What was the impact of Schrodingers equation and Heisenbergs uncertainty principle in quantum mechanics - by Ron Brown - Quora Question Review

This document contains a review of the answer by Ron Brown on the question in Quora: "What was the impact of Schrodingers equation and Heisenbergs uncertainty principle in quantum mechanics"
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• The text in italics is copied from the article.
  
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Contents

• 1. Answer Review

Reflection

• Reflection 1 - Question Review

1. Answer Review

It’s interesting that both the Schrödinger’s equation and Heisenberg’s uncertainty principle predict things that to our “classical” brains seem impossible to accept, and yet explain things that we cannot explain any other way.

Our classical brains will always accept the results of experiments, which can be repeated.

And the two notions are consistent in that Schrödinger’s wave theory predicts that reducing the uncertainty where a particle could possibly be, increases the uncertainty in its momentum.

Demonstrate this by means of an experiment.

Schrödinger’s equation is essentially an expression that the total energy of a particle is the sum of its kinetic energy and the potential energy associated with its environment - that is a very classical notion - but in a very different mathematical formalism.

Make this clear by means of an experiment.

Instead of solving for things like the position of an object as a function of time given the forces that act (as we do in classical physics), it establishes that there is a probability function that contains the information that we can know about a particle, and when solved for using the Schrödinger’s equation, one can determine the possible energies that a particle could have within the system being described, and for each of those possible energies, the probability of finding the particle at any particular location within that system. That is, one of the biggest differences between the classical approach to problem solving and the quantum theory as presented by Schrödinger is what is being solve for and the mathematical formalism to do so. But the differences in results are dramatically different - including requirements of quantization, which means there is a need for experimental observations to verify whether those results actually describe how nature behaves.

That what I mean all along. Start with an experiment and explain the issues involved.

One of the first examples usually presented in a quantum theory class is a particle trapped in a closed space.

How is this done with an experiment?

In one dimension, we call it the “infinite potential well” problem. But what it really means is that the particle being described cannot go to the left of one position on the x-axis or to the right of another position, but it can move freely between those two positions and will reflect off the boundaries and return. (Think of a bead on a long straight wire that is anchored at both ends. In effect if it can’t get out of that space, the potential energy is too great to overcome at each end.) What the Schrödinger’s equation solution says is that there are only certain energies the particle can have, zero is not one of those energies, and for each energy level, the probability of locating it in that space is different.

This again should be demonstrated by means of an experiment.

Well that solution makes no sense classically. That is, it says the particle couldn’t just be placed with zero speed at the center of the space and left alone. That is, it could not simultaneously have zero kinetic and zero potential energy. That, of course is completely consistent with Heisenberg’s uncertainty principle. If it had zero kinetic energy, we would know its momentum to be exactly zero while also knowing it is contained in that space. So ∆p∆x=0 in violation of Heisenberg’s uncertainty principle.

Is there any sort of experimental verification of this? Not directly as stated. But consider this: The three dimensional version of the same problem predicts the form of the density of states function for the free electrons in a metal (within some assumptions). That density of states function allows one to predict the temperature dependence of the electronic specific heat of metals - and that is experimentally verifiable. Sometimes, experimental verification of a theory depends on examining the implications of that theory.

Another version of the one-dimensional Schrödinger problem, but with a potential barrier rather than a potential well, predicts quantum tunneling.

How is that demonstrated?

As I student, that made so little sense to me I would have assumed that meant the theory was wrong.

A theory is wrong when the physical effect or behaviour can not be demonstrated by means of an experiment.

But there are electronic components in your cell phone that depend on quantum tunneling of electrons through barriers.

But the most dramatic success of this “improbable” theory was in solving for the possible energy levels and wave functions for the electron in an isolated hydrogen atom. It’s not trivial mathematically - and requires putting Schrödinger’s equation into spherical polar coordinates and solving. But it is a closed form solution that predicts the energy levels possible for an electron in the vicinity of a proton (i.e., the hydrogen atom) - and those energy levels predict the ionizaton energy of hydrogen as well as the wavelengths of the emission spectra of excited hydrogen gas … both of which had already been measured. And that information tells us that the stars are formed from hydrogen (and ultimately that those stars in distant galaxies are moving away from us). And the mathematical solution just for hydrogen, gives us the rules for the various other quantum numbers which lead to the organizing principles for more complicated atoms.

It has been nearly a century since all of that was developed. It is hard to overemphasize the significance of those developments. When I was fresh out of graduate school, my first academic appointment was at a college whose faculty included an emeritus professor (Vladimir Rojansky) who was the himself the first grad student of Nobel Laureate John Van Vleck, as young men at the time, they were friends with Paul Dirac. Rojansky talked about the excitement of the discussions (sometimes around a campfire while backpacking!), the speculations, and the pouring over the latest publications of Heisenberg and Schrödinger and the others as all of that was unfolding.

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