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The uncertainty principle

One naturally links the uncertainty principle with the probabilistic nature of quantum mechanics. They are connected but they involve different principles. The probabilistic nature of quantum mechanics comes from the lack of a physical state model. Only the evolution of probabilities is modeled. The uncertainty principle comes from the wave structure of these probability densities.

In classical physics one cannot simultaneously measure both the frequency and location of a wave. Figure 7.3 is the graph of a sine wave or pure tone. Objects that vibrate or resonate tend to move in a pattern resembling this figure. A musical instrument that produces a pure tone will be very close to the graph. Frequency in cycles per second is the number of peaks that occur each second. More complex waves can be regarded as the sum of many pure tones of different frequencies. Notice that the peaks and valleys are all the same size. A pure tone or pure frequency extends infinitely far in each direction with no fall off of amplitude. Thus no real sound is a pure tone.

Pure tones have an exact frequency and impulses have an exact position. An impulse is a vertical line at a single point. It is zero except at one point. An explosion will generate a sound approximating an impulse. There are no exact impulses just as there are no pure tones. Real explosions extend over some time. An impulse is the sum of all frequencies. Figure 7.4 shows how an impulse can be approximated by adding many pure tones.

A wave that is narrowly constrained in position has a broad range of frequencies. A wave that is constrained to a narrow range of frequencies extends over a large distance in position as Figure 7.4 illustrates. Mathematicians and scientists do not say the exact position or frequency of a wave is uncertain. Rather they are undefined or meaningless. There is no uncertainty in the structure of the wave or its evolution in time in either classical or quantum mechanics. However in quantum mechanics a wave is used to compute the probability that some event will be observed.

We can control the shape of the wave by the structure of the experiment. If we use a massive particle we can measure position accurately but not velocity because the wave function for a massive particle is focused in a narrow' range of positions but has much more variance in velocity. Mathematically this tradeoff is identical to the tradeoff between narrowly constraining a classical wave's location and narrowly constraining its frequency.

Uncertainty in quantum mechanics is not connected to the probabilistic nature of the wave function. It is inherent in any wave function including those in classical physics. The inability to assign exact position and momentum to a particle may mean that there is no such thing. The inability to make those assignments need not be an obstacle to deterministic predictions. For classical waves, frequency and location cannot be simultaneously assigned, but those models are completely deterministic.

We mentioned in Section 7.5 that a discrete model would not have point like particles but dynamically stable structures that approximate classical waves. Such structures would not have an exact position. However when they interact and transform their structure there would be a focal point of that transformation. Experimental conditions would determine how precise that focal point is just as the precision of the lenses in a camera determine how much detail is resolvable in pictures the camera takes.

The inability to simultaneously constrain a particle's position and momentum is fundamental to the wave structure of quantum mechanics. The question of predicting the outcome of experiments is an independent one. For a hypothetical model based on stable dynamic structures it might be possible to predict the exact outcome of experiments even though the exact position and momentum of particles is not known because it has no more meaning than the exact location and position of a classical wave. If the evolution of such structures is deterministic then one might, in some special cases, know enough to make exact predictions.

In 1932 the renowned mathematician and physicist von Neumann published a proof that no more complete theory could be consistent with the predictions of quantum mechanics[51][50]. Von Neumann's reputation was so great that the proof stood for thirty years in spite of Bohm's publication in 1952 of a more complete theory that was consistent with quantum mechanics [8]. Bohm thought at the time there were subtle differences in the predictions of the two theories and thus his result did not contradict von Neumann's proof.

In 1966 Bell published a paper revealing a problem with von Neumann's proof[6]. The mathematics was fine but the assumptions von Neumann made about the constraints a more complete theory had to meet were not justified. Bell went on to show that quantum mechanics was not a local theory. Bohm's theory was an explicitly nonlocal theory and Bohm's work was an important influence for Bell. This story continues in Section 8.4.

There are many properties like frequency and location that cannot be simultaneously measured with high accuracy. Such pairs are said to be non commuting. None of this says anything about uncertainty or lack of predictability. That is a separate issue. Bohm showed one way a more complete theory can be constructed. If there is one there are many.

Figure 7.3: A sine wave or pure tone
\includegraphics[]{../sumsine1}



A pure tone or pure sine wave extends infinitely far in both directions with no change in the size of the peaks or valleys. The frequency in cycles per second is the number of peaks that occur each second.

Figure 7.4: An impulse has all frequencies
\includegraphics[scale=.75]{../sumsine4}

4 sine waves and their sum.


\includegraphics[scale=.75]{../sumsine8}

8 sine waves and their sum.


\includegraphics[scale=.75]{../sumsine32}

The sum of 32 sine waves.


\includegraphics[scale=.75]{../sumsine128}

The sum of 128 sine waves.

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