Lecture 03

Lecture 03 - EEE 434/591Quantum Mechanics L3:1 David K....

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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L3:1 EEE 434/591—Quantum Mechanics David K. Ferry Regents’ Professor Arizona State University Alpbach, Austria
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L3:2 The Matrix Formulation of Quantum Mechanics In early 1925, Werner Heisenberg (a young postdoc) worked out a quantum theory based upon non- commuting operators, and assuming almost implicitly that he was dealing with the particle-like properties of various objects. This formulation was exceedingly difficult to work with in its basic form, due to its mathematical complexity. A colleague, Max Born, was the first to observe that this formulation of quantum mechanics was equivalent to matrix manipulations, and that a formulation in linear vector spaces was therefore possible. Born also formulated the probability density as being the squared magnitude of the wave function, but in matrix formulation. Paul Dirac also published a complete matrix based version of Heisenberg’s theory.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L3:3 In the winter of 1925, Erwin Schrödinger (an Austrian professor) worked out a wave formulation of quantum mechanics. It is this form which is intuitively simpler to understand, and which is almost universally taught first. He began from de Broglie’s wave picture, in which there were particles AND waves. But, he recognized that if you have waves, you must need a wave equation. But, there was no wave equation to be had. In the winter of 1925, he went to Arosa, Switzerland for several weeks of vacation (at his doctor’s cabin and with his doctor’s wife, but without his doctor). In this avowed erotic retreat, he developed the wave equation, and laid the groundwork to later show that it was completely equivalent to Heisenberg’s matrix approach.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L3:4 Now, there was a problem with waves, according to Heisenberg, Pauli, Born, et al . Waves were continuous, but the particle picture had significant discontinuities—the quantum jumps. Heisenberg did not accept the continuous wave picture and never came to tolerate the wave mechanics formulation! We will return to this a little later, but let us examine the wave equation. Schrödinger’s equation can not be derived (just as Maxwell’s equations cannot be derived). But, the formulation can be justified by noting that, in general, the energy can be written as
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L3:5 Last time, we showed that, in quantum mechanics, the momentum p is a differential operator so that the kinetic energy term becomes and the time independent Schrödinger equation becomes
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Lecture 03 - EEE 434/591Quantum Mechanics L3:1 David K....

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