notes02_qm2_schrodinger

# notes02_qm2_schrodinger - 2 Fundamentals of quantum...

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2 Fundamentals of quantum mechanics In the last section, we described the experimental motivations for a new type of physics at the atomic scale; one that captures the observed wave-like properties of elementary particles. Arguably the most intuitive underlying model of quantum mechanics uses a wavefunction to represent particles. The equation which governs how the wavefunction behaves is called Schr¨odinger’s equation. In this section we will focus on the Schr¨ odinger equation, and see how it can be used to determine the energy levels of some model potentials. Although we will concentrate on the wavefunction model of quantum mechanics, one should have a healthy skepticism of the aspects of the model which can not be observed. In particular, the wavefunction is a mathematical construction, and not something that can be directly observed. It is also good to keep in mind that there are several equivalent formulations of quantum mechanics, which have very diFerent representations. To give a few examples, the Heisenberg view focusses on how operators change in time and act on static particle states, the Bohmian view shows how a single point particle can describe a quantum particle when subject to quantum forces, and ±nally the ²eynman path integral view shows how the motion of a particle can be described by considering all possible paths that the particle can take. There is a beautiful (and short) book describing this last view, entitled “QED: The strange theory of light and matter,” which has been adapted from lectures that ²eynman gave. This would be a good book to read because it is accessible to a general audience, and it will give you a sense of how diFerent the underlying model of quantum mechanics can be, while still predicting the same observations as the standard models. 2.1 The Schr¨odinger view In the Schr¨odinger view, all particles are described by a wavefunction ψ .T h ew a v e - function can have positive, negative, and complex values. The wavefunction, itself, is not observable so one should not worry too much about the use of imaginary num- bers. All observable properties are real (not imaginary) and can be derived from the wavefunction. An example of an observable is the probability distribution of a particle ρ ( x )= | ψ ( x ) | 2 = ψ ( x ) 2 , (2.1) which is real. The association of

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## This note was uploaded on 08/30/2008 for the course CH 354L taught by Professor Henkelman during the Fall '06 term at University of Texas.

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notes02_qm2_schrodinger - 2 Fundamentals of quantum...

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