22.51 Quantum Theory of Radiation Interactions, 2012 Final
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چکیده
1. Propose a quantum mechanical model for the LC circuit and show that its (quantum) energy eigenvalues can be written in terms of the frequency ω = 1/ √ LC. 2. What is the meaning of the quantum eigenstates? [Hint: if you are not satisfied to find the quantum model by simple similarity with a very well-known model, consider ∂E ∂E that we have: = −Φ̇ and ̇ = Q, thus Φ and Q are generalized canonical variables] ∂Q ∂Φ Solution: Given the form of the classical hamiltonian, we expect the system to have the form of an harmonic oscillator, with Q and Φ playing the role of effective position an momentum. Since we usually have ω = k/m we can identify e.g. L → 1/k and Φ with the position, while C → m and Q with the momentum. The Hamiltonian is diagonalized as usual in terms of creation and annihilation operators, in this case, of (microwave) photons J
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