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Time-Dependent Schroedinger Equation
The glaringly missing part of the foundations of physics is the lack of a derivation of Schrödinger's equation from first principles. Schrödinger's equation is the widely accepted basis for quantum physics, but it is not derived; it is merely asserted. Here is the time-dependent Schrödinger equation
where i is the imaginary unit,
h is Planck's constant divided by 2π and H^ is the Hamiltonian operator for
the system under consideration. The dependent variable of the equation ψ, called the wave function,
was not defined by Schrödinger.
Schrödinger was responding to de Broglie's 1924 speculation that material particles have a wave nature. Schrödinger was trying to create an optical theory of matter waves analogous to the optics of radiation. For Schrödinger a particle had the nature of a wave packet, such as is shown below.
Or one with a higher energy
The Copenhagen School defined the wave function as such that its squared magnitude is the probability density function. This was an interpretation that Schrödinger disagreed with. Nevertheless, the Copenhagen Interpretation became the dominant view of quantum theory.
So, typically an attempted derivation of Schrödinger equation means a derivation of Schrödinger time-dependent equation with the Copenhagen Interpretation of the nature of its dependent variable. The derivation of Schrödinger time-independent equation from the time-dependent one is strictly a mathematical exercise and is not in dispute.
The attempted derivation of Schrödinger equation with the Copenhagen Interpretation fail because they always involve going from deterministic classical first principles to probability distributions without a justification.
Early in the development of quantum theory Niels Bohr stated that the validity of classical physics was well established and therefore for an element of quantum theory to be valid it must be such that when energy involved increases without bound the quantum solution must asymptotically approach the classical solution.
This must be modified somewhat. Below is shown the quantum theoretic solution for a particular case of a harmonic oscillator. It is derived from the solution of the time independent Schrödinger equation with the Copenhagen Interpretation. The thin line represents the quantum theoretic probability density function.
The heavy line is the time-spent probability density function for corresponding classical harmonic oscillator. As can be seen the spatial average of the quantum theoretic probability density function is at least aprroximately equal to the time-spent classical probability density function. From the mathematical solutions for harmonic oscillators it is easily shown that as energy increases without bound the spatial average of the quantum theoretic solution asymptotically approaches the classical time-spent probability density function. The quantum theoretic solution involves more and more dense fuctuations of probability density over space as the energy increases without bound.
Thus Bohr's version of the Correspondence Principle must modified to take into account the need for spatial averaging. Since physical observation cannot be for an instant in time or a point in space it necessarily involves some temporal or spatial averaging. Therefore the Correspondence Principle can be stated in terms of the observable quantum theoretic solution must asymptotically approach the classical solution. Here is a visual depiction of the situation.
The time-spent probability distribution for a particle executing a periodic trajectory is just the probability density of finding the particle in an interval ds at randomly chosen time. That is just the proportion of the time spent in that interval dt/T, where T is the total time required for traversing its trajectory. That is dt=(dx/|v|)/T where v is the velocity of the particle. Thus the time-spent probability density P(x) for a particle of energy E traversing a periodic path is
Let QE(x) be the probability density function that comes from the solution of the time independent Schrödinger equation of a system having a total energy E. Let QE(x) be the spatially averaged probability density function.
According to the modified Correspondence Principle
The above material gives a justification for interpreting Schrödinger's wave function in terms of a probability density function, it is not the probability density function of the Copenhagen Interpretation. Consider a rapidly rotating fan. It looks like an unchanging translucent disk. The Copenhagen Interpretation interprets the translucent disk of the rapidly rotating fan as a particle with a probability density function. It tries to say that the fan has no physical reality until it is subjected to physical measurement which acts like a wrench stuck into the translucent disk of a rapidly rotating fan and CLANG the fan acquires a physical reality.
Since for the classical time-spent probability density function probability density is inversely related to velocity and the quantum theoretic probability density functions asymptotically approach the time-spent probability density function there should be some inverse relationship between quantum theoretic probability density and particle velocity u(x); i.e.,
where k is a constant of proportionality.
This gives quantum motion as a sequence of intervals of slower motion and faster motion. The intervals of slower motion correspond to what are called the allowed states of the system. The states of rapid motion have velocities asymptotically approaching infinity.
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