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While the mathematical developments which constitute quantum mechanics have been outstandingly successful in describing all manner of observed properties of matter, it is fair to say that the conceptual basis of the theory is still somewhat obscure. I myself do not properly understand what it is that quantum theory tells us about the nature of the physical world, and by this I mean to imply that I do not think anybody else understands it either, though there are respectable scientists who write with confidence on the subject[39, p 95].

The above quote by Ian Lawrie is, no doubt, a minority opinion among professional physicists. But I suspect Lawrie may be more of a minority in his frankness than in his opinion. What quantum mechanics says is very strange. If you think you understand what it means you are almost certainly wrong. What one can understand is the structure of the mathematical theory and how experimental techniques are used to test the predictions of the model. In this and the next two sections we give an outline of that structure.

Models in Newtonian physics start with an initial state that is assumed or observed. To this state one applies a mathematical model that describes how the state evolves or changes over time. For example says the position of an object is given by the initial position plus the velocity (speed and direction) of the object multiplied by time. If a ball is one foot away from you and moving further away at 2 feet per second it will be 3 feet away one second later.

In classical physics the state of a particle is continuously described by an equation. We can directly map variables in the equation to physical quantities that we can measure. is position and is time in the model . The variables in quantum mechanics do not represent physical state but rather probability densities. What evolves over time is not the physical state but the probability density that the object will be observed in a given state.

Probability density gives the likelihood that a physical observable like position will assume a given value. Thus the probability density function must range over all possible values that the observable might be seen at. Consider a ball rolling down a frictionless inclined plane or ramp. A simple equation describes the ideal motion of the ball. However real balls are never perfectly round and real ramps are never frictionless. In an accurate enough experiment both these effects would be observable and we could no longer use an exact equation to predict the position of the ball. Instead we would need to use a probability density that would describe the most likely position of the ball as well as any position the ball might be found at no matter how unlikely.

A probability density function gives the relative probability that the ball will be at location . If it is twice as likely to be at as it is to be at then . The probability that the ball will be at any exact location is usually 0 since there are an infinite number of possible locations where the ball might be. Instead of asking for the probability that the ball will be at an exact location we can ask for the probability that it will be in some range for example between and .

To compute that we evaluate . An integral () is the limit of a sequence of additions. One can think of an integral as the computation of the area under a curve. One way to estimate the area is to break the region into rectangles that approximate the area. See Figure 7.2. The more rectangles we use the more accurately we can estimate the area. Some integrals can only be evaluated numerically by doing something similar to that shown in the figure. But for many functions we can compute an exact solution.

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