A First Approach to Quantum Mechanics
In this chapter, we try to understand the main ideas of quantum mechanics. In quantum mechanics, the outcome of a measurement cannot—even in principle—be predicted beforehand; only the probabilities for the outcome of the measurement can be predicted.
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In this chapter, we try to understand the main ideas of quantum mechanics. In quantum mechanics, the outcome of a measurement cannot—even in principle—be predicted beforehand; only the probabilities for the outcome of the measurement can be predicted. These probabilities are encoded in a wave function, which is a function of a position variable x ∈ Rn . The square of the absolute value of the wave function encodes the probabilities for the position of the particle. Meanwhile, the probabilities for the momentum of the particle are encoded in the frequency of oscillation of the wave function. The probabilities can be described using the position operator and the momentum operator. The time-evolution of the wave function is described by the Hamiltonian operator, which is analogous to the Hamiltonian (or energy) function in Hamilton’s equations.
3.1 Waves, Particles, and Probabilities There are two key ingredients to quantum theory, both of which arose from experiments. The first ingredient is wave–particle duality, in which objects are observed to have both wavelike and particlelike behavior. Light, for example, was thought to be a wave throughout much of the nineteenth century, but was observed in the early twentieth century to have particle behavior as well. Electrons, meanwhile, were originally thought to be particles, but were then observed to have wave behavior. B.C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, DOI 10.1007/978-1-4614-7116-5 3, © Springer Science+Business Media New York 2013
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3. A First Approach to Quantum Mechanics
The second ingredient of quantum theory is its probabilistic behavior. In the two-slit experiment, for example, electrons that are “identically prepared” do not all hit the screen at the same point. Quantum theory postulates that this randomness is fundamental to the way nature behaves. According to quantum mechanics, it is impossible (theoretically, not just in practice) to predict ahead of time what the outcome of an experiment will be. The best that can be done is to predict the probabilities for the outcome of an experiment. These two aspects of quantum theory come together in the wave function. The wave function is a function of a variable x ∈ Rn , which we interpret as describing the possible values of the position of a particle, and it evolves in time according to a wavelike equation (the Schr¨ odinger equation). The wave function and its time-evolution account for the wave aspect of quantum theory. The particle aspect of the theory comes from the interpretation of the wave function. Although it is tempting to interpret the wave function as a sort of cloud, where we have, say, a little bit of electron-cloud over here, and little bit of electron-cloud over there, this interpretation is not consistent with experiment. Whenever we attempt to measure the position of a single electron, we always find the electron at a single point. A single electron in the two-slit experiment is observed at a single point on the screen, not spread out over the
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