mathit{x}

\hat{v}

) , with the commutation relation [

\mathit{x}

\hat{v}

] =

\mathit{i}\hbar

\mathit{m}

, and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion

\mathit{K}

(

\mathit{x, v, t}

) of the classical system, The comparison with the quantization in the space (

\mathit{x, p}

) is done with the usual Schrödinger’s equation for the Hamiltonian

\mathit{H}

(

\mathit{x, p, t}

), and with the commutation relation [

\mathit{x}

\hat{p}

] =

\mathit{i}\hbar

. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (

\mathit{x}

\hat{v}

) quantization has fewer oscillations than the (

\mathit{x}

\hat{p}

) quantization, the average energy of the system is higher in (

\mathit{x}

\hat{p}

) quantization than on the (

\mathit{x}

\hat{v}

) quantization, and the Boltzmann- Shannon entropy on the (

\mathit{x}

\hat{p}

) quantization is higher than on the (

\mathit{x}

\hat{v}

) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.