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note07 - Introduction to Quantum and Statistical Mechanics...

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Introduction to Quantum and Statistical Mechanics Handout 7 Application of Harmonic Oscillator: Quantum LC Circuits and Phonons in Solids Textbook Reading: Chap. 11 (pp. 223 – 244); Appendix F (pp. 707-710) Additional reading: C. Kittel, Introduction to Solid State Physics, 6 th Ed., Appendix C, p. 611. 7.1 The Hamiltonian for the LC circuit The simple harmonic oscillator not only is important by itself to match with as well as to distinguish from classical mechanics - in both intuition and rigorous math, but also sheds light on the quantization of electromagnetic fields and lattice vibration in solids (or the elastic waves going through the solids induced by thermal vibration). The quantization of electromagnetic waves and lattice waves are named photons and phnons, respectively. With the limited time we have, however, we will focus more on the methodology and physical meaning, instead of the mathematical rigor in these two systems. You may ask why a circuit, or specifically the LC circuit here, will have quantization and hence will have to abide by the Uncertainty Principle in quantum mechanics. The intuitive answer is that capacitance (and the voltage across it) is just related to the position of the charges, while the inductor (and the current through it) is just related to the velocity (or momentum) of the charge movement. Charge is a quantized particle. Position and momentum will then need to obey the Uncertainty Principle. Both the voltage and current oscillate like the classical case of pendulum. The close analogy between SHO and LC circuits should not be a surprise. We can follow the previous note to define the total energy in the classical mechanics, and then try to match the quantum operators in circuits that will have the correct expectation values. This is summarized in Table 1. Table 1. Classical analogy between harmonic oscillator and LC circuits SHO LC circuits Basic classical equations ) ( ) ( ) ( ) ( 1 ) ( t kx x F t p dt d t p m t x dt d - = = = ) ( 1 ) ( ) ( 1 ) ( t v L t i dt d t i C t v dt d - = = Classical solution 0 0 0 0 0 0 0 0 ) cos( ) ( ) sin( ) ( x m p m k t p t p t x t x ϖ ϖ χ ϖ χ ϖ = + = + = 0 0 0 0 0 0 0 0 1 ) cos( ) ( ) sin( ) ( V C I LC t I t i t V t v ϖ ϖ χ ϖ θ ϖ = + = + = 1
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Total energy 2 2 2 2 0 2 x m m p E ϖ + = 2 ) ( 2 ) ( 2 2 t Cv t Li E + = Maximum energy 2 2 2 0 2 0 2 0 x m m p E ϖ = = 2 2 2 0 2 0 LI CV E = = ( Exercise 7.1 ) Remember our discussion of a wave propagation with linear dispersion in Chap. 1. For a segment of the transmission line containing the LC, what is the wave form that propagates on the transmission line? How do you view the wave-particle duality there? Do you expect the quantization to happen or all eigenvalues are allowed? Although it is straightforward to make an analogy between x/p with v/i , our next task is to identify the analogy of operators. In the x/p case, we have: k i x x i k p = - = = ˆ ˆ C C (7.1) Before we just use straightforward imitation to put down the voltage and current operators, there is a formal method to identify the operators, the Dirac’s canonical conjugates, which will make the correspondence complete and share the Fourier transform properties.
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