In this note, we wish to consider a derivation of sorts of time independent Schrodinger type equations from general principles related to a function which is the square root of a density. This derivation is not aimed at quantum mechanics, in fact it is classical, yet it appears to apply to quantum mechanics and the oscillator potential of statistical mechanics and not to the 1/v(x) density of classical mechanics. We try to consider the reason for this.
In this note, we wish to consider a derivation of sorts of time independent Schrodinger type equations from general principles related to a function which is the square root of a density. This derivation is not aimed at quantum mechanics, in fact it is classical, yet it appears to apply to quantum mechanics and the oscillator potential of statistical mechanics and not to the 1/v(x) density of classical mechanics. We try to consider the reason for this.
In the literature, stochastic approaches employed to derive the Schrodinger equation seem to focus on the idea of a classical spatial density which depends on time and space ie. density(x,t). As a result, d/dt (partial) density appears in various equations such as the continuity and Focker Planck equations. At one point in the derivation the quantum wavefunction is introduced “mathematically” as W(x,t)=sqrt(density(x,t)) exp(i S(x,t)) where S is related to a velocity field through u(x,t)=velocity= d/dx S. From quantum mechanics, -Et becomes a term of S the velocity field. The stochastic/hydrodynamical approaches employ the term du/dt partial = d/dt (partial) d/dx S and then “pull out”” d/dx leaving d/dt (partial) S which yields -E average energy from the -Et term. In this note we argue that for a bound quantum state density(x) so d/dt (partial) density = 0, yet the statistical ideas of quantum mechanics should be present in such a case. Furthermore for the free particle wavefunction exp(ipx-iEt), density=1. In fact, the Schrodinger equation should follow for a free particle even the case of density=1. Stochastic processes seem to be based on equations which consider a change in classical spatial density at a point x in time due to flow in or out or stochastic changes. Quantum processes seem to deal with periodic probability changes from one dimension to another exp(ipx) (2 dimensions) which are somehow related to a physical process of momentum flow. Ultimately one links this process to an energy conservation which does not seem to be the same as a classical stochastic/hydrodynamical approach. In this note we argue that a free particle with a constant momentum p has periodic stochastic processes associated with it through its complex probability change in the region of its wavelength, but that these are linked to creating the momentum p. Furthermore momentum is linked to energy so we argue that the Schrodinger equation is a stochastic equation linking energy and momentum and that time and space are separated i.e. there is no situation of following a change in spatial density in time for a free particle with momentum p. For a bound state, one has a distribution of free particles, but again time is removed from the problem in the sense that it only appears in exp(-iEt) in the wavefunction to again establish a relationship between energy and average momentum and potential. There is again no nonzero d/dt (partial) density term which seems to be key in stochastic equations. Only if one takes a linear combination of energy eigenstates Sum over En an(En) exp(-iEnt) Wn(x) does one obtain the form density(x,t), but we argue that quantum features should already appear in exp(ipx) and the bound state scenario with a zero d/dt (partial) density(x,t). We argue, however, that the Schrodinger equation and fundamental stochastic ideas of quantum mechanics should already be present at the level of exp(ipx).
The time-independent Schrodinger equation is written terms of energy, namely using V(x) the potential and E, the energy eigenvalue. In fact, the equation can be written as KE average(x) + V(x) = E and so can be expressed in terms of a classical energy conservation equation at each point. KE average, however, in such a case is complicated and seems to contain various kinds of kinetic energy motion which only yields on average the classical kinetic energy. The purpose of this note, is to try to determine the form of the time-independent Schrodinger equation without any concept of energy. Only at the end of the discussion, will comparisons be made with classical conservation of energy. In such a discussion, it is assumed that particles undergo a zitterbewegung type motion which can be modeled by exp(ikx) hbar k being the momentum and that there are forces (related to a V(x)) in the one dimensional problem. The objective is to try to understand microscopically what is happening without immediate resort to a conservation equation.
In classical mechanics, the spatial density d(x) of a particle moving under a conservation of energy law .5mv(x)v(x) + V(x) = E is proportional to 1/v(x). This is obtained from d(x)= dt/T, i.e. the probability to be at x equals the time spent at x divided by the cycle time T. Given that dx=vdt to first order, density is proportional to 1/v(x). Now, d(x) is supposed to match the time independent quantum mechanical density W(x)W(x), where W(x) is the wavefunction, for high energy levels. Thus, one would expect the time a p wave (plane wave) spends at a point should be part of time independent quantum mechanics, but it does not seem to be overtly present. Furthermore, there is the question of T for the quantum mechanical problem. It is known that even for a time-independent solution W(x) has a factor exp(iEt). Thus, it seems time is present, even if it disappears in the density W*(x)W(x). It is argued that W*(x)W(x) = P(x) where P is probability, but W(x) is related to conditional probabilities such as P(x/p) or P(p/x). Thus, these conditional probabilities should be related to time. In an earlier note, we tried to show the time-independent Schrodinger equation implies all p plane waves are cycling with the same frequency E. In this note, we try to find a quantum mechanical representation for P(x,p)=dt/T where dt(x,p) is the time spent at x by a plane wave p and also try to evaluate T.
A derivation of the time-dependent Schrödinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent Schrödinger equation. It is shown that the standard time-dependent Schrödinger equation can be obtained for the semiclassical clock only. For elucidation of the physical meaning of the equation obtained in this way, various types of clocks are discussed. In addition, the corresponding equation for the density matrix and formulas for the mean values of operators are derived.
Previously a 2x2 matrix equation, linear in energy, was obtained from Einstein’s energy momentum equation E2=p2+m2 which reproduced Schrodinger zitterbewegung [1]. A similar equation was applied to the case of a gas with two allowable energy levels and to a gas in a potential field, the goal beng to obtain zitterbewegung results in statistical mechanics. The starting point for applying the 2x2 matrix equation to statistical mechanics was the fact that the 2x2 matrix approach yields an eigenvector which is the square root of two probabilities which sum to one. In this note, we wish to see if we can extend ideas further. First, we wish to to see if there are stronger reasons for applying the 2x2 matrix approach to statistical mechanics, and second, we wish to find further analogies between the approach and statistical mechanics. We consider relationships between physical constraint equations and the 2x2 matrix approach., We also consider functions analogous to the rest mass and energy in the statistical case, where the analogous energy also represents the frequency in zitterbewegung.
Density distribution function of classical statistical mechanics is generally generalized as a product of a general complex function and its complex Hermitian conjugate function, and the average of classical statistical mechanics is generalized as the average of the quantum mechanics. Furthermore, this paper derives three ones of the five axiom presumptions of quantum mechanics, e.g., deduces Schrȍdinger equation by two general ways, makes the three axiom presumptions into three theorems of quantum mechanics, not only solves the crisis to hard understand, but also gets new theories and new discoveries, e.g., this paper solves the crisis of the origin of the wave-particle duality (i.e., complementary principle), derives operators, eigenvalues and eigenstates, deduces commutation relations for coordinate and momentum as well as the time and energy, and discovers quantum mechanics is just a generalization ( mechanics ) theory of the complex square root of ( real density function of ) classical statistical mechanics, which will make people renew thinking modern physics development. In addition, this paper discovers the reason that Schrȍdinger equation doesn’t takes the time derivative of space coordinates. Therefore, this paper gives solution to the Crisis of the first quantization origin.