The state of a quantum system

The state of a quantum system - one translates to an...

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The state of a quantum system Let us first look at how we specify the state for a classical system. Once again, we use the ubiquitous billiard ball. As any player knows, there are three important aspects to its motion: position, velocity and spin (angular momentum around its centre). Knowing these quantities we can in principle (no friction) predict its motion for all times. We have argued before that quantum mechanics involves an element of uncertainty. We cannot predict a state as in classical mechanics, we need to predict a probability. We want to be able to predict the outcome of a measurement of, say, position. Since position is a continuous variable, we cannot just deal with a discrete probability, we need a probability density, To understand this fact look at the probability that we measure to be between and . If is small enough, this probability is directly proportional to the length of the interval (3.1) Here is called the probability density. The standard statement that the total probability is
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Unformatted text preview: one translates to an integral statement, (3.2) (Here I am lazy and use the lower case where I have used before; this a standard practice in QM.) Since probabilities are always positive, we require . Now let us try to look at some aspects of classical waves, and see whether they can help us to guess how to derive a probability density from a wave equation. The standard example of a classical wave is the motion of a string. Typically a string can move up and down, and the standard solution to the wave equation (3.3) can be positive as well as negative. Actually the square of the wave function is a possible choice for the probability (this is proportional to the intensity for radiation). Now let us try to argue what wave equation describes the quantum analog of classical mechanics, i.e., quantum mechanics. The starting point is a propagating wave. In standard wave problems this is given by a plane wave, i.e.,...
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