The Schrodinger equation, as it is used most of the time, is the 3D wave equations with stationary solutions. In every practical application you use dynamic models where the resonances and states are part of the constraints. The vacuum states usually require working with the nonlinear Schrodinger solutions. And most scattering and resonance problems require solitons and wavelets. The representation of the solutions – sufficiently efficient to solve in a human lifetime – are many. I spent close to 50 years following quantum chemistry – which is just Schrodinger solutions – simplified in various ways to apply to practical problems of modeling and simulation chemical structure, nuclear structure, molecular and atomic structure. And the reactions, and the possible products and properties.
Every application of one of these sets of representations of real world problems requires calibrating the model. So, in that sense, the Schrodinger equation was born of the practical problem of representing – in a reasonably concise way – the properties of large numbers of particles and states and materials. I worked in industrial chemistry for a few years. When you build new industrial processes for chemical and nuclear materials, you cannot do it by talking. You have to “do the numbers”. It is too costly and dangerous to guess or “just build it”. So we simulate and model and estimate and check and calibrate and keep working until we find economic and financially useful solutions.
So I am saying that “statistics” as a nice theoretical framework might sound nice, but all practical applications and economically important things are driven by real problems – that need calibrated models and practical calculations.
My uncle, Royal E Collins, was a physicist. He spent his life on practical problems. In his later years, he summarized his life’s work, saying that all of quantum mechanics has a statistical basis. I somewhat agree with him. If you look at the real world quantitatively – there is always measurement, statistical summaries, and now more and more, stochastic, evolutionary, “AI” and machine learning methods – that are all at their heart statistical.
Did you know that magnetic and electric fields can modify the rate of clocks – simply by their energy density? We have the Zeeman and Kerr and many variations of fields modifying the frequencies of transitions between states (back to Schrodinger and representations of data about those kinds of measurements). But gravitational potential fields, velocity potential fields, magnetic and electric potential fields — all can be used in the time dilation equations (used for very practical purposes every day for GPS and solar system wide synchronization) to model the rate of time.
I have to get back to something else. I noticed your question. I wish you would pick something practical to work on.
The human species needs “atomic fuels” – safe, reliable, economic and energy dense enough to power earth to orbit, earth to mars and solar system wide transport. The binding energies for chemical fuels are roughly 10 electron volts per bond. The binding energy for “nuclear” are millions of electron volts per bond. And the “atomic” are intermediate. At 10 keV per bond, an atomic fuel would be a thousand times more energy dense than chemical fuels. That 100 meter tall fuel tank would shrink to 10 centimeters.
And the equations and models needed for that kind of chemistry and nuclear reactions generally – are modifications and variations on the basic nonlinear Schrodinger equation for the physical vacuum ( a vacuum with specific and measurable properties). And the first order solutions (magnetic dipole terms) are sufficient to estimate and design fuels and new materials. Then you use the full solutions (better representations of all the details) to make practical and safe new things.
I don’t know if you care about the human species over the new few hundred years. “Solar system colonization” will require increasing the human economy by a thousand fold. Just adding “atomic fuels” will change the game for every industry and part of society. And it comes down to models and simulations, calibrations and practical solutions to real problems. Yes, statistics is part of that. But it is more often than not just “let’s do it”. If you want to modify time, there are practical ways to do that too.
Richard Collins, Director, The Internet Foundation