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lecture_19 - 103 Applying the Concepts of Matter Waves Once...

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103 Applying the Concepts of Matter Waves Once the concept of matter waves was advanced, it was quite easy to rationalize the ad hoc quantization of angular momentum that Bohr had introduced: stationary states occurred when an integral number of deBroglie waves could fit exactly on the circumference of the orbit: 2 π R = n mv h n = 1, 2, 3, 4, … Panels (a) and (b) show cases where 4 or 5 deBroglie waves fit exactly. We say that standing waves corresponding to complete constructive interference are formed. However, when we attempt to fit a non-integral multiple of deBroglie wavelengths on the circle, as in panel (c), complete destructive interference occurs quickly. The Heisenberg Uncertainty Principle Ascribing the properties of waves to matter comes at a price. It is a fundamental property of waves that it is impossible to determine the position of a wave and its momentum simultaneously with arbitrarily high precision. This is true for light waves as well as matter waves. When this idea is applied to matter waves, the Heisenberg Uncertainty Principle results: Let x be the uncertainty in the measurement of the position of a particle. Let p be the corresponding uncertainty in the measurement of momentum of a particle: The conclusion: stationary states are a consequence of constructive interference of matter waves in a fixed region of space. This was the conceptual foundation for the “New Quantum Theory”, Schrödinger’s wave mechanics.
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104 Then, the Heisenberg Uncertainty Principle states that 2 x p h We can apply this idea to the circular orbits in the Bohr atom. Effectively, the Uncertainly Principle means that we cannot speak of the electron’s motion in terms of a well-defined circular trajectory with a precise radius. The electron’s motion is “fuzzy”, and all we can do is talk about the probability of finding the electron in a region of space. An interesting example:
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lecture_19 - 103 Applying the Concepts of Matter Waves Once...

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