In this article we study the nature of time in Mechanics.The fundamental principle, according to which a mechanical system evolves governed by a second order differential equation, implies the existence of an absolute time-duration in the sense of Newton.There is a second notion of time for conservative systems which makes the Hamiltonian action evolve at a constant rate.In Quantum Mechanics the absolute time loses its sense as it does the notion of trajectory.Then, we propose two different ways to reach the time dependent Schrödinger equation.One way consists of considering a "time constraint" on a free system.The other way is based on the point of view of Hertz, by considering the system as a projection of a free system.In the later manner, the "time" appearing in the Schrödinger equation is a linear combination of the time-duration with the "time" quotient of the action by the energy on each solution of the Hamilton-Jacobi equation.Both of them are based on a rule of quantization that we explain in Section 4. Contents 1. Introduction 1 2. Time as duration and its absolute character 3 3. Classical Mechanics 6 4. Quantization of contravariant tensors.Dequantization of differential operators 10 5. Time dependent Schrödinger equation from a time constraint 14 6.Waves in Classical Mechanics 17 7.Time dependent Schrödinger equation from a Hertz constraint 21 8.
In this article we study the nature of time in Mechanics.The fundamental principle, according to which a mechanical system evolves governed by a second order differential equation, implies the existence of an absolute time-duration in the sense of Newton.There is a second notion of time for conservative systems which makes the Hamiltonian action evolve at a constant rate.In Quantum Mechanics the absolute time loses its sense as it does the notion of trajectory.Then, we propose two different ways to reach the time dependent Schrödinger equation.One way consists of considering a "time constraint" on a free system.The other way is based on the point of view of Hertz, by considering the system as a projection of a free system.In the later manner, the "time" appearing in the Schrödinger equation is a linear combination of the time-duration with the "time" quotient of the action by the energy on each solution of the Hamilton-Jacobi equation.Both of them are based on a rule of quantization that we explain in Section 4. Contents 1. Introduction 1 2. Time as duration and its absolute character 3 3. Classical Mechanics 6 4. Quantization of contravariant tensors.Dequantization of differential operators 10 5. Time dependent Schrödinger equation from a time constraint 14 6.Waves in Classical Mechanics 17 7.Time dependent Schrödinger equation from a Hertz constraint 21 8.
In this article we study the nature of time in Mechanics. The fundamental principle, according to which a mechanical system evolves governed by a second order differential equation , implies the existence of an absolute time-duration in the sense of Newton. There is a second notion of time for conservative systems which makes the Hamiltonian action evolves at a constant rate. In Quantum Mechanics the absolute time loses its sense as it does the notion of trajectory. Then, we propose two different ways to reach the time dependent Schrödinger equation. One way consists of considering a “time constraint” on a free system. The other way is based on the point of view of Hertz, by considering the system as a projection of a free system. In the later manner, the “time” appearing in the Schrödinger equation is a linear combination of the time-duration with the “time” quotient of the action by the energy on each solution of the Hamilton-Jacobi equation. Both of them are based on a rule of quantization that we explain in Section 4.
The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → i ℏ(∂/∂ t ) and momentum → (ℏ/ i ) operator relations. In the present paper we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational: namely 'a particle is what a particle detector detects.' This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation.
This paper focuses on the principle of time dependent Schrodinger equation (TDSE) which simplified so many reviewed ideas, class room teaching, research ideologies and personal study meant for advanced knowledge. In this research work, the enlightenment of the basic concept of Schrodinger wave equation to improve knowledge about simple ways for mathematical understanding in deriving TDSE using different technique in more comprehensive approach. The research shows clearly that TDSE can be derived using time independent equation, wave mechanics, classical & Hamilton-Jacobi’s equations. Different methods and ways by different researchers/scholars have been used in the past. In this paper, we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational. Surely, a particle is what a particle detector detects. This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation. However, the result provides a comprehensive well derived derivation, derived using various approaches, which would make this research a unique one from different areas of specializations.
The time-independent Schrodinger equation: En= -1/2m [d/dx dWn/dx] / Wn(x) + V(x) ((1)) matches a classical conservation of energy equation where -1/2m [d/dx dWn/dx] / Wn(x) = KEclassical(x). In classical physics, however, one has x(t) so each x point is associated with a specific t for a given energy En. In the quantum time-independent equation, one has exp(-iEnt) as the time dependent factor and does not consider any link with x although if one is comparing ((1)) to a classical conservation equation then a link between x and t could be made to complete the analogy. This is usually not considered, but we examine this point as we argue it becomes important in the time-dependent situation. For the time-dependent Schrodinger case we consider two approaches. First, a linear combination of time independent solutions is taken i.e. W(x,t) = Sum over n an(En) exp(-iEnt) Wn(x) ((2)) Again each Wn(x) has an association with KEclassical(x) and so one may think in terms of (x,t) classical pairs for a given En. The second approach is to consider the function: exp(-iEnt + m Integral (0,x) v(y) dy) ((3)). This satisfies the time-dependent Schrodinger equation and yields a classical conservation of energy equation with v(x) being the classical velocity. Noting that En= .5m v(x)v(x) + V(x), ((3)) may be rewritten as: exp(i Integral L dt) where L = T-V is the Langrangian with T=.5mv(x)v(x). At this point En has been removed from the picture, but is replaced with: x(t)= X g(t)/g(T) ((4)) where (X,T) are a space-time pair corresponding to a particular energy E value. Using exp(i Action) ((5)) in the time-dependent Schrodinger equation yields a conservation equation which holds for a particular classical energy as long as (x,t) values in the equation are classically linked to the appropriate classical (X,T) values i.e. for a classical trajectory there is a specific family of X at time T for a given energy. For another energy, there is another family of (X,T) pairs. Thus the equation obtained represents energy conservation for many different energy values depending on which simultaneous values one uses for x and t. There is, however, no overlap as a specific (X,T) family corresponds to an energy E and another nonoverlapping family of (X,T) to another energy. This, however, is the same situation as ((2)) for which there are also families of (X,T) corresponding to each En. The only difference is that the En are quantized i.e. discrete. ((3)) and ((5)) are both solutions to the time-dependent Schrodinger equation which represent a variety of energies and families of (X,T) associated to the classical solution of the energy in question. We thus suggest that ((2)) and ((4)) are equivalent for this reason without using the idea of a complete set of basis functions represented by Wn(x). We consider these ideas as the Schrodinger propagator is given by ((5)), but with ((4)) generalized to: x(t)= Xi g(tf-t)/g(tf-ti) + Xf g(t-ti)/g(tf-ti) for the case that g(0)=0 (2) so that Xi,Xf, tf and ti enter the picture. This then should be equivalent to: Sum over n exp(-iEntf) Wn(xf) Wn* (xi) ((6)) In this note, we suggest that one may link ((5)) and ((6)) based on ideas of space-time families associated with different energies as well as the two being solutions of the time-dependent Schrodinger equation.
This study examines integrating the Schrödinger equation with classical mechanics using a virtual axis-to-dimensional expansion. One-dimensional material fluctuations are viewed in a two-dimensional plane, explaining the random nature of these fluctuations and their spatial and temporal trajectories. A quantum-consistent force field is proposed, with its strength determined by the Planck constant and inversely proportional to the distance from the stationary point. Newton's second law is applied to establish a second-order linear differential equation for material fluctuations, from which the standard one-dimensional Schrödinger equation is derived, showing their equivalence. The study extends the three-dimensional Schrödinger equation to include external forces and explains quantum phenomena like energy levels and transitions through particle trajectory changes. This approach connects classical mechanics and quantum mechanics, offering a concise and intuitive formulation with clear physical significance.
The concept of time dependent Schrödinger equation (TDSE) illustrated in literature and even during class room teaching is mostly either complex or meant for advanced learners. This article is intended to enlighten the concept to the beginners in the field and further to improve knowledge about detailed steps for abstract mathematical formulation used which helps in understanding to derive TDSE using various tools and in more comprehensible manner. It is shown that TDSE may be derived using wave mechanics, time independent equation, classical & Hamilton-Jacobi’s equations. Similar attempts have been done earlier by some researchers. However, this article provides a comprehensive, lucid and well derived derivation, derived using various approaches, which would make this article unique.