Consider the Lagrangian of the one-dimensional harmonic oscillator, m L = 12 2 - m 2 (i) Write down the Euler-Lagrange equation for the system. (ii) Write down the Hamiltonian of the system, and the corresponding Hamilton equations. (iii) Consider the following complex quantities a == mw 2 ip mw + a* := mw 2 (x — —³) · ip mw Calculate aa* and re-express the Hamiltonian in terms of a and a*. (iv) For two given functions A(x,p) and B(x,p), the Poisson bracket {A, B} is defined as ДА ӘВ ДА ДВ {A, B} == Эх др др дх Calculate the Poisson brackets {a, a*}, {a, H}, and {a*, H}. (v) Write down the time evolution equation for a(t), and its solution.

Consider the Lagrangian of the one-dimensional harmonic oscillator, m L = 12 2 - m 2 (i) Write down the Euler-Lagrange equation for the system. (ii) Write down the Hamiltonian of the system, and the corresponding Hamilton equations. (iii) Consider the following complex quantities a == mw 2 ip mw + a* := mw 2 (x — —³) · ip mw Calculate aa* and re-express the Hamiltonian in terms of a and a. (iv) For two given functions A(x,p) and B(x,p), the Poisson bracket {A, B} is defined as ДА ӘВ ДА ДВ {A, B} == Эх др др дх Calculate the Poisson brackets {a, a}, {a, H}, and {a*, H}. (v) Write down the time evolution equation for a(t), and its solution.