A Corbeled Gallery work

Wednesday, August 31, 2005

Deterministic Universe (step three) Probability Clouds, and wave function

The future of all events above the single atom level will happen in an exact way. For every second that exists, an infinite amount of variables will impact an atom, and this atom will in turn act according to classical physics. Therefore, if we knew every single variable, and was able to observe the atom without being involved with the atom, then we would know exactly where this atom will be a billion years from now. The only thing that will ever change this is if electrons do not behave in a predictable way.

The nature of electrons is odd. They seem to exist in different places at different points in time, but it is impossible to say where the electron will be at a given time. At time t1 it is at point A, then at time t2 it is at point B, yet without moving from A to B. It seems to appear in different places without describing a trajectory. Therefore, even if t1 and A can be pinpointed, it is impossible to derive t2 and B from this measurement. In other words: There seems to be no causal relation between any two positions. The concept of causality cannot be applied to what is observed. In case of the electrons of an atom, the closest we can get with describing the electron's position is by giving a number for the probability of it being at a particular place. Moreover, particles have other "disturbing" properties: They have a tendency to decay into other particles or into energy, and sometimes -under special circumstances- they merge to form new particles. They do so after indeterminate time spans. Although we can make statistical assertions about a particle's lifetime, it is impossible to predict the fate of an individual particle.

As I take a more objective standpoint on physics, I do not like functions. However to better understand what I was dealing with, I had to understand wave function. The wave function is used in the Schrödinger equation. This equation plays the role of Newton's laws and conservation of energy, in regards to classical physics. The purpose of such function is to predict the future behavior of a dynamic system, which is exactly what I am aiming to do. The detailed outcome depends on change, but given a large number of events, the Schrödinger equation will predict the distribution of results.

In the wave function, each "particle" is represented by a wave function ψ (position, time) such that ψ *ψ equals the probability of finding the particle at a new position at that time. ψ *ψ summed over all space = 1. This means, that if a particle exists, the probability of finding it somewhere in the universe is 100%. ψ is calculated in a three dimensional distribution. It permits calculation of an expectation value. A free particle is a sine wave, implying a determined momentum and totally uncertain position.

I dislike working with mathematics. A long line of scientists and mathematicians have forged the Wave Function, so I do not believe I need to explain this, so long as you assume it's existence. To this end, from now on I will explain that I use the Wave Function to derive an answer, and a reader must assume it's accuracy, or at least existence.

To better understand the wave function and probability clouds, imagine a large, irregular thundercloud that fills up the sky. The darker the thundercloud, the greater the concentration of water vapor and dust at that point. By simply looking at a thundercloud, we can estimate the probability of finding large concentrations of water and dust in certain parts of the sky. This thundercloud may be compared to a single electron's wave function. Like our thundercloud, it filles up all of the given space. The greater its value at a point (the darker the cloud), the greater the probability of finding the electron there (or water particles). Imagine a wave function applying to you. As you read, there is a certain probability you exist in your chair right now. If you were to see this function, it would resemble a cloud very much like the shape of your body. Some of this cloud would spread out to the moon, and further. Although the likelihood of you being at the moon at the present time has asemtoped to 0, a chance still exists. This means that there is a very large possibility that you are sitting in your chair right now, and not at the moon. Although part of your wave function has spread even beyond our galaxy, there is an infinitesimal chance that you are sitting in another galaxy.

Now that the wave function is explain (and hopefully assumed), I will now explain Barrier Penetration, otherwise known as the element of Tunneling.

According to classical physics, a particle of energy E less than the height U0 of a barrier could not penetrate - the region inside the barrier is classically forbidden. But the wave function associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. The wave function must also be continuous on the far side of the barrier, so there is a finite probability that the particle will tunnel through the barrier. As a particle approaches the barrier, it is described by a free particle wave function. When it reaches the barrier, it must satisfy the Schrödinger equation in the form (see picture for better understanding).



Now, barrier penetration takes the wave function, and shows that an electron has the possibility of appearing elsewhere in the universe, much like our example before, where you have the chance of being out at the moon.

Tunneling is the quantum-mechanical effect of transitioning through a barrier that was forbidden by classical physics. Consider rolling a ball up a hill. If the ball is not given enough push, then the ball will not make it to the other side of the hill. In this case the ball does not have enough energy to roll over the hill. But in quantum mechanics, things are not inherently classical particles (balls). In quantum mechanics things are fundamentally probability waves of finite extent. The implication is that in the analogous quantum situation of a quantum particle moving against a potential hill, some of the probability wave can extend all the way through to the other side of the potential hill. Having some of the wave on the other side of the hill means that there is a probability the quantum particle can be on the other side of the hill. The quantum particle can not travel over the hill, but it can possibly tunnel through the hill.

In the early 1900's radioactive materials were known to have characteristic exponential decay rates or half lives and the radiation emissions were known to have certain characteristic energies. By 1928 George Gamow solved the theory of the alpha-decay of a nucleus via tunneling. Classically the particle is confined to the nucleus because of a very strong potential. Classically it takes an awesome amount of energy to pull apart the nucleus. But because of Quantum Mechanics there is a probability the particle can tunnel through the potential and escape. Gamow solved a model potential for the nucleus and derived a relationship between the half life of the particle and the energy of the emission.

Alpha-decay via tunneling was also solved concurrently by Ronald Gurney and Edward Condon. Shortly thereafter both groups had also considered that particles could also tunnel into the nucleus.

After attending a seminar by Gamow it was Max Born that recognized the generality of quantum mechanical tunneling. The tunneling phenomena was not restricted to nuclear physics, but was a general result of Quantum Mechanics that applies to many different systems. Today tunneling is even applied to the early cosmology of the universe.
Therefore, we can understand that the basic building block of our universe, a particle, can never be accurately located. The future is not unable to be predicted, because we do not know exactly where an electron is, nor do we know how it will act.

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