4_quantum_mechanics.pdf - Part 4 Quantum mechanics PHYS...

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Part 4. Quantum mechanics PHYS 2022, Yi Wang, Department of Physics, HKUST This part is an introduction of quantum mechanics. We start from This part is essential. the key observations that why the world has to be quantized. Af- terwards we introduce the wave-particle duality, measurements and uncertainty principle, the Schrödinger equation, square potential, op- erators and the harmonic oscillator. We end with an overview on the interpretations of quantum mechanics. 1 Clouds over the classical world “It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, we had everything before us, we had nothing before us, we were all going direct to Heaven, we were all going direct the other way in short, the period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, in the superlative degree of comparison only.” – Charles Dickens, “A Tale of Two Cities” I wanted to cite the first sentence of the above, but I cannot help to cite a whole paragraph. This is how in the 1900s the physicists felt – All the dream, love and faith of their beautiful, harmonic and pre- dictable classical world had collapsed. Nobody would have imagined how the real world behaves if the new experimental discoveries were not thrown onto their faces. 1.1 The UV catastrophe The equipartition rule To start to understand how bad things can go, let us first review Boltamann’s equipartition rule. For the theory of ideal gas, each The situation is similar to what if I send we-chat random red pocket to the class. Say I prepare $100 and there are 100 of you. Then each of you get statistically $1. degree of freedom (dof) gets on average an energy of 1 2 k B T . That is to say, in a thermal equilibrium, everybody get statistically equal share of energy. This rule is expected in general because lower energy dof tends to get kicked by the higher energy dof and gets statistically equal share of energy. For example, for monatomic ideal gas, the internal energy is 3 2 Nk B T , because each of the N particles has 3 directions to move. There are thus 3 N dof.
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part 4. quantum mechanics 2 What about let us apply this equipartition rule to light, i.e. E&M radiation? The question is inevitable because hot objects indeed emit E&M radiation. How do hot objects emit E&M radiation? The UV catastrophe of radiation Assuming each degree of freedom get the same share, the problem reduces to counting the degrees of freedom of E&M waves. For sim- plicity, let us consider 1-dimensional waves in a box, and neglect the issue of polarization. We will comment the 3-dimensional case later qualitatively.
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