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Quantum Mechanics

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from the website http://www.bottomlayer.com/bottom/argument/Argument4.pdf

We have seen one aspect of the measurement effect, which is that measurement (or observation) appears to determine whether a quantum unit is displayed or projected to the user (as a "particle"), or whether instead the phenomenon remains inchoate, unobserved, behaving according to a mathematical algorithm (as a "wave").

There is another aspect of measurement that relates to the observed properties of the particle-like phenomenon as it is detected. This is the famous Heisenberg uncertainty principle.

As with all aspects of quantum mechanics, the uncertainty principle is not a statement of philosophy, but rather a mathematical model which is exacting and precise. That is, we can be certain of many quantum measurements in many situations, and we can be completely certain that our results will conform to quantum mechanical principles.

In quantum mechanics, the "uncertainty principle" has a specific meaning, and it describes the relationship between two properties which are "complementary," that is, which are linked in a quantum mechanical sense (they "complement" each other, i.e., they are counterparts, each of which makes the other "complete").

The original example of complementary properties was the relationship between position and momentum. According to classical Newtonian physics and to common sense, if an object simply exists we should be able to measure both where it is and how fast it is moving.

Measuring these two properties would allow us to predict where the object will be in the future. In practice, it turns out that both position and momentum cannot be exactly determined at the same moment – a discovery that threw a monkey wrench into the clockwork predictability of the universe.

Put simply, the uncertainty relationship is this: for any two complementary properties, any increase in the certainty of knowledge of one property will necessarily lead to a decrease in the certainty of knowledge of the other property.

The uncertainty principle was originally thought to be more statement of experimental error than an actual principle of any great importance. When scientists were measuring the location and the speed (or, more precisely, the momentum) of a quantum unit – two properties which turn out to be complementary – they found that they could not pin down both at once. That is, after measuring momentum, they would determine position; but then they found that the momentum had changed.

The obvious explanation was that, in determining position, they had bumped the quantum unit and thereby changed its momentum. What they needed (so they thought) were better, less intrusive instruments. On closer inspection, however, this did not turn out to be the case.

The measurements did not so much change the momentum, as they made the momentum less certain, less predictable. On re-measurement, the momentum might be the same, faster, or slower. What is more, the range of uncertainty of momentum increased in direct proportion to the accuracy of the measurement of location.

In 1925, Werner Heisenberg conducted a mathematical analysis of the position and momentum of quantum units. His results were surprising, in that they showed a mathematical incompatibility between the two properties.

Heisenberg was able to state that there was a mathematical relationship between the properties p (position) and m (momentum), such that the more precise your knowledge of the one, the less precise your knowledge of the other.

This "uncertainty" followed a formula which, itself, was quite certain. Heisenberg's mathematical formula accounted for the experimental results far, far more accurately than any notion of needing better equipment in the laboratory.

It seems, then, that uncertainty in the knowledge of two complementary properties is more than a laboratory phenomenon -- it is a law of nature which can be expressed mathematically.

A good way to understand the uncertainty principle is to take the extreme cases. As we will discuss later on, a distinguishing feature of quantum units is that many of their properties come in whole units and whole units only. That is, many quantum properties have an either/or quality such that there is no in between: the quantum unit must be either one way or the other.

We say that these properties are "quantized," meaning that the property must be one specific value (quantity) or another, but never anything else. When the uncertainty principle is applied to two complementary properties which are themselves quantized, the result is stark.

Think about it. If a property is quantized, it can only be one way or the other; therefore, if we know anything about this property, we know everything about this property.

There are few, if any, properties in our day to day lives that can be only one way or the other, never in between…  For a "complementary" property … let us use marital status. (In law, you are either married or not married, with important consequences for bigamy prosecutions.)

The logical consequence of knowing everything about one complementary property is that, as a law of nature, we then would know nothing about the other complementary property.

For our example, we must imagine that, by learning whether a married woman is pregnant, we thereby no longer know whether she is married. We don't forget what we once knew; we just can no longer be certain that we will get any particular answer the next time we check on her marital status.

The mathematical statement is that, by knowing pregnancy, you do not know whether she is married; and, by knowing marital status, you do not know whether she is pregnant.

In order to make this statement true, if you once know her marital status, and you then learn her pregnancy status (without having you forget your prior knowledge of marital status), the very fact of her marital status must become random yes or no. A definite maybe.

What is controlling is your state of certainty about one property or the other. In just this way, the experimentalist sees an electron or some other quantum unit whose properties depend on the experimentalist’s knowledge or certainty of some other complementary property.

a computer’s data

If we cease to think of the quantum unit as a "thing," and begin to imagine it as a pixel, that is, as a display of information in graphic (or other sensory) form, it is far easier to conceive of how the uncertainty principle might work.

The "properties" we measure are variables which are computed for the purpose of display, which is to say, for the purpose of giving the user knowledge via the interface. A computed variable will display according to the underlying algorithm each time it is computed, and while the algorithm remains stable, the results of a particular calculation can be made to depend on some other factor, including another variable.

It would be far easier to understand our changing impressions of the hypothetical woman if we knew that, although she appeared to be a person like ourselves, in fact she was a computer projection. As a computer projection, she could be pregnant or not pregnant, married or single, according to whatever rules the computer might be using to create her image.

Complementary properties are simply paired variables, the calculation of which depends on the state of the other. Perhaps they share a memory location, so that when one variable is calculated and stored, it displaces whatever value formerly occupied that location; then the other variable would have to be calculated anew the next time it was called for.

In this way, or in some analogous way, we can see that the appearance of a property does not need to be related to the previously displayed value of the property, but only to the underlying algorithm.